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* 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.math3.geometry.euclidean.threed;
import org.apache.commons.math3.exception.MathArithmeticException;
import org.apache.commons.math3.exception.util.LocalizedFormats;
import org.apache.commons.math3.geometry.Point;
import org.apache.commons.math3.geometry.Vector;
import org.apache.commons.math3.geometry.euclidean.oned.Euclidean1D;
import org.apache.commons.math3.geometry.euclidean.oned.Vector1D;
import org.apache.commons.math3.geometry.euclidean.twod.Euclidean2D;
import org.apache.commons.math3.geometry.euclidean.twod.PolygonsSet;
import org.apache.commons.math3.geometry.euclidean.twod.Vector2D;
import org.apache.commons.math3.geometry.partitioning.Embedding;
import org.apache.commons.math3.geometry.partitioning.Hyperplane;
import org.apache.commons.math3.util.FastMath;
/** The class represent planes in a three dimensional space.
* @since 3.0
*/
public class Plane implements Hyperplane<Euclidean3D>, Embedding<Euclidean3D, Euclidean2D> {
/** Default value for tolerance. */
private static final double DEFAULT_TOLERANCE = 1.0e-10;
/** Offset of the origin with respect to the plane. */
private double originOffset;
/** Origin of the plane frame. */
private Vector3D origin;
/** First vector of the plane frame (in plane). */
private Vector3D u;
/** Second vector of the plane frame (in plane). */
private Vector3D v;
/** Third vector of the plane frame (plane normal). */
private Vector3D w;
/** Tolerance below which points are considered identical. */
private final double tolerance;
/** Build a plane normal to a given direction and containing the origin.
* @param normal normal direction to the plane
* @param tolerance tolerance below which points are considered identical
* @exception MathArithmeticException if the normal norm is too small
* @since 3.3
*/
public Plane(final Vector3D normal, final double tolerance)
throws MathArithmeticException {
setNormal(normal);
this.tolerance = tolerance;
originOffset = 0;
setFrame();
}
/** Build a plane from a point and a normal.
* @param p point belonging to the plane
* @param normal normal direction to the plane
* @param tolerance tolerance below which points are considered identical
* @exception MathArithmeticException if the normal norm is too small
* @since 3.3
*/
public Plane(final Vector3D p, final Vector3D normal, final double tolerance)
throws MathArithmeticException {
setNormal(normal);
this.tolerance = tolerance;
originOffset = -p.dotProduct(w);
setFrame();
}
/** Build a plane from three points.
* <p>The plane is oriented in the direction of
* {@code (p2-p1) ^ (p3-p1)}</p>
* @param p1 first point belonging to the plane
* @param p2 second point belonging to the plane
* @param p3 third point belonging to the plane
* @param tolerance tolerance below which points are considered identical
* @exception MathArithmeticException if the points do not constitute a plane
* @since 3.3
*/
public Plane(final Vector3D p1, final Vector3D p2, final Vector3D p3, final double tolerance)
throws MathArithmeticException {
this(p1, p2.subtract(p1).crossProduct(p3.subtract(p1)), tolerance);
}
/** Build a plane normal to a given direction and containing the origin.
* @param normal normal direction to the plane
* @exception MathArithmeticException if the normal norm is too small
* @deprecated as of 3.3, replaced with {@link #Plane(Vector3D, double)}
*/
@Deprecated
public Plane(final Vector3D normal) throws MathArithmeticException {
this(normal, DEFAULT_TOLERANCE);
}
/** Build a plane from a point and a normal.
* @param p point belonging to the plane
* @param normal normal direction to the plane
* @exception MathArithmeticException if the normal norm is too small
* @deprecated as of 3.3, replaced with {@link #Plane(Vector3D, Vector3D, double)}
*/
@Deprecated
public Plane(final Vector3D p, final Vector3D normal) throws MathArithmeticException {
this(p, normal, DEFAULT_TOLERANCE);
}
/** Build a plane from three points.
* <p>The plane is oriented in the direction of
* {@code (p2-p1) ^ (p3-p1)}</p>
* @param p1 first point belonging to the plane
* @param p2 second point belonging to the plane
* @param p3 third point belonging to the plane
* @exception MathArithmeticException if the points do not constitute a plane
* @deprecated as of 3.3, replaced with {@link #Plane(Vector3D, Vector3D, Vector3D, double)}
*/
@Deprecated
public Plane(final Vector3D p1, final Vector3D p2, final Vector3D p3)
throws MathArithmeticException {
this(p1, p2, p3, DEFAULT_TOLERANCE);
}
/** Copy constructor.
