[GitHub] [commons-geometry] aherbert commented on a diff in pull request #218: Refactor hull

["aherbert (via GitHub)" <gi...@apache.org>]

aherbert commented on code in PR #218:
URL: https://github.com/apache/commons-geometry/pull/218#discussion_r1268708345


##########
commons-geometry-euclidean/src/main/java/org/apache/commons/geometry/euclidean/twod/hull/ConvexHull2D.java:
##########


Review Comment:
   This is showing as a new file, but has code from the deleted file in the old hull module. A git move and then supplementing the builder code would preserve the commit history.



##########
commons-geometry-euclidean/src/main/java/org/apache/commons/geometry/euclidean/twod/hull/ConvexHull2D.java:
##########
@@ -0,0 +1,469 @@
+/*
+ * Licensed to the Apache Software Foundation (ASF) under one or more
+ * contributor license agreements.  See the NOTICE file distributed with
+ * this work for additional information regarding copyright ownership.
+ * The ASF licenses this file to You under the Apache License, Version 2.0
+ * (the "License"); you may not use this file except in compliance with
+ * the License.  You may obtain a copy of the License at
+ *
+ *      http://www.apache.org/licenses/LICENSE-2.0
+ *
+ * Unless required by applicable law or agreed to in writing, software
+ * distributed under the License is distributed on an "AS IS" BASIS,
+ * WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.
+ * See the License for the specific language governing permissions and
+ * limitations under the License.
+ */
+
+package org.apache.commons.geometry.euclidean.twod.hull;
+
+import java.util.ArrayList;
+import java.util.Collection;
+import java.util.Collections;
+import java.util.Iterator;
+import java.util.List;
+
+import org.apache.commons.geometry.core.ConvexHull;
+import org.apache.commons.geometry.euclidean.EuclideanCollections;
+import org.apache.commons.geometry.euclidean.twod.ConvexArea;
+import org.apache.commons.geometry.euclidean.twod.Lines;
+import org.apache.commons.geometry.euclidean.twod.Vector2D;
+import org.apache.commons.geometry.euclidean.twod.path.LinePath;
+import org.apache.commons.numbers.core.Precision;
+
+/**
+ * This class represents a convex hull in two-dimensional Euclidean space.
+ */
+public final class ConvexHull2D implements ConvexHull<Vector2D> {
+
+    /** Vertices for the convex hull, in order. */
+    private final List<Vector2D> vertices;
+
+    /** Polyline path for the convex hull. */
+    private final LinePath path;
+
+    /** Simple constructor; no validation is performed.
+     * @param vertices the vertices of the convex hull; callers are responsible for ensuring that
+     *      the given vertices are in order, unique, and define a convex hull.
+     * @param precision precision context used to compare floating point numbers
+     */
+    ConvexHull2D(final Collection<Vector2D> vertices, final Precision.DoubleEquivalence precision) {
+        this.vertices = Collections.unmodifiableList(new ArrayList<>(vertices));
+        this.path = buildHullPath(vertices, precision);
+    }
+
+    /** {@inheritDoc} */
+    @Override
+    public List<Vector2D> getVertices() {
+        return vertices;
+    }
+
+    /** Get a path defining the convex hull. The path will contain
+     * <ul>
+     *      <li>zero segments if the hull consists of only a single point,</li>
+     *      <li>one segment if the hull consists of two points,</li>
+     *      <li>three or more segments defining a closed loop if the hull consists of more than
+     *          two non-collinear points.</li>
+     * </ul>
+     * @return polyline path defining the convex hull
+     */
+    public LinePath getPath() {
+        return path;
+    }
+
+    /** {@inheritDoc} */
+    @Override
+    public ConvexArea getRegion() {
+        return path.isClosed() ?
+                ConvexArea.convexPolygonFromPath(path) :
+                null;
+    }
+
+    /** {@inheritDoc} */
+    @Override
+    public String toString() {
+        final StringBuilder sb = new StringBuilder();
+        sb.append(getClass().getSimpleName())
+            .append("[vertices= ")
+            .append(getVertices())
+            .append(']');
+
+        return sb.toString();
+    }
+
+    /** Build a polyline representing the path for a convex hull.
+     * @param vertices convex hull vertices
+     * @param precision precision context used to compare floating point values
+     * @return path for the convex hull defined by the given vertices
+     */
+    private static LinePath buildHullPath(final Collection<Vector2D> vertices,
+            final Precision.DoubleEquivalence precision) {
+        if (vertices.size() < 2) {
+            return LinePath.empty();
+        }
+
+        final boolean closeLoop = vertices.size() > 2;
+
+        return LinePath.builder(precision)
+                .appendVertices(vertices)
+                .build(closeLoop);
+    }
+
+    /** Class used to build convex hulls. The builder is based on the Akl-Toussaint
+     * heuristic to construct the hull. The heuristic is based on the idea of a
+     * convex quadrilateral, which is formed by four points with the lowest and
+     * highest x / y coordinates. Any point that lies inside this quadrilateral can
+     * not be part of the convex hull and can thus be safely discarded before
+     * generating the convex hull itself.
+     * <p>
+     * The complexity of the operation is O(n), and may greatly improve the time it
+     * takes to construct the convex hull afterwards, depending on the point
+     * distribution.
+     *
+     * @see <a href=
+     *      "http://en.wikipedia.org/wiki/Convex_hull_algorithms#Akl-Toussaint_heuristic">
+     *      Akl-Toussaint heuristic (Wikipedia)</a>
+     */
+    public static final class Builder {
+
+        /** Corner of triangle with minimal x coordinate. */

Review Comment:
   Q. `triangle` to `quadrilateral`?



