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* 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.math.geometry;
import java.io.Serializable;
import org.apache.commons.math.exception.MathArithmeticException;
import org.apache.commons.math.exception.util.LocalizedFormats;
import org.apache.commons.math.util.MathUtils;
import org.apache.commons.math.util.FastMath;
/**
* This class implements vectors in a three-dimensional space.
* Instance of this class are guaranteed to be immutable.
* @version $Revision$ $Date$
* @since 1.2
*/
public class Vector3D implements Serializable {
/** Null vector (coordinates: 0, 0, 0). */
public static final Vector3D ZERO = new Vector3D(0, 0, 0);
/** First canonical vector (coordinates: 1, 0, 0). */
public static final Vector3D PLUS_I = new Vector3D(1, 0, 0);
/** Opposite of the first canonical vector (coordinates: -1, 0, 0). */
public static final Vector3D MINUS_I = new Vector3D(-1, 0, 0);
/** Second canonical vector (coordinates: 0, 1, 0). */
public static final Vector3D PLUS_J = new Vector3D(0, 1, 0);
/** Opposite of the second canonical vector (coordinates: 0, -1, 0). */
public static final Vector3D MINUS_J = new Vector3D(0, -1, 0);
/** Third canonical vector (coordinates: 0, 0, 1). */
public static final Vector3D PLUS_K = new Vector3D(0, 0, 1);
/** Opposite of the third canonical vector (coordinates: 0, 0, -1). */
public static final Vector3D MINUS_K = new Vector3D(0, 0, -1);
// CHECKSTYLE: stop ConstantName
/** A vector with all coordinates set to NaN. */
public static final Vector3D NaN = new Vector3D(Double.NaN, Double.NaN, Double.NaN);
// CHECKSTYLE: resume ConstantName
/** A vector with all coordinates set to positive infinity. */
public static final Vector3D POSITIVE_INFINITY =
new Vector3D(Double.POSITIVE_INFINITY, Double.POSITIVE_INFINITY, Double.POSITIVE_INFINITY);
/** A vector with all coordinates set to negative infinity. */
public static final Vector3D NEGATIVE_INFINITY =
new Vector3D(Double.NEGATIVE_INFINITY, Double.NEGATIVE_INFINITY, Double.NEGATIVE_INFINITY);
/** Default format. */
private static final Vector3DFormat DEFAULT_FORMAT =
Vector3DFormat.getInstance();
/** Serializable version identifier. */
private static final long serialVersionUID = 5133268763396045979L;
/** Abscissa. */
private final double x;
/** Ordinate. */
private final double y;
/** Height. */
private final double z;
/** Simple constructor.
* Build a vector from its coordinates
* @param x abscissa
* @param y ordinate
* @param z height
* @see #getX()
* @see #getY()
* @see #getZ()
*/
public Vector3D(double x, double y, double z) {
this.x = x;
this.y = y;
this.z = z;
}
/** Simple constructor.
* Build a vector from its azimuthal coordinates
* @param alpha azimuth (α) around Z
* (0 is +X, π/2 is +Y, π is -X and 3π/2 is -Y)
* @param delta elevation (δ) above (XY) plane, from -π/2 to +π/2
* @see #getAlpha()
* @see #getDelta()
*/
public Vector3D(double alpha, double delta) {
double cosDelta = FastMath.cos(delta);
this.x = FastMath.cos(alpha) * cosDelta;
this.y = FastMath.sin(alpha) * cosDelta;
this.z = FastMath.sin(delta);
}
/** Multiplicative constructor
* Build a vector from another one and a scale factor.
* The vector built will be a * u
* @param a scale factor
* @param u base (unscaled) vector
*/
public Vector3D(double a, Vector3D u) {
this.x = a * u.x;
this.y = a * u.y;
this.z = a * u.z;
}
/** Linear constructor
* Build a vector from two other ones and corresponding scale factors.