* <p>The instance created is completely independant of the original
* one. A deep copy is used, none of the underlying object are
* shared.</p>
* @param plane plane to copy
*/
public Plane(final Plane plane) {
originOffset = plane.originOffset;
origin = plane.origin;
u = plane.u;
v = plane.v;
w = plane.w;
tolerance = plane.tolerance;
}
/** Copy the instance.
* <p>The instance created is completely independant of the original
* one. A deep copy is used, none of the underlying objects are
* shared (except for immutable objects).</p>
* @return a new hyperplane, copy of the instance
*/
public Plane copySelf() {
return new Plane(this);
}
/** Reset the instance as if built from a point and a normal.
* @param p point belonging to the plane
* @param normal normal direction to the plane
* @exception MathArithmeticException if the normal norm is too small
*/
public void reset(final Vector3D p, final Vector3D normal) throws MathArithmeticException {
setNormal(normal);
originOffset = -p.dotProduct(w);
setFrame();
}
/** Reset the instance from another one.
* <p>The updated instance is completely independant of the original
* one. A deep reset is used none of the underlying object is
* shared.</p>
* @param original plane to reset from
*/
public void reset(final Plane original) {
originOffset = original.originOffset;
origin = original.origin;
u = original.u;
v = original.v;
w = original.w;
}
/** Set the normal vactor.
* @param normal normal direction to the plane (will be copied)
* @exception MathArithmeticException if the normal norm is too small
*/
private void setNormal(final Vector3D normal) throws MathArithmeticException {
final double norm = normal.getNorm();
if (norm < 1.0e-10) {
throw new MathArithmeticException(LocalizedFormats.ZERO_NORM);
}
w = new Vector3D(1.0 / norm, normal);
}
/** Reset the plane frame.
*/
private void setFrame() {
origin = new Vector3D(-originOffset, w);
u = w.orthogonal();
v = Vector3D.crossProduct(w, u);
}
/** Get the origin point of the plane frame.
* <p>The point returned is the orthogonal projection of the
* 3D-space origin in the plane.</p>
* @return the origin point of the plane frame (point closest to the
* 3D-space origin)
*/
public Vector3D getOrigin() {
return origin;
}
/** Get the normalized normal vector.
* <p>The frame defined by ({@link #getU getU}, {@link #getV getV},
* {@link #getNormal getNormal}) is a rigth-handed orthonormalized
* frame).</p>
* @return normalized normal vector
* @see #getU
* @see #getV
*/
public Vector3D getNormal() {
return w;
}
/** Get the plane first canonical vector.
* <p>The frame defined by ({@link #getU getU}, {@link #getV getV},
* {@link #getNormal getNormal}) is a rigth-handed orthonormalized
* frame).</p>
* @return normalized first canonical vector
* @see #getV
* @see #getNormal
*/
public Vector3D getU() {
return u;
}
/** Get the plane second canonical vector.
* <p>The frame defined by ({@link #getU getU}, {@link #getV getV},
* {@link #getNormal getNormal}) is a rigth-handed orthonormalized
* frame).</p>
* @return normalized second canonical vector
* @see #getU
* @see #getNormal
*/
public Vector3D getV() {
return v;
}
/** {@inheritDoc}
* @since 3.3
*/
public Point<Euclidean3D> project(Point<Euclidean3D> point) {
return toSpace(toSubSpace(point));
}
/** {@inheritDoc}
* @since 3.3
*/
public double getTolerance() {
return tolerance;
}
/** Revert the plane.
* <p>Replace the instance by a similar plane with opposite orientation.</p>
* <p>The new plane frame is chosen in such a way that a 3D point that had
* {@code (x, y)} in-plane coordinates and {@code z} offset with
* respect to the plane and is unaffected by the change will have
* {@code (y, x)} in-plane coordinates and {@code -z} offset with
* respect to the new plane. This means that the {@code u} and {@code v}
* vectors returned by the {@link #getU} and {@link #getV} methods are exchanged,
* and the {@code w} vector returned by the {@link #getNormal} method is
* reversed.</p>
*/
public void revertSelf() {
final Vector3D tmp = u;
u = v;
v = tmp;
w = w.negate();
originOffset = -originOffset;
}
/** Transform a space point into a sub-space point.