##########
commons-geometry-euclidean/src/main/java/org/apache/commons/geometry/euclidean/twod/hull/ConvexHull2D.java:
##########
@@ -0,0 +1,469 @@
+/*
+ * Licensed to the Apache Software Foundation (ASF) under one or more
+ * contributor license agreements.  See the NOTICE file distributed with
+ * this work for additional information regarding copyright ownership.
+ * The ASF licenses this file to You under the Apache License, Version 2.0
+ * (the "License"); you may not use this file except in compliance with
+ * the License.  You may obtain a copy of the License at
+ *
+ *      http://www.apache.org/licenses/LICENSE-2.0
+ *
+ * Unless required by applicable law or agreed to in writing, software
+ * distributed under the License is distributed on an "AS IS" BASIS,
+ * WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.
+ * See the License for the specific language governing permissions and
+ * limitations under the License.
+ */
+
+package org.apache.commons.geometry.euclidean.twod.hull;
+
+import java.util.ArrayList;
+import java.util.Collection;
+import java.util.Collections;
+import java.util.Iterator;
+import java.util.List;
+
+import org.apache.commons.geometry.core.ConvexHull;
+import org.apache.commons.geometry.euclidean.EuclideanCollections;
+import org.apache.commons.geometry.euclidean.twod.ConvexArea;
+import org.apache.commons.geometry.euclidean.twod.Lines;
+import org.apache.commons.geometry.euclidean.twod.Vector2D;
+import org.apache.commons.geometry.euclidean.twod.path.LinePath;
+import org.apache.commons.numbers.core.Precision;
+
+/**
+ * This class represents a convex hull in two-dimensional Euclidean space.
+ */
+public final class ConvexHull2D implements ConvexHull<Vector2D> {
+
+    /** Vertices for the convex hull, in order. */
+    private final List<Vector2D> vertices;
+
+    /** Polyline path for the convex hull. */
+    private final LinePath path;
+
+    /** Simple constructor; no validation is performed.
+     * @param vertices the vertices of the convex hull; callers are responsible for ensuring that
+     *      the given vertices are in order, unique, and define a convex hull.
+     * @param precision precision context used to compare floating point numbers
+     */
+    ConvexHull2D(final Collection<Vector2D> vertices, final Precision.DoubleEquivalence precision) {
+        this.vertices = Collections.unmodifiableList(new ArrayList<>(vertices));
+        this.path = buildHullPath(vertices, precision);
+    }
+
+    /** {@inheritDoc} */
+    @Override
+    public List<Vector2D> getVertices() {
+        return vertices;
+    }
+
+    /** Get a path defining the convex hull. The path will contain
+     * <ul>
+     *      <li>zero segments if the hull consists of only a single point,</li>
+     *      <li>one segment if the hull consists of two points,</li>
+     *      <li>three or more segments defining a closed loop if the hull consists of more than
+     *          two non-collinear points.</li>
+     * </ul>
+     * @return polyline path defining the convex hull
+     */
+    public LinePath getPath() {
+        return path;
+    }
+
+    /** {@inheritDoc} */
+    @Override
+    public ConvexArea getRegion() {
+        return path.isClosed() ?
+                ConvexArea.convexPolygonFromPath(path) :
+                null;
+    }
+
+    /** {@inheritDoc} */
+    @Override
+    public String toString() {
+        final StringBuilder sb = new StringBuilder();
+        sb.append(getClass().getSimpleName())
+            .append("[vertices= ")
+            .append(getVertices())
+            .append(']');
+
+        return sb.toString();
+    }
+
+    /** Build a polyline representing the path for a convex hull.
+     * @param vertices convex hull vertices
+     * @param precision precision context used to compare floating point values
+     * @return path for the convex hull defined by the given vertices
+     */
+    private static LinePath buildHullPath(final Collection<Vector2D> vertices,
+            final Precision.DoubleEquivalence precision) {
+        if (vertices.size() < 2) {
+            return LinePath.empty();
+        }
+
+        final boolean closeLoop = vertices.size() > 2;
+
+        return LinePath.builder(precision)
+                .appendVertices(vertices)
+                .build(closeLoop);
+    }
+
+    /** Class used to build convex hulls. The builder is based on the Akl-Toussaint
+     * heuristic to construct the hull. The heuristic is based on the idea of a
+     * convex quadrilateral, which is formed by four points with the lowest and
+     * highest x / y coordinates. Any point that lies inside this quadrilateral can
+     * not be part of the convex hull and can thus be safely discarded before
+     * generating the convex hull itself.
+     * <p>
+     * The complexity of the operation is O(n), and may greatly improve the time it
+     * takes to construct the convex hull afterwards, depending on the point
+     * distribution.
+     *
+     * @see <a href=
+     *      "http://en.wikipedia.org/wiki/Convex_hull_algorithms#Akl-Toussaint_heuristic">
+     *      Akl-Toussaint heuristic (Wikipedia)</a>
+     */
+    public static final class Builder {
+
+        /** Corner of triangle with minimal x coordinate. */
+        private Vector2D minX;
+
+        /** Corner of triangle with maximal x coordinate. */
+        private Vector2D maxX;
+
+        /** Corner of triangle with minimal y coordinate. */
+        private Vector2D minY;
+
+        /** Corner of triangle with maximal y coordinate. */
+        private Vector2D maxY;
+
+        /** Collection of all remaining candidates for a convex hull. */
+        private final Collection<Vector2D> candidates;
+
+        /** A precision context for comparing points. */
+        private final Precision.DoubleEquivalence precision;
+
+        /** Indicates if collinear points on the hull shall be present in the output.
+         * If {@code false}, only the extreme points are added to the hull.
+         */
+        private final boolean includeCollinearPoints;
+
+        /**Return a {@link Builder} instance configured with the given precision
+         * context. The precision context is used when comparing points.
+         *
+         * @param builderPrecision       precision context to use when building a convex
+         *                               hull from raw vertices; may be null if raw
+         *                               vertices are not used.
+         * @param includeCollinearPoints whether collinear points shall be added as hull
+         *                               vertices
+         */
+        public Builder(final boolean includeCollinearPoints, final Precision.DoubleEquivalence builderPrecision) {
+            this.precision = builderPrecision;
+            this.includeCollinearPoints = includeCollinearPoints;
+            candidates = EuclideanCollections.pointSet2D(builderPrecision);
+        }
+
+        /** Appends the given point to a collection of possible hull points, if and only
+         * if the given point is outside of a constructed quadrilateral of extreme properties.
+         *
+         * @param point a given point.
+         * @return this instance.
+         */
+        public Builder append(Vector2D point) {
+
+            //Checks if the given point supersedes one of the corners.
+            checkCorners(point);
+
+            //Only proceed if the quadrilateral is complete.
+            if (candidates.size() < 4) {
+                return this;
+            }
+
+            final List<Vector2D> quadrilateral = buildQuadrilateral(minY, maxX, maxY, minX);
+            // if the quadrilateral is not well formed, e.g. only 2 points, do not attempt to reduce
+            if (quadrilateral.size() < 3) {
+                return this;
+            }
+
+            // check all points if they are within the quadrilateral
+            // in which case they can not be part of the convex hull
+            if (!insideQuadrilateral(point, quadrilateral)) {
+                candidates.add(point);
+            }
+
+            return this;
+        }
+
+        /** Appends the given points to a collection of possible hull points, if and only
+         * if the given points are outside of a constructed quadrilateral of extreme
+         * properties.
+         *
+         * @param points a given collection of points.
+         * @throws NullPointerException if points is {@code null}.
+         * @return this instance.
+         */
+        public Builder append(Collection<Vector2D> points) {
+            points.forEach(this::append);
+            return this;
+        }
+
+        /**
+         * Build a convex hull from the set appended points.
+         *
+         * @return the convex hull
+         * @throws IllegalStateException if generator fails to generate a convex hull for
+         *      the given set of input points
+         */
+        public ConvexHull2D build() {
+            Collection<Vector2D> hullVertices;
+            if (candidates.size() < 2) {
+                hullVertices = candidates;
+            } else {
+                hullVertices = findHullVertices(candidates);
+            }
+
+            if (!isConvex(hullVertices)) {
+                throw new IllegalStateException("Convex hull algorithm failed to generate solution");
+            }
+
+            return new ConvexHull2D(hullVertices, precision);
+        }
+
+        /** Build the convex quadrilateral with the found corner points (with min/max x/y
+         * coordinates).
+         *
+         * @param points the respective points with min/max x/y coordinate
+         * @return the quadrilateral
+         */
+        private static List<Vector2D> buildQuadrilateral(final Vector2D... points) {
+            final List<Vector2D> quadrilateral = new ArrayList<>();
+            for (final Vector2D p : points) {
+                if (!quadrilateral.contains(p)) {
+                    quadrilateral.add(p);
+                }
+            }
+            return quadrilateral;
+        }
+
+        /** Checks if the given point supersedes one of the corners. If it does the old
+         * corner is removed and the point added to the collection of points.
+         *
+         * @param point a given point.
+         */
+        private void checkCorners(Vector2D point) {
+            if (minX == null || point.getX() < minX.getX()) {

Review Comment:
   Here we only require one null check. If minX is null then the other corners are also null and all can be set to the point:
   ```Java
   if (minX == null) {
       minX = maxX = minY = maxY = point;
       candidates.add(point);
       return;
   }
   ```
   IIUC this condition will also be met if `candidate.isEmpty()`. However using the null check does indicate that for the rest of the method the corners are not null.