* The vector built will be a1 * u1 + a2 * u2
* @param a1 first scale factor
* @param u1 first base (unscaled) vector
* @param a2 second scale factor
* @param u2 second base (unscaled) vector
*/
public Vector3D(double a1, Vector3D u1, double a2, Vector3D u2) {
this.x = a1 * u1.x + a2 * u2.x;
this.y = a1 * u1.y + a2 * u2.y;
this.z = a1 * u1.z + a2 * u2.z;
}
/** Linear constructor
* Build a vector from three other ones and corresponding scale factors.
* The vector built will be a1 * u1 + a2 * u2 + a3 * u3
* @param a1 first scale factor
* @param u1 first base (unscaled) vector
* @param a2 second scale factor
* @param u2 second base (unscaled) vector
* @param a3 third scale factor
* @param u3 third base (unscaled) vector
*/
public Vector3D(double a1, Vector3D u1, double a2, Vector3D u2,
double a3, Vector3D u3) {
this.x = a1 * u1.x + a2 * u2.x + a3 * u3.x;
this.y = a1 * u1.y + a2 * u2.y + a3 * u3.y;
this.z = a1 * u1.z + a2 * u2.z + a3 * u3.z;
}
/** Linear constructor
* Build a vector from four other ones and corresponding scale factors.
* The vector built will be a1 * u1 + a2 * u2 + a3 * u3 + a4 * u4
* @param a1 first scale factor
* @param u1 first base (unscaled) vector
* @param a2 second scale factor
* @param u2 second base (unscaled) vector
* @param a3 third scale factor
* @param u3 third base (unscaled) vector
* @param a4 fourth scale factor
* @param u4 fourth base (unscaled) vector
*/
public Vector3D(double a1, Vector3D u1, double a2, Vector3D u2,
double a3, Vector3D u3, double a4, Vector3D u4) {
this.x = a1 * u1.x + a2 * u2.x + a3 * u3.x + a4 * u4.x;
this.y = a1 * u1.y + a2 * u2.y + a3 * u3.y + a4 * u4.y;
this.z = a1 * u1.z + a2 * u2.z + a3 * u3.z + a4 * u4.z;
}
/** Get the abscissa of the vector.
* @return abscissa of the vector
* @see #Vector3D(double, double, double)
*/
public double getX() {
return x;
}
/** Get the ordinate of the vector.
* @return ordinate of the vector
* @see #Vector3D(double, double, double)
*/
public double getY() {
return y;
}
/** Get the height of the vector.
* @return height of the vector
* @see #Vector3D(double, double, double)
*/
public double getZ() {
return z;
}
/** Get the L1 norm for the vector.
* @return L1 norm for the vector
*/
public double getNorm1() {
return FastMath.abs(x) + FastMath.abs(y) + FastMath.abs(z);
}
/** Get the L2 norm for the vector.
* @return euclidian norm for the vector
*/
public double getNorm() {
return FastMath.sqrt (x * x + y * y + z * z);
}
/** Get the square of the norm for the vector.
* @return square of the euclidian norm for the vector
*/
public double getNormSq() {
return x * x + y * y + z * z;
}
/** Get the L∞ norm for the vector.
* @return L∞ norm for the vector
*/
public double getNormInf() {
return FastMath.max(FastMath.max(FastMath.abs(x), FastMath.abs(y)), FastMath.abs(z));
}
/** Get the azimuth of the vector.
* @return azimuth (α) of the vector, between -π and +π
* @see #Vector3D(double, double)
*/
public double getAlpha() {
return FastMath.atan2(y, x);
}
/** Get the elevation of the vector.
* @return elevation (δ) of the vector, between -π/2 and +π/2
* @see #Vector3D(double, double)
*/
public double getDelta() {
return FastMath.asin(z / getNorm());
}
/** Add a vector to the instance.
* @param v vector to add
* @return a new vector
*/
public Vector3D add(Vector3D v) {
return new Vector3D(x + v.x, y + v.y, z + v.z);
}
/** Add a scaled vector to the instance.