* @param vector n-dimension point of the space
* @return (n-1)-dimension point of the sub-space corresponding to
* the specified space point
*/
public Vector2D toSubSpace(Vector<Euclidean3D> vector) {
return toSubSpace((Point<Euclidean3D>) vector);
}
/** Transform a sub-space point into a space point.
* @param vector (n-1)-dimension point of the sub-space
* @return n-dimension point of the space corresponding to the
* specified sub-space point
*/
public Vector3D toSpace(Vector<Euclidean2D> vector) {
return toSpace((Point<Euclidean2D>) vector);
}
/** Transform a 3D space point into an in-plane point.
* @param point point of the space (must be a {@link Vector3D
* Vector3D} instance)
* @return in-plane point (really a {@link
* org.apache.commons.math3.geometry.euclidean.twod.Vector2D Vector2D} instance)
* @see #toSpace
*/
public Vector2D toSubSpace(final Point<Euclidean3D> point) {
final Vector3D p3D = (Vector3D) point;
return new Vector2D(p3D.dotProduct(u), p3D.dotProduct(v));
}
/** Transform an in-plane point into a 3D space point.
* @param point in-plane point (must be a {@link
* org.apache.commons.math3.geometry.euclidean.twod.Vector2D Vector2D} instance)
* @return 3D space point (really a {@link Vector3D Vector3D} instance)
* @see #toSubSpace
*/
public Vector3D toSpace(final Point<Euclidean2D> point) {
final Vector2D p2D = (Vector2D) point;
return new Vector3D(p2D.getX(), u, p2D.getY(), v, -originOffset, w);
}
/** Get one point from the 3D-space.
* @param inPlane desired in-plane coordinates for the point in the
* plane
* @param offset desired offset for the point
* @return one point in the 3D-space, with given coordinates and offset
* relative to the plane
*/
public Vector3D getPointAt(final Vector2D inPlane, final double offset) {
return new Vector3D(inPlane.getX(), u, inPlane.getY(), v, offset - originOffset, w);
}
/** Check if the instance is similar to another plane.
* <p>Planes are considered similar if they contain the same
* points. This does not mean they are equal since they can have
* opposite normals.</p>
* @param plane plane to which the instance is compared
* @return true if the planes are similar
*/
public boolean isSimilarTo(final Plane plane) {
final double angle = Vector3D.angle(w, plane.w);
return ((angle < 1.0e-10) && (FastMath.abs(originOffset - plane.originOffset) < tolerance)) ||
((angle > (FastMath.PI - 1.0e-10)) && (FastMath.abs(originOffset + plane.originOffset) < tolerance));
}
/** Rotate the plane around the specified point.
* <p>The instance is not modified, a new instance is created.</p>
* @param center rotation center
* @param rotation vectorial rotation operator
* @return a new plane
*/
public Plane rotate(final Vector3D center, final Rotation rotation) {
final Vector3D delta = origin.subtract(center);
final Plane plane = new Plane(center.add(rotation.applyTo(delta)),
rotation.applyTo(w), tolerance);
// make sure the frame is transformed as desired
plane.u = rotation.applyTo(u);
plane.v = rotation.applyTo(v);
return plane;
}
/** Translate the plane by the specified amount.
* <p>The instance is not modified, a new instance is created.</p>
* @param translation translation to apply
* @return a new plane
*/
public Plane translate(final Vector3D translation) {
final Plane plane = new Plane(origin.add(translation), w, tolerance);
// make sure the frame is transformed as desired
plane.u = u;
plane.v = v;
return plane;
}
/** Get the intersection of a line with the instance.
* @param line line intersecting the instance
* @return intersection point between between the line and the
* instance (null if the line is parallel to the instance)
*/
public Vector3D intersection(final Line line) {
final Vector3D direction = line.getDirection();
final double dot = w.dotProduct(direction);
if (FastMath.abs(dot) < 1.0e-10) {
return null;
}
final Vector3D point = line.toSpace((Point<Euclidean1D>) Vector1D.ZERO);
final double k = -(originOffset + w.dotProduct(point)) / dot;
return new Vector3D(1.0, point, k, direction);
}
/** Build the line shared by the instance and another plane.