##########
commons-geometry-euclidean/src/main/java/org/apache/commons/geometry/euclidean/twod/hull/ConvexHull2D.java:
##########
@@ -0,0 +1,469 @@
+/*
+ * Licensed to the Apache Software Foundation (ASF) under one or more
+ * contributor license agreements.  See the NOTICE file distributed with
+ * this work for additional information regarding copyright ownership.
+ * The ASF licenses this file to You under the Apache License, Version 2.0
+ * (the "License"); you may not use this file except in compliance with
+ * the License.  You may obtain a copy of the License at
+ *
+ *      http://www.apache.org/licenses/LICENSE-2.0
+ *
+ * Unless required by applicable law or agreed to in writing, software
+ * distributed under the License is distributed on an "AS IS" BASIS,
+ * WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.
+ * See the License for the specific language governing permissions and
+ * limitations under the License.
+ */
+
+package org.apache.commons.geometry.euclidean.twod.hull;
+
+import java.util.ArrayList;
+import java.util.Collection;
+import java.util.Collections;
+import java.util.Iterator;
+import java.util.List;
+
+import org.apache.commons.geometry.core.ConvexHull;
+import org.apache.commons.geometry.euclidean.EuclideanCollections;
+import org.apache.commons.geometry.euclidean.twod.ConvexArea;
+import org.apache.commons.geometry.euclidean.twod.Lines;
+import org.apache.commons.geometry.euclidean.twod.Vector2D;
+import org.apache.commons.geometry.euclidean.twod.path.LinePath;
+import org.apache.commons.numbers.core.Precision;
+
+/**
+ * This class represents a convex hull in two-dimensional Euclidean space.
+ */
+public final class ConvexHull2D implements ConvexHull<Vector2D> {
+
+    /** Vertices for the convex hull, in order. */
+    private final List<Vector2D> vertices;
+
+    /** Polyline path for the convex hull. */
+    private final LinePath path;
+
+    /** Simple constructor; no validation is performed.
+     * @param vertices the vertices of the convex hull; callers are responsible for ensuring that
+     *      the given vertices are in order, unique, and define a convex hull.
+     * @param precision precision context used to compare floating point numbers
+     */
+    ConvexHull2D(final Collection<Vector2D> vertices, final Precision.DoubleEquivalence precision) {
+        this.vertices = Collections.unmodifiableList(new ArrayList<>(vertices));
+        this.path = buildHullPath(vertices, precision);
+    }
+
+    /** {@inheritDoc} */
+    @Override
+    public List<Vector2D> getVertices() {
+        return vertices;
+    }
+
+    /** Get a path defining the convex hull. The path will contain
+     * <ul>
+     *      <li>zero segments if the hull consists of only a single point,</li>
+     *      <li>one segment if the hull consists of two points,</li>
+     *      <li>three or more segments defining a closed loop if the hull consists of more than
+     *          two non-collinear points.</li>
+     * </ul>
+     * @return polyline path defining the convex hull
+     */
+    public LinePath getPath() {
+        return path;
+    }
+
+    /** {@inheritDoc} */
+    @Override
+    public ConvexArea getRegion() {
+        return path.isClosed() ?
+                ConvexArea.convexPolygonFromPath(path) :
+                null;
+    }
+
+    /** {@inheritDoc} */
+    @Override
+    public String toString() {
+        final StringBuilder sb = new StringBuilder();
+        sb.append(getClass().getSimpleName())
+            .append("[vertices= ")
+            .append(getVertices())
+            .append(']');
+
+        return sb.toString();
+    }
+
+    /** Build a polyline representing the path for a convex hull.
+     * @param vertices convex hull vertices
+     * @param precision precision context used to compare floating point values
+     * @return path for the convex hull defined by the given vertices
+     */
+    private static LinePath buildHullPath(final Collection<Vector2D> vertices,
+            final Precision.DoubleEquivalence precision) {
+        if (vertices.size() < 2) {
+            return LinePath.empty();
+        }
+
+        final boolean closeLoop = vertices.size() > 2;
+
+        return LinePath.builder(precision)
+                .appendVertices(vertices)
+                .build(closeLoop);
+    }
+
+    /** Class used to build convex hulls. The builder is based on the Akl-Toussaint
+     * heuristic to construct the hull. The heuristic is based on the idea of a
+     * convex quadrilateral, which is formed by four points with the lowest and
+     * highest x / y coordinates. Any point that lies inside this quadrilateral can
+     * not be part of the convex hull and can thus be safely discarded before
+     * generating the convex hull itself.
+     * <p>
+     * The complexity of the operation is O(n), and may greatly improve the time it
+     * takes to construct the convex hull afterwards, depending on the point
+     * distribution.
+     *
+     * @see <a href=
+     *      "http://en.wikipedia.org/wiki/Convex_hull_algorithms#Akl-Toussaint_heuristic">
+     *      Akl-Toussaint heuristic (Wikipedia)</a>
+     */
+    public static final class Builder {
+
+        /** Corner of triangle with minimal x coordinate. */
+        private Vector2D minX;
+
+        /** Corner of triangle with maximal x coordinate. */
+        private Vector2D maxX;
+
+        /** Corner of triangle with minimal y coordinate. */
+        private Vector2D minY;
+
+        /** Corner of triangle with maximal y coordinate. */
+        private Vector2D maxY;
+
+        /** Collection of all remaining candidates for a convex hull. */
+        private final Collection<Vector2D> candidates;
+
+        /** A precision context for comparing points. */
+        private final Precision.DoubleEquivalence precision;
+
+        /** Indicates if collinear points on the hull shall be present in the output.
+         * If {@code false}, only the extreme points are added to the hull.
+         */
+        private final boolean includeCollinearPoints;
+
+        /**Return a {@link Builder} instance configured with the given precision
+         * context. The precision context is used when comparing points.
+         *
+         * @param builderPrecision       precision context to use when building a convex
+         *                               hull from raw vertices; may be null if raw
+         *                               vertices are not used.
+         * @param includeCollinearPoints whether collinear points shall be added as hull
+         *                               vertices
+         */
+        public Builder(final boolean includeCollinearPoints, final Precision.DoubleEquivalence builderPrecision) {
+            this.precision = builderPrecision;
+            this.includeCollinearPoints = includeCollinearPoints;
+            candidates = EuclideanCollections.pointSet2D(builderPrecision);
+        }
+
+        /** Appends the given point to a collection of possible hull points, if and only
+         * if the given point is outside of a constructed quadrilateral of extreme properties.
+         *
+         * @param point a given point.
+         * @return this instance.
+         */
+        public Builder append(Vector2D point) {
+
+            //Checks if the given point supersedes one of the corners.
+            checkCorners(point);
+
+            //Only proceed if the quadrilateral is complete.
+            if (candidates.size() < 4) {
+                return this;
+            }
+
+            final List<Vector2D> quadrilateral = buildQuadrilateral(minY, maxX, maxY, minX);
+            // if the quadrilateral is not well formed, e.g. only 2 points, do not attempt to reduce
+            if (quadrilateral.size() < 3) {
+                return this;
+            }
+
+            // check all points if they are within the quadrilateral
+            // in which case they can not be part of the convex hull
+            if (!insideQuadrilateral(point, quadrilateral)) {
+                candidates.add(point);
+            }
+
+            return this;
+        }
+
+        /** Appends the given points to a collection of possible hull points, if and only
+         * if the given points are outside of a constructed quadrilateral of extreme
+         * properties.
+         *
+         * @param points a given collection of points.
+         * @throws NullPointerException if points is {@code null}.
+         * @return this instance.
+         */
+        public Builder append(Collection<Vector2D> points) {
+            points.forEach(this::append);
+            return this;
+        }
+
+        /**
+         * Build a convex hull from the set appended points.
+         *
+         * @return the convex hull
+         * @throws IllegalStateException if generator fails to generate a convex hull for
+         *      the given set of input points
+         */
+        public ConvexHull2D build() {
+            Collection<Vector2D> hullVertices;
+            if (candidates.size() < 2) {
+                hullVertices = candidates;
+            } else {
+                hullVertices = findHullVertices(candidates);
+            }
+
+            if (!isConvex(hullVertices)) {
+                throw new IllegalStateException("Convex hull algorithm failed to generate solution");
+            }
+
+            return new ConvexHull2D(hullVertices, precision);
+        }
+
+        /** Build the convex quadrilateral with the found corner points (with min/max x/y
+         * coordinates).
+         *
+         * @param points the respective points with min/max x/y coordinate
+         * @return the quadrilateral
+         */
+        private static List<Vector2D> buildQuadrilateral(final Vector2D... points) {
+            final List<Vector2D> quadrilateral = new ArrayList<>();
+            for (final Vector2D p : points) {
+                if (!quadrilateral.contains(p)) {
+                    quadrilateral.add(p);
+                }
+            }
+            return quadrilateral;
+        }
+
+        /** Checks if the given point supersedes one of the corners. If it does the old
+         * corner is removed and the point added to the collection of points.
+         *
+         * @param point a given point.
+         */
+        private void checkCorners(Vector2D point) {
+            if (minX == null || point.getX() < minX.getX()) {
+                minX = point;
+                candidates.add(point);
+            }
+            if (maxX == null || point.getX() > maxX.getX()) {
+                maxX = point;
+                candidates.add(point);
+            }
+            if (minY == null || point.getY() < minY.getY()) {
+                minY = point;
+                candidates.add(point);
+            }
+            if (maxY == null || point.getY() > maxY.getY()) {
+                maxY = point;
+                candidates.add(point);
+            }
+        }
+
+        /** Checks if the given point is located within the convex quadrilateral.
+         * @param point the point to check
+         * @param quadrilateralPoints the convex quadrilateral, represented by 4 points
+         * @return {@code true} if the point is inside the quadrilateral, {@code false} otherwise
+         */
+        private boolean insideQuadrilateral(final Vector2D point,
+                                                   final List<? extends Vector2D> quadrilateralPoints) {
+
+            Vector2D v1 = quadrilateralPoints.get(0);
+            Vector2D v2 = quadrilateralPoints.get(1);
+
+            if (point.equals(v1) || point.equals(v2)) {
+                return true;
+            }
+
+            // get the location of the point relative to the first two vertices
+            final double last = signedAreaPoints(v1, v2, point);
+
+            // If the area is zero then this means the given point is on a boundary line.
+            // and must be included as collinear point.
+            if (precision.eq(last, 0.0) && includeCollinearPoints) {
+                return false;
+            }
+
+            final int size = quadrilateralPoints.size();
+            // loop through the rest of the vertices
+            for (int i = 1; i < size; i++) {
+                v1 = v2;
+                v2 = quadrilateralPoints.get((i + 1) == size ? 0 : i + 1);
+
+                if (point.equals(v2)) {
+                    return true;
+                }
+
+                // do side of line test: multiply the last location with this location
+                // if they are the same sign then the operation will yield a positive result
+                // -x * -y = +xy, x * y = +xy, -x * y = -xy, x * -y = -xy
+                if (last * signedAreaPoints(v1, v2, point) < 0) {
+                    return false;
+                }
+            }
+            return true;
+        }
+
+        /** Compute the signed area of the parallelogram formed by vectors between the given points. The first
+         * vector points from {@code p0} to {@code p1} and the second from {@code p0} to {@code p3}.
+         * @param p0 first point
+         * @param p1 second point
+         * @param p2 third point
+         * @return signed area of parallelogram formed by vectors between the given points
+         */
+        private static double signedAreaPoints(final Vector2D p0, final Vector2D p1, final Vector2D p2) {
+            return p0.vectorTo(p1).signedArea(p0.vectorTo(p2));
+        }
+
+        /**
+         * Find the convex hull vertices from the set of input points.
+         * @param points the set of input points
+         * @return the convex hull vertices in CCW winding
+         */
+        private Collection<Vector2D> findHullVertices(final Collection<Vector2D> points) {
+
+            final List<Vector2D> pointsSortedByXAxis = new ArrayList<>(points);
+
+            // sort the points in increasing order on the x-axis
+            pointsSortedByXAxis.sort((o1, o2) -> {
+                // need to take the tolerance value into account, otherwise collinear points
+                // will not be handled correctly when building the upper/lower hull
+                final int cmp = precision.compare(o1.getX(), o2.getX());
+                if (cmp == 0) {
+                    return precision.compare(o1.getY(), o2.getY());
+                } else {
+                    return cmp;
+                }
+            });
+
+            // build lower hull
+            final List<Vector2D> lowerHull = new ArrayList<>();
+            for (final Vector2D p : pointsSortedByXAxis) {
+                updateHull(p, lowerHull);
+            }
+
+            // build upper hull
+            final List<Vector2D> upperHull = new ArrayList<>();
+            for (int idx = pointsSortedByXAxis.size() - 1; idx >= 0; idx--) {
+                final Vector2D p = pointsSortedByXAxis.get(idx);
+                updateHull(p, upperHull);
+            }
+
+            // concatenate the lower and upper hulls
+            // the last point of each list is omitted as it is repeated at the beginning of the other list
+            final List<Vector2D> hullVertices = new ArrayList<>(lowerHull.size() + upperHull.size() - 2);
+            for (int idx = 0; idx < lowerHull.size() - 1; idx++) {
+                hullVertices.add(lowerHull.get(idx));
+            }
+            for (int idx = 0; idx < upperHull.size() - 1; idx++) {
+                hullVertices.add(upperHull.get(idx));
+            }
+
+            // special case: if the lower and upper hull may contain only 1 point if all are identical
+            if (hullVertices.isEmpty() && !lowerHull.isEmpty()) {
+                hullVertices.add(lowerHull.get(0));
+            }
+
+            return hullVertices;
+        }
+
+        /**
+         * Update the partial hull with the current point.
+         *
+         * @param point the current point
+         * @param hull the partial hull
+         */
+        private void updateHull(final Vector2D point, final List<Vector2D> hull) {
+            if (hull.size() == 1) {
+                // ensure that we do not add an identical point
+                final Vector2D p1 = hull.get(0);
+                if (p1.eq(point, precision)) {
+                    return;
+                }
+            }
+
+            while (hull.size() >= 2) {
+                final int size = hull.size();
+                final Vector2D p1 = hull.get(size - 2);
+                final Vector2D p2 = hull.get(size - 1);
+
+                final double offset = Lines.fromPoints(p1, p2, precision).offset(point);
+                if (precision.eqZero(offset)) {
+                    // the point is collinear to the line (p1, p2)
+
+                    final double distanceToCurrent = p1.distance(point);
+                    if (precision.eqZero(distanceToCurrent) || precision.eqZero(p2.distance(point))) {
+                        // the point is assumed to be identical to either p1 or p2
+                        return;
+                    }
+
+                    final double distanceToLast = p1.distance(p2);
+                    if (includeCollinearPoints) {
+                        final int index = distanceToCurrent < distanceToLast ? size - 1 : size;
+                        hull.add(index, point);
+                    } else {
+                        if (distanceToCurrent > distanceToLast) {
+                            hull.remove(size - 1);
+                            hull.add(point);
+                        }
+                    }
+                    return;
+                } else if (offset > 0) {
+                    hull.remove(size - 1);
+                } else {
+                    break;
+                }
+            }
+            hull.add(point);
+        }
+
+        /** Return true if the given vertices define a convex hull.
+         * @param vertices the hull vertices
+         * @return {@code true} if the vertices form a convex hull, {@code false} otherwise
+         */
+        private boolean isConvex(final Collection<Vector2D> vertices) {
+            final int size = vertices.size();
+
+            if (size < 3) {
+                // 1 or 2 points always define a convex set
+                return true;
+            }
+
+            final Iterator<Vector2D> it = vertices.iterator();
+
+            Vector2D p1 = it.next();
+            Vector2D p2 = it.next();
+            Vector2D p3;
+
+            Vector2D v1;
+            Vector2D v2;
+
+            while (it.hasNext()) {
+                p3 = it.next();
+
+                v1 = p1.vectorTo(p2);