* @param factor scale factor to apply to v before adding it
* @param v vector to add
* @return a new vector
*/
public Vector3D add(double factor, Vector3D v) {
return new Vector3D(x + factor * v.x, y + factor * v.y, z + factor * v.z);
}
/** Subtract a vector from the instance.
* @param v vector to subtract
* @return a new vector
*/
public Vector3D subtract(Vector3D v) {
return new Vector3D(x - v.x, y - v.y, z - v.z);
}
/** Subtract a scaled vector from the instance.
* @param factor scale factor to apply to v before subtracting it
* @param v vector to subtract
* @return a new vector
*/
public Vector3D subtract(double factor, Vector3D v) {
return new Vector3D(x - factor * v.x, y - factor * v.y, z - factor * v.z);
}
/** Get a normalized vector aligned with the instance.
* @return a new normalized vector
* @exception ArithmeticException if the norm is zero
*/
public Vector3D normalize() {
double s = getNorm();
if (s == 0) {
throw new MathArithmeticException(LocalizedFormats.CANNOT_NORMALIZE_A_ZERO_NORM_VECTOR);
}
return scalarMultiply(1 / s);
}
/** Get a vector orthogonal to the instance.
* There are an infinite number of normalized vectors orthogonal
* to the instance. This method picks up one of them almost
* arbitrarily. It is useful when one needs to compute a reference
* frame with one of the axes in a predefined direction. The
* following example shows how to build a frame having the k axis
* aligned with the known vector u :
*
* Vector3D k = u.normalize();
* Vector3D i = k.orthogonal();
* Vector3D j = Vector3D.crossProduct(k, i);
*
* @return a new normalized vector orthogonal to the instance
* @exception ArithmeticException if the norm of the instance is null
*/
public Vector3D orthogonal() {
double threshold = 0.6 * getNorm();
if (threshold == 0) {
throw new MathArithmeticException(LocalizedFormats.ZERO_NORM);
}
if ((x >= -threshold) && (x <= threshold)) {
double inverse = 1 / FastMath.sqrt(y * y + z * z);
return new Vector3D(0, inverse * z, -inverse * y);
} else if ((y >= -threshold) && (y <= threshold)) {
double inverse = 1 / FastMath.sqrt(x * x + z * z);
return new Vector3D(-inverse * z, 0, inverse * x);
}
double inverse = 1 / FastMath.sqrt(x * x + y * y);
return new Vector3D(inverse * y, -inverse * x, 0);
}
/** Compute the angular separation between two vectors.
* This method computes the angular separation between two
* vectors using the dot product for well separated vectors and the
* cross product for almost aligned vectors. This allows to have a
* good accuracy in all cases, even for vectors very close to each
* other.
* @param v1 first vector
* @param v2 second vector
* @return angular separation between v1 and v2
* @exception ArithmeticException if either vector has a null norm
*/
public static double angle(Vector3D v1, Vector3D v2) {
double normProduct = v1.getNorm() * v2.getNorm();
if (normProduct == 0) {
throw new MathArithmeticException(LocalizedFormats.ZERO_NORM);
}
double dot = dotProduct(v1, v2);
double threshold = normProduct * 0.9999;
if ((dot < -threshold) || (dot > threshold)) {
// the vectors are almost aligned, compute using the sine
Vector3D v3 = crossProduct(v1, v2);
if (dot >= 0) {
return FastMath.asin(v3.getNorm() / normProduct);
}
return FastMath.PI - FastMath.asin(v3.getNorm() / normProduct);
}
// the vectors are sufficiently separated to use the cosine
return FastMath.acos(dot / normProduct);
}
/** Get the opposite of the instance.