* @param other other plane
* @return line at the intersection of the instance and the
* other plane (really a {@link Line Line} instance)
*/
public Line intersection(final Plane other) {
final Vector3D direction = Vector3D.crossProduct(w, other.w);
if (direction.getNorm() < tolerance) {
return null;
}
final Vector3D point = intersection(this, other, new Plane(direction, tolerance));
return new Line(point, point.add(direction), tolerance);
}
/** Get the intersection point of three planes.
* @param plane1 first plane1
* @param plane2 second plane2
* @param plane3 third plane2
* @return intersection point of three planes, null if some planes are parallel
*/
public static Vector3D intersection(final Plane plane1, final Plane plane2, final Plane plane3) {
// coefficients of the three planes linear equations
final double a1 = plane1.w.getX();
final double b1 = plane1.w.getY();
final double c1 = plane1.w.getZ();
final double d1 = plane1.originOffset;
final double a2 = plane2.w.getX();
final double b2 = plane2.w.getY();
final double c2 = plane2.w.getZ();
final double d2 = plane2.originOffset;
final double a3 = plane3.w.getX();
final double b3 = plane3.w.getY();
final double c3 = plane3.w.getZ();
final double d3 = plane3.originOffset;
// direct Cramer resolution of the linear system
// (this is still feasible for a 3x3 system)
final double a23 = b2 * c3 - b3 * c2;
final double b23 = c2 * a3 - c3 * a2;
final double c23 = a2 * b3 - a3 * b2;
final double determinant = a1 * a23 + b1 * b23 + c1 * c23;
if (FastMath.abs(determinant) < 1.0e-10) {
return null;
}
final double r = 1.0 / determinant;
return new Vector3D(
(-a23 * d1 - (c1 * b3 - c3 * b1) * d2 - (c2 * b1 - c1 * b2) * d3) * r,
(-b23 * d1 - (c3 * a1 - c1 * a3) * d2 - (c1 * a2 - c2 * a1) * d3) * r,
(-c23 * d1 - (b1 * a3 - b3 * a1) * d2 - (b2 * a1 - b1 * a2) * d3) * r);
}
/** Build a region covering the whole hyperplane.
* @return a region covering the whole hyperplane
*/
public SubPlane wholeHyperplane() {
return new SubPlane(this, new PolygonsSet(tolerance));
}
/** Build a region covering the whole space.
* @return a region containing the instance (really a {@link
* PolyhedronsSet PolyhedronsSet} instance)
*/
public PolyhedronsSet wholeSpace() {
return new PolyhedronsSet(tolerance);
}
/** Check if the instance contains a point.
* @param p point to check
* @return true if p belongs to the plane
*/
public boolean contains(final Vector3D p) {
return FastMath.abs(getOffset(p)) < tolerance;
}
/** Get the offset (oriented distance) of a parallel plane.
* <p>This method should be called only for parallel planes otherwise
* the result is not meaningful.</p>
* <p>The offset is 0 if both planes are the same, it is
* positive if the plane is on the plus side of the instance and
* negative if it is on the minus side, according to its natural
* orientation.</p>
* @param plane plane to check
* @return offset of the plane
*/
public double getOffset(final Plane plane) {
return originOffset + (sameOrientationAs(plane) ? -plane.originOffset : plane.originOffset);
}
/** Get the offset (oriented distance) of a vector.
* @param vector vector to check
* @return offset of the vector
*/
public double getOffset(Vector<Euclidean3D> vector) {
return getOffset((Point<Euclidean3D>) vector);
}
/** Get the offset (oriented distance) of a point.
* <p>The offset is 0 if the point is on the underlying hyperplane,
* it is positive if the point is on one particular side of the
* hyperplane, and it is negative if the point is on the other side,
* according to the hyperplane natural orientation.</p>
* @param point point to check
* @return offset of the point
*/
public double getOffset(final Point<Euclidean3D> point) {
return ((Vector3D) point).dotProduct(w) + originOffset;
}
/** Check if the instance has the same orientation as another hyperplane.
* @param other other hyperplane to check against the instance
* @return true if the instance and the other hyperplane have
* the same orientation
*/
public boolean sameOrientationAs(final Hyperplane<Euclidean3D> other) {
return (((Plane) other).w).dotProduct(w) > 0.0;
}
}