Review Comment:
   Precompute `v2` before the loop and then here use `v1 = v2`. By shifting the vectors you save the vector computation, and the need to track p1.



##########
commons-geometry-euclidean/src/main/java/org/apache/commons/geometry/euclidean/twod/hull/ConvexHull2D.java:
##########
@@ -0,0 +1,469 @@
+/*
+ * Licensed to the Apache Software Foundation (ASF) under one or more
+ * contributor license agreements.  See the NOTICE file distributed with
+ * this work for additional information regarding copyright ownership.
+ * The ASF licenses this file to You under the Apache License, Version 2.0
+ * (the "License"); you may not use this file except in compliance with
+ * the License.  You may obtain a copy of the License at
+ *
+ *      http://www.apache.org/licenses/LICENSE-2.0
+ *
+ * Unless required by applicable law or agreed to in writing, software
+ * distributed under the License is distributed on an "AS IS" BASIS,
+ * WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.
+ * See the License for the specific language governing permissions and
+ * limitations under the License.
+ */
+
+package org.apache.commons.geometry.euclidean.twod.hull;
+
+import java.util.ArrayList;
+import java.util.Collection;
+import java.util.Collections;
+import java.util.Iterator;
+import java.util.List;
+
+import org.apache.commons.geometry.core.ConvexHull;
+import org.apache.commons.geometry.euclidean.EuclideanCollections;
+import org.apache.commons.geometry.euclidean.twod.ConvexArea;
+import org.apache.commons.geometry.euclidean.twod.Lines;
+import org.apache.commons.geometry.euclidean.twod.Vector2D;
+import org.apache.commons.geometry.euclidean.twod.path.LinePath;
+import org.apache.commons.numbers.core.Precision;
+
+/**
+ * This class represents a convex hull in two-dimensional Euclidean space.
+ */
+public final class ConvexHull2D implements ConvexHull<Vector2D> {
+
+    /** Vertices for the convex hull, in order. */
+    private final List<Vector2D> vertices;
+
+    /** Polyline path for the convex hull. */
+    private final LinePath path;
+
+    /** Simple constructor; no validation is performed.
+     * @param vertices the vertices of the convex hull; callers are responsible for ensuring that
+     *      the given vertices are in order, unique, and define a convex hull.
+     * @param precision precision context used to compare floating point numbers
+     */
+    ConvexHull2D(final Collection<Vector2D> vertices, final Precision.DoubleEquivalence precision) {
+        this.vertices = Collections.unmodifiableList(new ArrayList<>(vertices));
+        this.path = buildHullPath(vertices, precision);
+    }
+
+    /** {@inheritDoc} */
+    @Override
+    public List<Vector2D> getVertices() {
+        return vertices;
+    }
+
+    /** Get a path defining the convex hull. The path will contain
+     * <ul>
+     *      <li>zero segments if the hull consists of only a single point,</li>
+     *      <li>one segment if the hull consists of two points,</li>
+     *      <li>three or more segments defining a closed loop if the hull consists of more than
+     *          two non-collinear points.</li>
+     * </ul>
+     * @return polyline path defining the convex hull
+     */
+    public LinePath getPath() {
+        return path;
+    }
+
+    /** {@inheritDoc} */
+    @Override
+    public ConvexArea getRegion() {
+        return path.isClosed() ?
+                ConvexArea.convexPolygonFromPath(path) :
+                null;
+    }
+
+    /** {@inheritDoc} */
+    @Override
+    public String toString() {
+        final StringBuilder sb = new StringBuilder();
+        sb.append(getClass().getSimpleName())
+            .append("[vertices= ")
+            .append(getVertices())
+            .append(']');
+
+        return sb.toString();
+    }
+
+    /** Build a polyline representing the path for a convex hull.
+     * @param vertices convex hull vertices
+     * @param precision precision context used to compare floating point values
+     * @return path for the convex hull defined by the given vertices
+     */
+    private static LinePath buildHullPath(final Collection<Vector2D> vertices,
+            final Precision.DoubleEquivalence precision) {
+        if (vertices.size() < 2) {
+            return LinePath.empty();
+        }
+
+        final boolean closeLoop = vertices.size() > 2;
+
+        return LinePath.builder(precision)
+                .appendVertices(vertices)
+                .build(closeLoop);
+    }
+
+    /** Class used to build convex hulls. The builder is based on the Akl-Toussaint
+     * heuristic to construct the hull. The heuristic is based on the idea of a
+     * convex quadrilateral, which is formed by four points with the lowest and
+     * highest x / y coordinates. Any point that lies inside this quadrilateral can
+     * not be part of the convex hull and can thus be safely discarded before
+     * generating the convex hull itself.
+     * <p>
+     * The complexity of the operation is O(n), and may greatly improve the time it
+     * takes to construct the convex hull afterwards, depending on the point
+     * distribution.
+     *
+     * @see <a href=
+     *      "http://en.wikipedia.org/wiki/Convex_hull_algorithms#Akl-Toussaint_heuristic">
+     *      Akl-Toussaint heuristic (Wikipedia)</a>
+     */
+    public static final class Builder {
+
+        /** Corner of triangle with minimal x coordinate. */
+        private Vector2D minX;
+
+        /** Corner of triangle with maximal x coordinate. */
+        private Vector2D maxX;
+
+        /** Corner of triangle with minimal y coordinate. */
+        private Vector2D minY;
+
+        /** Corner of triangle with maximal y coordinate. */
+        private Vector2D maxY;
+
+        /** Collection of all remaining candidates for a convex hull. */
+        private final Collection<Vector2D> candidates;
+
+        /** A precision context for comparing points. */
+        private final Precision.DoubleEquivalence precision;
+
+        /** Indicates if collinear points on the hull shall be present in the output.
+         * If {@code false}, only the extreme points are added to the hull.
+         */
+        private final boolean includeCollinearPoints;
+
+        /**Return a {@link Builder} instance configured with the given precision
+         * context. The precision context is used when comparing points.
+         *
+         * @param builderPrecision       precision context to use when building a convex
+         *                               hull from raw vertices; may be null if raw
+         *                               vertices are not used.
+         * @param includeCollinearPoints whether collinear points shall be added as hull
+         *                               vertices
+         */
+        public Builder(final boolean includeCollinearPoints, final Precision.DoubleEquivalence builderPrecision) {
+            this.precision = builderPrecision;
+            this.includeCollinearPoints = includeCollinearPoints;
+            candidates = EuclideanCollections.pointSet2D(builderPrecision);
+        }
+
+        /** Appends the given point to a collection of possible hull points, if and only
+         * if the given point is outside of a constructed quadrilateral of extreme properties.
+         *
+         * @param point a given point.
+         * @return this instance.
+         */
+        public Builder append(Vector2D point) {
+
+            //Checks if the given point supersedes one of the corners.

Review Comment:
   Whitespace after `//`. Same again on the next comment below.



##########
commons-geometry-euclidean/src/main/java/org/apache/commons/geometry/euclidean/twod/hull/ConvexHull2D.java:
##########
@@ -0,0 +1,469 @@
+/*
+ * Licensed to the Apache Software Foundation (ASF) under one or more
+ * contributor license agreements.  See the NOTICE file distributed with
+ * this work for additional information regarding copyright ownership.
+ * The ASF licenses this file to You under the Apache License, Version 2.0
+ * (the "License"); you may not use this file except in compliance with
+ * the License.  You may obtain a copy of the License at
+ *
+ *      http://www.apache.org/licenses/LICENSE-2.0
+ *
+ * Unless required by applicable law or agreed to in writing, software
+ * distributed under the License is distributed on an "AS IS" BASIS,
+ * WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.
+ * See the License for the specific language governing permissions and
+ * limitations under the License.
+ */
+
+package org.apache.commons.geometry.euclidean.twod.hull;
+
+import java.util.ArrayList;
+import java.util.Collection;
+import java.util.Collections;
+import java.util.Iterator;
+import java.util.List;
+
+import org.apache.commons.geometry.core.ConvexHull;
+import org.apache.commons.geometry.euclidean.EuclideanCollections;
+import org.apache.commons.geometry.euclidean.twod.ConvexArea;
+import org.apache.commons.geometry.euclidean.twod.Lines;
+import org.apache.commons.geometry.euclidean.twod.Vector2D;
+import org.apache.commons.geometry.euclidean.twod.path.LinePath;
+import org.apache.commons.numbers.core.Precision;
+
+/**
+ * This class represents a convex hull in two-dimensional Euclidean space.
+ */
+public final class ConvexHull2D implements ConvexHull<Vector2D> {
+
+    /** Vertices for the convex hull, in order. */
+    private final List<Vector2D> vertices;
+
+    /** Polyline path for the convex hull. */
+    private final LinePath path;
+
+    /** Simple constructor; no validation is performed.
+     * @param vertices the vertices of the convex hull; callers are responsible for ensuring that
+     *      the given vertices are in order, unique, and define a convex hull.
+     * @param precision precision context used to compare floating point numbers
+     */
+    ConvexHull2D(final Collection<Vector2D> vertices, final Precision.DoubleEquivalence precision) {
+        this.vertices = Collections.unmodifiableList(new ArrayList<>(vertices));
+        this.path = buildHullPath(vertices, precision);
+    }
+
+    /** {@inheritDoc} */
+    @Override
+    public List<Vector2D> getVertices() {
+        return vertices;
+    }
+
+    /** Get a path defining the convex hull. The path will contain
+     * <ul>
+     *      <li>zero segments if the hull consists of only a single point,</li>
+     *      <li>one segment if the hull consists of two points,</li>
+     *      <li>three or more segments defining a closed loop if the hull consists of more than
+     *          two non-collinear points.</li>
+     * </ul>
+     * @return polyline path defining the convex hull
+     */
+    public LinePath getPath() {
+        return path;
+    }
+
+    /** {@inheritDoc} */
+    @Override
+    public ConvexArea getRegion() {
+        return path.isClosed() ?
+                ConvexArea.convexPolygonFromPath(path) :
+                null;
+    }
+
+    /** {@inheritDoc} */
+    @Override
+    public String toString() {
+        final StringBuilder sb = new StringBuilder();
+        sb.append(getClass().getSimpleName())
+            .append("[vertices= ")
+            .append(getVertices())
+            .append(']');
+
+        return sb.toString();
+    }
+
+    /** Build a polyline representing the path for a convex hull.
+     * @param vertices convex hull vertices
+     * @param precision precision context used to compare floating point values
+     * @return path for the convex hull defined by the given vertices
+     */
+    private static LinePath buildHullPath(final Collection<Vector2D> vertices,
+            final Precision.DoubleEquivalence precision) {
+        if (vertices.size() < 2) {
+            return LinePath.empty();
+        }
+
+        final boolean closeLoop = vertices.size() > 2;
+
+        return LinePath.builder(precision)
+                .appendVertices(vertices)
+                .build(closeLoop);
+    }
+
+    /** Class used to build convex hulls. The builder is based on the Akl-Toussaint
+     * heuristic to construct the hull. The heuristic is based on the idea of a
+     * convex quadrilateral, which is formed by four points with the lowest and
+     * highest x / y coordinates. Any point that lies inside this quadrilateral can
+     * not be part of the convex hull and can thus be safely discarded before
+     * generating the convex hull itself.
+     * <p>
+     * The complexity of the operation is O(n), and may greatly improve the time it
+     * takes to construct the convex hull afterwards, depending on the point
+     * distribution.
+     *
+     * @see <a href=
+     *      "http://en.wikipedia.org/wiki/Convex_hull_algorithms#Akl-Toussaint_heuristic">
+     *      Akl-Toussaint heuristic (Wikipedia)</a>
+     */
+    public static final class Builder {
+
+        /** Corner of triangle with minimal x coordinate. */
+        private Vector2D minX;
+
+        /** Corner of triangle with maximal x coordinate. */
+        private Vector2D maxX;
+
+        /** Corner of triangle with minimal y coordinate. */
+        private Vector2D minY;
+
+        /** Corner of triangle with maximal y coordinate. */
+        private Vector2D maxY;
+
+        /** Collection of all remaining candidates for a convex hull. */
+        private final Collection<Vector2D> candidates;
+
+        /** A precision context for comparing points. */
+        private final Precision.DoubleEquivalence precision;
+
+        /** Indicates if collinear points on the hull shall be present in the output.
+         * If {@code false}, only the extreme points are added to the hull.
+         */
+        private final boolean includeCollinearPoints;
+
+        /**Return a {@link Builder} instance configured with the given precision
+         * context. The precision context is used when comparing points.
+         *
+         * @param builderPrecision       precision context to use when building a convex
+         *                               hull from raw vertices; may be null if raw
+         *                               vertices are not used.
+         * @param includeCollinearPoints whether collinear points shall be added as hull
+         *                               vertices
+         */
+        public Builder(final boolean includeCollinearPoints, final Precision.DoubleEquivalence builderPrecision) {
+            this.precision = builderPrecision;
+            this.includeCollinearPoints = includeCollinearPoints;
+            candidates = EuclideanCollections.pointSet2D(builderPrecision);
+        }
+
+        /** Appends the given point to a collection of possible hull points, if and only
+         * if the given point is outside of a constructed quadrilateral of extreme properties.
+         *
+         * @param point a given point.
+         * @return this instance.
+         */
+        public Builder append(Vector2D point) {
+
+            //Checks if the given point supersedes one of the corners.
+            checkCorners(point);
+
+            //Only proceed if the quadrilateral is complete.
+            if (candidates.size() < 4) {
+                return this;
+            }
+
+            final List<Vector2D> quadrilateral = buildQuadrilateral(minY, maxX, maxY, minX);
+            // if the quadrilateral is not well formed, e.g. only 2 points, do not attempt to reduce
+            if (quadrilateral.size() < 3) {
+                return this;
+            }
+
+            // check all points if they are within the quadrilateral
+            // in which case they can not be part of the convex hull
+            if (!insideQuadrilateral(point, quadrilateral)) {
+                candidates.add(point);
+            }
+
+            return this;
+        }
+
+        /** Appends the given points to a collection of possible hull points, if and only
+         * if the given points are outside of a constructed quadrilateral of extreme
+         * properties.
+         *
+         * @param points a given collection of points.
+         * @throws NullPointerException if points is {@code null}.
+         * @return this instance.
+         */
+        public Builder append(Collection<Vector2D> points) {
+            points.forEach(this::append);
+            return this;
+        }
+
+        /**
+         * Build a convex hull from the set appended points.
+         *
+         * @return the convex hull
+         * @throws IllegalStateException if generator fails to generate a convex hull for
+         *      the given set of input points
+         */
+        public ConvexHull2D build() {
+            Collection<Vector2D> hullVertices;
+            if (candidates.size() < 2) {
+                hullVertices = candidates;
+            } else {
+                hullVertices = findHullVertices(candidates);
+            }
+
+            if (!isConvex(hullVertices)) {
+                throw new IllegalStateException("Convex hull algorithm failed to generate solution");
+            }
+
+            return new ConvexHull2D(hullVertices, precision);
+        }
+
+        /** Build the convex quadrilateral with the found corner points (with min/max x/y
+         * coordinates).
+         *
+         * @param points the respective points with min/max x/y coordinate
+         * @return the quadrilateral
+         */
+        private static List<Vector2D> buildQuadrilateral(final Vector2D... points) {
+            final List<Vector2D> quadrilateral = new ArrayList<>();
+            for (final Vector2D p : points) {
+                if (!quadrilateral.contains(p)) {
+                    quadrilateral.add(p);
+                }
+            }
+            return quadrilateral;
+        }
+
+        /** Checks if the given point supersedes one of the corners. If it does the old
+         * corner is removed and the point added to the collection of points.
+         *
+         * @param point a given point.
+         */
+        private void checkCorners(Vector2D point) {
+            if (minX == null || point.getX() < minX.getX()) {
+                minX = point;
+                candidates.add(point);
+            }
+            if (maxX == null || point.getX() > maxX.getX()) {
+                maxX = point;
+                candidates.add(point);
+            }
+            if (minY == null || point.getY() < minY.getY()) {
+                minY = point;
+                candidates.add(point);
+            }
+            if (maxY == null || point.getY() > maxY.getY()) {
+                maxY = point;
+                candidates.add(point);
+            }
+        }
+
+        /** Checks if the given point is located within the convex quadrilateral.
+         * @param point the point to check
+         * @param quadrilateralPoints the convex quadrilateral, represented by 4 points
+         * @return {@code true} if the point is inside the quadrilateral, {@code false} otherwise
+         */
+        private boolean insideQuadrilateral(final Vector2D point,
+                                                   final List<? extends Vector2D> quadrilateralPoints) {

Review Comment:
   Indentation



##########
commons-geometry-euclidean/src/main/java/org/apache/commons/geometry/euclidean/twod/hull/ConvexHull2D.java:
##########
@@ -0,0 +1,469 @@
+/*
+ * Licensed to the Apache Software Foundation (ASF) under one or more
+ * contributor license agreements.  See the NOTICE file distributed with
+ * this work for additional information regarding copyright ownership.
+ * The ASF licenses this file to You under the Apache License, Version 2.0
+ * (the "License"); you may not use this file except in compliance with
+ * the License.  You may obtain a copy of the License at
+ *
+ *      http://www.apache.org/licenses/LICENSE-2.0
+ *
+ * Unless required by applicable law or agreed to in writing, software
+ * distributed under the License is distributed on an "AS IS" BASIS,
+ * WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.
+ * See the License for the specific language governing permissions and
+ * limitations under the License.
+ */
+
+package org.apache.commons.geometry.euclidean.twod.hull;
+
+import java.util.ArrayList;
+import java.util.Collection;
+import java.util.Collections;
+import java.util.Iterator;
+import java.util.List;
+
+import org.apache.commons.geometry.core.ConvexHull;
+import org.apache.commons.geometry.euclidean.EuclideanCollections;
+import org.apache.commons.geometry.euclidean.twod.ConvexArea;
+import org.apache.commons.geometry.euclidean.twod.Lines;
+import org.apache.commons.geometry.euclidean.twod.Vector2D;
+import org.apache.commons.geometry.euclidean.twod.path.LinePath;
+import org.apache.commons.numbers.core.Precision;
+
+/**
+ * This class represents a convex hull in two-dimensional Euclidean space.
+ */
+public final class ConvexHull2D implements ConvexHull<Vector2D> {
+
+    /** Vertices for the convex hull, in order. */
+    private final List<Vector2D> vertices;
+
+    /** Polyline path for the convex hull. */
+    private final LinePath path;
+
+    /** Simple constructor; no validation is performed.
+     * @param vertices the vertices of the convex hull; callers are responsible for ensuring that
+     *      the given vertices are in order, unique, and define a convex hull.
+     * @param precision precision context used to compare floating point numbers
+     */
+    ConvexHull2D(final Collection<Vector2D> vertices, final Precision.DoubleEquivalence precision) {
+        this.vertices = Collections.unmodifiableList(new ArrayList<>(vertices));
+        this.path = buildHullPath(vertices, precision);
+    }
+
+    /** {@inheritDoc} */
+    @Override
+    public List<Vector2D> getVertices() {
+        return vertices;
+    }
+
+    /** Get a path defining the convex hull. The path will contain
+     * <ul>
+     *      <li>zero segments if the hull consists of only a single point,</li>
+     *      <li>one segment if the hull consists of two points,</li>
+     *      <li>three or more segments defining a closed loop if the hull consists of more than
+     *          two non-collinear points.</li>
+     * </ul>
+     * @return polyline path defining the convex hull
+     */
+    public LinePath getPath() {
+        return path;
+    }
+
+    /** {@inheritDoc} */
+    @Override
+    public ConvexArea getRegion() {
+        return path.isClosed() ?
+                ConvexArea.convexPolygonFromPath(path) :
+                null;
+    }
+
+    /** {@inheritDoc} */
+    @Override
+    public String toString() {
+        final StringBuilder sb = new StringBuilder();
+        sb.append(getClass().getSimpleName())
+            .append("[vertices= ")
+            .append(getVertices())
+            .append(']');
+
+        return sb.toString();
+    }
+
+    /** Build a polyline representing the path for a convex hull.
+     * @param vertices convex hull vertices
+     * @param precision precision context used to compare floating point values
+     * @return path for the convex hull defined by the given vertices
+     */
+    private static LinePath buildHullPath(final Collection<Vector2D> vertices,
+            final Precision.DoubleEquivalence precision) {
+        if (vertices.size() < 2) {
+            return LinePath.empty();
+        }
+
+        final boolean closeLoop = vertices.size() > 2;
+
+        return LinePath.builder(precision)
+                .appendVertices(vertices)
+                .build(closeLoop);
+    }
+
+    /** Class used to build convex hulls. The builder is based on the Akl-Toussaint
+     * heuristic to construct the hull. The heuristic is based on the idea of a
+     * convex quadrilateral, which is formed by four points with the lowest and
+     * highest x / y coordinates. Any point that lies inside this quadrilateral can
+     * not be part of the convex hull and can thus be safely discarded before
+     * generating the convex hull itself.
+     * <p>
+     * The complexity of the operation is O(n), and may greatly improve the time it
+     * takes to construct the convex hull afterwards, depending on the point
+     * distribution.
+     *
+     * @see <a href=
+     *      "http://en.wikipedia.org/wiki/Convex_hull_algorithms#Akl-Toussaint_heuristic">
+     *      Akl-Toussaint heuristic (Wikipedia)</a>
+     */
+    public static final class Builder {
+
+        /** Corner of triangle with minimal x coordinate. */
+        private Vector2D minX;
+
+        /** Corner of triangle with maximal x coordinate. */
+        private Vector2D maxX;
+
+        /** Corner of triangle with minimal y coordinate. */
+        private Vector2D minY;
+
+        /** Corner of triangle with maximal y coordinate. */
+        private Vector2D maxY;
+
+        /** Collection of all remaining candidates for a convex hull. */
+        private final Collection<Vector2D> candidates;
+
+        /** A precision context for comparing points. */
+        private final Precision.DoubleEquivalence precision;
+
+        /** Indicates if collinear points on the hull shall be present in the output.
+         * If {@code false}, only the extreme points are added to the hull.
+         */
+        private final boolean includeCollinearPoints;
+
+        /**Return a {@link Builder} instance configured with the given precision

Review Comment:
   Whitespace after `/**`



##########
commons-geometry-euclidean/src/main/java/org/apache/commons/geometry/euclidean/twod/hull/ConvexHull2D.java:
##########
@@ -0,0 +1,469 @@
+/*
+ * Licensed to the Apache Software Foundation (ASF) under one or more
+ * contributor license agreements.  See the NOTICE file distributed with
+ * this work for additional information regarding copyright ownership.
+ * The ASF licenses this file to You under the Apache License, Version 2.0
+ * (the "License"); you may not use this file except in compliance with
+ * the License.  You may obtain a copy of the License at
+ *
+ *      http://www.apache.org/licenses/LICENSE-2.0
+ *
+ * Unless required by applicable law or agreed to in writing, software
+ * distributed under the License is distributed on an "AS IS" BASIS,
+ * WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.
+ * See the License for the specific language governing permissions and
+ * limitations under the License.
+ */
+
+package org.apache.commons.geometry.euclidean.twod.hull;
+
+import java.util.ArrayList;
+import java.util.Collection;
+import java.util.Collections;
+import java.util.Iterator;
+import java.util.List;
+
+import org.apache.commons.geometry.core.ConvexHull;
+import org.apache.commons.geometry.euclidean.EuclideanCollections;
+import org.apache.commons.geometry.euclidean.twod.ConvexArea;
+import org.apache.commons.geometry.euclidean.twod.Lines;
+import org.apache.commons.geometry.euclidean.twod.Vector2D;
+import org.apache.commons.geometry.euclidean.twod.path.LinePath;
+import org.apache.commons.numbers.core.Precision;
+
+/**
+ * This class represents a convex hull in two-dimensional Euclidean space.
+ */
+public final class ConvexHull2D implements ConvexHull<Vector2D> {
+
+    /** Vertices for the convex hull, in order. */
+    private final List<Vector2D> vertices;
+
+    /** Polyline path for the convex hull. */
+    private final LinePath path;
+
+    /** Simple constructor; no validation is performed.
+     * @param vertices the vertices of the convex hull; callers are responsible for ensuring that
+     *      the given vertices are in order, unique, and define a convex hull.
+     * @param precision precision context used to compare floating point numbers
+     */
+    ConvexHull2D(final Collection<Vector2D> vertices, final Precision.DoubleEquivalence precision) {
+        this.vertices = Collections.unmodifiableList(new ArrayList<>(vertices));
+        this.path = buildHullPath(vertices, precision);
+    }
+
+    /** {@inheritDoc} */
+    @Override
+    public List<Vector2D> getVertices() {
+        return vertices;
+    }
+
+    /** Get a path defining the convex hull. The path will contain
+     * <ul>
+     *      <li>zero segments if the hull consists of only a single point,</li>
+     *      <li>one segment if the hull consists of two points,</li>
+     *      <li>three or more segments defining a closed loop if the hull consists of more than
+     *          two non-collinear points.</li>
+     * </ul>
+     * @return polyline path defining the convex hull
+     */
+    public LinePath getPath() {
+        return path;
+    }
+
+    /** {@inheritDoc} */
+    @Override
+    public ConvexArea getRegion() {
+        return path.isClosed() ?
+                ConvexArea.convexPolygonFromPath(path) :
+                null;
+    }
+
+    /** {@inheritDoc} */
+    @Override
+    public String toString() {
+        final StringBuilder sb = new StringBuilder();
+        sb.append(getClass().getSimpleName())
+            .append("[vertices= ")
+            .append(getVertices())
+            .append(']');
+
+        return sb.toString();
+    }
+
+    /** Build a polyline representing the path for a convex hull.
+     * @param vertices convex hull vertices
+     * @param precision precision context used to compare floating point values
+     * @return path for the convex hull defined by the given vertices
+     */
+    private static LinePath buildHullPath(final Collection<Vector2D> vertices,
+            final Precision.DoubleEquivalence precision) {
+        if (vertices.size() < 2) {
+            return LinePath.empty();
+        }
+
+        final boolean closeLoop = vertices.size() > 2;
+
+        return LinePath.builder(precision)
+                .appendVertices(vertices)
+                .build(closeLoop);
+    }
+
+    /** Class used to build convex hulls. The builder is based on the Akl-Toussaint
+     * heuristic to construct the hull. The heuristic is based on the idea of a
+     * convex quadrilateral, which is formed by four points with the lowest and
+     * highest x / y coordinates. Any point that lies inside this quadrilateral can
+     * not be part of the convex hull and can thus be safely discarded before
+     * generating the convex hull itself.
+     * <p>
+     * The complexity of the operation is O(n), and may greatly improve the time it
+     * takes to construct the convex hull afterwards, depending on the point
+     * distribution.
+     *
+     * @see <a href=
+     *      "http://en.wikipedia.org/wiki/Convex_hull_algorithms#Akl-Toussaint_heuristic">
+     *      Akl-Toussaint heuristic (Wikipedia)</a>
+     */
+    public static final class Builder {
+
+        /** Corner of triangle with minimal x coordinate. */
+        private Vector2D minX;
+
+        /** Corner of triangle with maximal x coordinate. */
+        private Vector2D maxX;
+
+        /** Corner of triangle with minimal y coordinate. */
+        private Vector2D minY;
+
+        /** Corner of triangle with maximal y coordinate. */
+        private Vector2D maxY;
+
+        /** Collection of all remaining candidates for a convex hull. */
+        private final Collection<Vector2D> candidates;
+
+        /** A precision context for comparing points. */
+        private final Precision.DoubleEquivalence precision;
+
+        /** Indicates if collinear points on the hull shall be present in the output.
+         * If {@code false}, only the extreme points are added to the hull.
+         */
+        private final boolean includeCollinearPoints;
+
+        /**Return a {@link Builder} instance configured with the given precision
+         * context. The precision context is used when comparing points.
+         *
+         * @param builderPrecision       precision context to use when building a convex
+         *                               hull from raw vertices; may be null if raw
+         *                               vertices are not used.
+         * @param includeCollinearPoints whether collinear points shall be added as hull
+         *                               vertices
+         */
+        public Builder(final boolean includeCollinearPoints, final Precision.DoubleEquivalence builderPrecision) {
+            this.precision = builderPrecision;
+            this.includeCollinearPoints = includeCollinearPoints;
+            candidates = EuclideanCollections.pointSet2D(builderPrecision);
+        }
+
+        /** Appends the given point to a collection of possible hull points, if and only
+         * if the given point is outside of a constructed quadrilateral of extreme properties.
+         *
+         * @param point a given point.
+         * @return this instance.
+         */
+        public Builder append(Vector2D point) {
+
+            //Checks if the given point supersedes one of the corners.
+            checkCorners(point);
+
+            //Only proceed if the quadrilateral is complete.
+            if (candidates.size() < 4) {
+                return this;
+            }
+
+            final List<Vector2D> quadrilateral = buildQuadrilateral(minY, maxX, maxY, minX);
+            // if the quadrilateral is not well formed, e.g. only 2 points, do not attempt to reduce
+            if (quadrilateral.size() < 3) {
+                return this;
+            }
+
+            // check all points if they are within the quadrilateral
+            // in which case they can not be part of the convex hull
+            if (!insideQuadrilateral(point, quadrilateral)) {
+                candidates.add(point);
+            }
+
+            return this;
+        }
+
+        /** Appends the given points to a collection of possible hull points, if and only
+         * if the given points are outside of a constructed quadrilateral of extreme
+         * properties.
+         *
+         * @param points a given collection of points.
+         * @throws NullPointerException if points is {@code null}.
+         * @return this instance.
+         */
+        public Builder append(Collection<Vector2D> points) {
+            points.forEach(this::append);
+            return this;
+        }
+
+        /**
+         * Build a convex hull from the set appended points.
+         *
+         * @return the convex hull
+         * @throws IllegalStateException if generator fails to generate a convex hull for
+         *      the given set of input points
+         */
+        public ConvexHull2D build() {
+            Collection<Vector2D> hullVertices;
+            if (candidates.size() < 2) {
+                hullVertices = candidates;
+            } else {
+                hullVertices = findHullVertices(candidates);
+            }
+
+            if (!isConvex(hullVertices)) {
+                throw new IllegalStateException("Convex hull algorithm failed to generate solution");
+            }
+
+            return new ConvexHull2D(hullVertices, precision);
+        }
+
+        /** Build the convex quadrilateral with the found corner points (with min/max x/y
+         * coordinates).
+         *
+         * @param points the respective points with min/max x/y coordinate
+         * @return the quadrilateral
+         */
+        private static List<Vector2D> buildQuadrilateral(final Vector2D... points) {
+            final List<Vector2D> quadrilateral = new ArrayList<>();
+            for (final Vector2D p : points) {
+                if (!quadrilateral.contains(p)) {
+                    quadrilateral.add(p);
+                }
+            }
+            return quadrilateral;
+        }
+
+        /** Checks if the given point supersedes one of the corners. If it does the old
+         * corner is removed and the point added to the collection of points.
+         *
+         * @param point a given point.
+         */
+        private void checkCorners(Vector2D point) {
+            if (minX == null || point.getX() < minX.getX()) {
+                minX = point;
+                candidates.add(point);
+            }
+            if (maxX == null || point.getX() > maxX.getX()) {
+                maxX = point;
+                candidates.add(point);
+            }
+            if (minY == null || point.getY() < minY.getY()) {
+                minY = point;
+                candidates.add(point);
+            }
+            if (maxY == null || point.getY() > maxY.getY()) {
+                maxY = point;
+                candidates.add(point);
+            }
+        }
+
+        /** Checks if the given point is located within the convex quadrilateral.
+         * @param point the point to check
+         * @param quadrilateralPoints the convex quadrilateral, represented by 4 points
+         * @return {@code true} if the point is inside the quadrilateral, {@code false} otherwise
+         */
+        private boolean insideQuadrilateral(final Vector2D point,
+                                                   final List<? extends Vector2D> quadrilateralPoints) {
+
+            Vector2D v1 = quadrilateralPoints.get(0);

Review Comment:
   Since this walks a circle around the quadrilateral we can start anywhere. IIUC this can be changed to:
   ```java
   final int size = quadrilateralPoints.size();
   Vector2D v1 = quadrilateralPoints.get(size - 1);
   Vector2D v2 = quadrilateralPoints.get(0);
   
   // ...
   
   for (int i = 1; i < size; i++) {
       v1 = v2;
       v2 = quadrilateralPoints.get(i);
   ```
   



##########
commons-geometry-euclidean/src/main/java/org/apache/commons/geometry/euclidean/twod/hull/ConvexHull2D.java:
##########
@@ -0,0 +1,469 @@
+/*
+ * Licensed to the Apache Software Foundation (ASF) under one or more
+ * contributor license agreements.  See the NOTICE file distributed with
+ * this work for additional information regarding copyright ownership.
+ * The ASF licenses this file to You under the Apache License, Version 2.0
+ * (the "License"); you may not use this file except in compliance with
+ * the License.  You may obtain a copy of the License at
+ *
+ *      http://www.apache.org/licenses/LICENSE-2.0
+ *
+ * Unless required by applicable law or agreed to in writing, software
+ * distributed under the License is distributed on an "AS IS" BASIS,
+ * WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.
+ * See the License for the specific language governing permissions and
+ * limitations under the License.
+ */
+
+package org.apache.commons.geometry.euclidean.twod.hull;
+
+import java.util.ArrayList;
+import java.util.Collection;
+import java.util.Collections;
+import java.util.Iterator;
+import java.util.List;
+
+import org.apache.commons.geometry.core.ConvexHull;
+import org.apache.commons.geometry.euclidean.EuclideanCollections;
+import org.apache.commons.geometry.euclidean.twod.ConvexArea;
+import org.apache.commons.geometry.euclidean.twod.Lines;
+import org.apache.commons.geometry.euclidean.twod.Vector2D;
+import org.apache.commons.geometry.euclidean.twod.path.LinePath;
+import org.apache.commons.numbers.core.Precision;
+
+/**
+ * This class represents a convex hull in two-dimensional Euclidean space.
+ */
+public final class ConvexHull2D implements ConvexHull<Vector2D> {
+
+    /** Vertices for the convex hull, in order. */
+    private final List<Vector2D> vertices;
+
+    /** Polyline path for the convex hull. */
+    private final LinePath path;
+
+    /** Simple constructor; no validation is performed.
+     * @param vertices the vertices of the convex hull; callers are responsible for ensuring that
+     *      the given vertices are in order, unique, and define a convex hull.
+     * @param precision precision context used to compare floating point numbers
+     */
+    ConvexHull2D(final Collection<Vector2D> vertices, final Precision.DoubleEquivalence precision) {
+        this.vertices = Collections.unmodifiableList(new ArrayList<>(vertices));
+        this.path = buildHullPath(vertices, precision);
+    }
+
+    /** {@inheritDoc} */
+    @Override
+    public List<Vector2D> getVertices() {
+        return vertices;
+    }
+
+    /** Get a path defining the convex hull. The path will contain
+     * <ul>
+     *      <li>zero segments if the hull consists of only a single point,</li>
+     *      <li>one segment if the hull consists of two points,</li>
+     *      <li>three or more segments defining a closed loop if the hull consists of more than
+     *          two non-collinear points.</li>
+     * </ul>
+     * @return polyline path defining the convex hull
+     */
+    public LinePath getPath() {
+        return path;
+    }
+
+    /** {@inheritDoc} */
+    @Override
+    public ConvexArea getRegion() {
+        return path.isClosed() ?
+                ConvexArea.convexPolygonFromPath(path) :
+                null;
+    }
+
+    /** {@inheritDoc} */
+    @Override
+    public String toString() {
+        final StringBuilder sb = new StringBuilder();
+        sb.append(getClass().getSimpleName())
+            .append("[vertices= ")
+            .append(getVertices())
+            .append(']');
+
+        return sb.toString();
+    }
+
+    /** Build a polyline representing the path for a convex hull.
+     * @param vertices convex hull vertices
+     * @param precision precision context used to compare floating point values
+     * @return path for the convex hull defined by the given vertices
+     */
+    private static LinePath buildHullPath(final Collection<Vector2D> vertices,
+            final Precision.DoubleEquivalence precision) {
+        if (vertices.size() < 2) {
+            return LinePath.empty();
+        }
+
+        final boolean closeLoop = vertices.size() > 2;
+
+        return LinePath.builder(precision)
+                .appendVertices(vertices)
+                .build(closeLoop);
+    }
+
+    /** Class used to build convex hulls. The builder is based on the Akl-Toussaint
+     * heuristic to construct the hull. The heuristic is based on the idea of a
+     * convex quadrilateral, which is formed by four points with the lowest and
+     * highest x / y coordinates. Any point that lies inside this quadrilateral can
+     * not be part of the convex hull and can thus be safely discarded before
+     * generating the convex hull itself.
+     * <p>
+     * The complexity of the operation is O(n), and may greatly improve the time it
+     * takes to construct the convex hull afterwards, depending on the point
+     * distribution.
+     *
+     * @see <a href=
+     *      "http://en.wikipedia.org/wiki/Convex_hull_algorithms#Akl-Toussaint_heuristic">
+     *      Akl-Toussaint heuristic (Wikipedia)</a>
+     */
+    public static final class Builder {
+
+        /** Corner of triangle with minimal x coordinate. */
+        private Vector2D minX;
+
+        /** Corner of triangle with maximal x coordinate. */
+        private Vector2D maxX;
+
+        /** Corner of triangle with minimal y coordinate. */
+        private Vector2D minY;
+
+        /** Corner of triangle with maximal y coordinate. */
+        private Vector2D maxY;
+
+        /** Collection of all remaining candidates for a convex hull. */
+        private final Collection<Vector2D> candidates;
+
+        /** A precision context for comparing points. */
+        private final Precision.DoubleEquivalence precision;
+
+        /** Indicates if collinear points on the hull shall be present in the output.
+         * If {@code false}, only the extreme points are added to the hull.
+         */
+        private final boolean includeCollinearPoints;
+
+        /**Return a {@link Builder} instance configured with the given precision
+         * context. The precision context is used when comparing points.
+         *
+         * @param builderPrecision       precision context to use when building a convex
+         *                               hull from raw vertices; may be null if raw
+         *                               vertices are not used.
+         * @param includeCollinearPoints whether collinear points shall be added as hull
+         *                               vertices
+         */
+        public Builder(final boolean includeCollinearPoints, final Precision.DoubleEquivalence builderPrecision) {
+            this.precision = builderPrecision;
+            this.includeCollinearPoints = includeCollinearPoints;
+            candidates = EuclideanCollections.pointSet2D(builderPrecision);
+        }
+
+        /** Appends the given point to a collection of possible hull points, if and only
+         * if the given point is outside of a constructed quadrilateral of extreme properties.
+         *
+         * @param point a given point.
+         * @return this instance.
+         */
+        public Builder append(Vector2D point) {
+
+            //Checks if the given point supersedes one of the corners.
+            checkCorners(point);
+
+            //Only proceed if the quadrilateral is complete.
+            if (candidates.size() < 4) {
+                return this;
+            }
+
+            final List<Vector2D> quadrilateral = buildQuadrilateral(minY, maxX, maxY, minX);
+            // if the quadrilateral is not well formed, e.g. only 2 points, do not attempt to reduce
+            if (quadrilateral.size() < 3) {
+                return this;
+            }
+
+            // check all points if they are within the quadrilateral
+            // in which case they can not be part of the convex hull
+            if (!insideQuadrilateral(point, quadrilateral)) {
+                candidates.add(point);
+            }
+
+            return this;
+        }
+
+        /** Appends the given points to a collection of possible hull points, if and only
+         * if the given points are outside of a constructed quadrilateral of extreme
+         * properties.
+         *
+         * @param points a given collection of points.
+         * @throws NullPointerException if points is {@code null}.
+         * @return this instance.
+         */
+        public Builder append(Collection<Vector2D> points) {
+            points.forEach(this::append);
+            return this;
+        }
+
+        /**
+         * Build a convex hull from the set appended points.
+         *
+         * @return the convex hull
+         * @throws IllegalStateException if generator fails to generate a convex hull for
+         *      the given set of input points
+         */
+        public ConvexHull2D build() {
+            Collection<Vector2D> hullVertices;
+            if (candidates.size() < 2) {
+                hullVertices = candidates;
+            } else {
+                hullVertices = findHullVertices(candidates);
+            }
+
+            if (!isConvex(hullVertices)) {
+                throw new IllegalStateException("Convex hull algorithm failed to generate solution");
+            }
+
+            return new ConvexHull2D(hullVertices, precision);
+        }
+
+        /** Build the convex quadrilateral with the found corner points (with min/max x/y
+         * coordinates).
+         *
+         * @param points the respective points with min/max x/y coordinate
+         * @return the quadrilateral
+         */
+        private static List<Vector2D> buildQuadrilateral(final Vector2D... points) {
+            final List<Vector2D> quadrilateral = new ArrayList<>();
+            for (final Vector2D p : points) {
+                if (!quadrilateral.contains(p)) {
+                    quadrilateral.add(p);
+                }
+            }
+            return quadrilateral;
+        }
+
+        /** Checks if the given point supersedes one of the corners. If it does the old
+         * corner is removed and the point added to the collection of points.
+         *
+         * @param point a given point.
+         */
+        private void checkCorners(Vector2D point) {
+            if (minX == null || point.getX() < minX.getX()) {
+                minX = point;
+                candidates.add(point);
+            }
+            if (maxX == null || point.getX() > maxX.getX()) {
+                maxX = point;
+                candidates.add(point);
+            }
+            if (minY == null || point.getY() < minY.getY()) {
+                minY = point;
+                candidates.add(point);
+            }
+            if (maxY == null || point.getY() > maxY.getY()) {
+                maxY = point;
+                candidates.add(point);
+            }
+        }
+
+        /** Checks if the given point is located within the convex quadrilateral.
+         * @param point the point to check
+         * @param quadrilateralPoints the convex quadrilateral, represented by 4 points
+         * @return {@code true} if the point is inside the quadrilateral, {@code false} otherwise
+         */
+        private boolean insideQuadrilateral(final Vector2D point,
+                                                   final List<? extends Vector2D> quadrilateralPoints) {
+
+            Vector2D v1 = quadrilateralPoints.get(0);
+            Vector2D v2 = quadrilateralPoints.get(1);
+
+            if (point.equals(v1) || point.equals(v2)) {
+                return true;
+            }
+
+            // get the location of the point relative to the first two vertices
+            final double last = signedAreaPoints(v1, v2, point);
+
+            // If the area is zero then this means the given point is on a boundary line.
+            // and must be included as collinear point.
+            if (precision.eq(last, 0.0) && includeCollinearPoints) {
+                return false;
+            }
+
+            final int size = quadrilateralPoints.size();
+            // loop through the rest of the vertices
+            for (int i = 1; i < size; i++) {
+                v1 = v2;
+                v2 = quadrilateralPoints.get((i + 1) == size ? 0 : i + 1);
+
+                if (point.equals(v2)) {
+                    return true;
+                }
+
+                // do side of line test: multiply the last location with this location
+                // if they are the same sign then the operation will yield a positive result
+                // -x * -y = +xy, x * y = +xy, -x * y = -xy, x * -y = -xy
+                if (last * signedAreaPoints(v1, v2, point) < 0) {
+                    return false;
+                }
+            }
+            return true;
+        }
+
+        /** Compute the signed area of the parallelogram formed by vectors between the given points. The first
+         * vector points from {@code p0} to {@code p1} and the second from {@code p0} to {@code p3}.
+         * @param p0 first point
+         * @param p1 second point
+         * @param p2 third point
+         * @return signed area of parallelogram formed by vectors between the given points
+         */
+        private static double signedAreaPoints(final Vector2D p0, final Vector2D p1, final Vector2D p2) {
+            return p0.vectorTo(p1).signedArea(p0.vectorTo(p2));
+        }
+
+        /**
+         * Find the convex hull vertices from the set of input points.
+         * @param points the set of input points
+         * @return the convex hull vertices in CCW winding
+         */
+        private Collection<Vector2D> findHullVertices(final Collection<Vector2D> points) {
+
+            final List<Vector2D> pointsSortedByXAxis = new ArrayList<>(points);
+
+            // sort the points in increasing order on the x-axis
+            pointsSortedByXAxis.sort((o1, o2) -> {
+                // need to take the tolerance value into account, otherwise collinear points
+                // will not be handled correctly when building the upper/lower hull
+                final int cmp = precision.compare(o1.getX(), o2.getX());
+                if (cmp == 0) {
+                    return precision.compare(o1.getY(), o2.getY());
+                } else {
+                    return cmp;
+                }
+            });
+
+            // build lower hull
+            final List<Vector2D> lowerHull = new ArrayList<>();
+            for (final Vector2D p : pointsSortedByXAxis) {
+                updateHull(p, lowerHull);
+            }
+
+            // build upper hull
+            final List<Vector2D> upperHull = new ArrayList<>();
+            for (int idx = pointsSortedByXAxis.size() - 1; idx >= 0; idx--) {
+                final Vector2D p = pointsSortedByXAxis.get(idx);
+                updateHull(p, upperHull);
+            }
+
+            // concatenate the lower and upper hulls
+            // the last point of each list is omitted as it is repeated at the beginning of the other list
+            final List<Vector2D> hullVertices = new ArrayList<>(lowerHull.size() + upperHull.size() - 2);
+            for (int idx = 0; idx < lowerHull.size() - 1; idx++) {

Review Comment:
   Can these loops be changed to omit the first point of each half:
   ```java
   for (int idx = 1; idx < lowerHull.size(); idx++) {
   ```



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