* @return a new vector which is opposite to the instance
*/
public Vector3D negate() {
return new Vector3D(-x, -y, -z);
}
/** Multiply the instance by a scalar
* @param a scalar
* @return a new vector
*/
public Vector3D scalarMultiply(double a) {
return new Vector3D(a * x, a * y, a * z);
}
/**
* Returns true if any coordinate of this vector is NaN; false otherwise
* @return true if any coordinate of this vector is NaN; false otherwise
*/
public boolean isNaN() {
return Double.isNaN(x) || Double.isNaN(y) || Double.isNaN(z);
}
/**
* Returns true if any coordinate of this vector is infinite and none are NaN;
* false otherwise
* @return true if any coordinate of this vector is infinite and none are NaN;
* false otherwise
*/
public boolean isInfinite() {
return !isNaN() && (Double.isInfinite(x) || Double.isInfinite(y) || Double.isInfinite(z));
}
/**
* Test for the equality of two 3D vectors.
*
* If all coordinates of two 3D vectors are exactly the same, and none are
* Double.NaN, the two 3D vectors are considered to be equal.
*
*
* NaN coordinates are considered to affect globally the vector
* and be equals to each other - i.e, if either (or all) coordinates of the
* 3D vector are equal to Double.NaN, the 3D vector is equal to
* {@link #NaN}.
*
*
* @param other Object to test for equality to this
* @return true if two 3D vector objects are equal, false if
* object is null, not an instance of Vector3D, or
* not equal to this Vector3D instance
*
*/
@Override
public boolean equals(Object other) {
if (this == other) {
return true;
}
if (other instanceof Vector3D) {
final Vector3D rhs = (Vector3D)other;
if (rhs.isNaN()) {
return this.isNaN();
}
return (x == rhs.x) && (y == rhs.y) && (z == rhs.z);
}
return false;
}
/**
* Get a hashCode for the 3D vector.
*
* All NaN values have the same hash code.
*
* @return a hash code value for this object
*/
@Override
public int hashCode() {
if (isNaN()) {
return 8;
}
return 31 * (23 * MathUtils.hash(x) + 19 * MathUtils.hash(y) + MathUtils.hash(z));
}
/** Compute the dot-product of two vectors.
* @param v1 first vector
* @param v2 second vector
* @return the dot product v1.v2
*/
public static double dotProduct(Vector3D v1, Vector3D v2) {
return v1.x * v2.x + v1.y * v2.y + v1.z * v2.z;
}
/** Compute the cross-product of two vectors.
* @param v1 first vector
* @param v2 second vector
* @return the cross product v1 ^ v2 as a new Vector
*/
public static Vector3D crossProduct(final Vector3D v1, final Vector3D v2) {
// rescale both vectors without losing precision,
// to ensure their norm are the same order of magnitude
// we reduce cancellation errors by preconditioning,
// we replace v1 by v3 = v1 - rho v2 with rho chosen in order to compute
// v3 without loss of precision. See Kahan lecture
// "Computing Cross-Products and Rotations in 2- and 3-Dimensional Euclidean Spaces"
// available at http://www.cs.berkeley.edu/~wkahan/MathH110/Cross.pdf
// compute rho as an 8 bits approximation of v1.v2 / v2.v2
// compute cross product from v3 and v2 instead of v1 and v2
return new Vector3D(v1.y * v2.z - v1.z * v2.y, v1.z * v2.x - v1.x * v2.z, v1.x * v2.y - v1.y * v2.x);
}
/** Compute the distance between two vectors according to the L1 norm.
* Calling this method is equivalent to calling:
* v1.subtract(v2).getNorm1() except that no intermediate
* vector is built
* @param v1 first vector
* @param v2 second vector
* @return the distance between v1 and v2 according to the L1 norm
*/
public static double distance1(Vector3D v1, Vector3D v2) {
final double dx = FastMath.abs(v2.x - v1.x);
final double dy = FastMath.abs(v2.y - v1.y);
final double dz = FastMath.abs(v2.z - v1.z);
return dx + dy + dz;
}
/** Compute the distance between two vectors according to the L2 norm.
* Calling this method is equivalent to calling:
* v1.subtract(v2).getNorm() except that no intermediate
* vector is built
* @param v1 first vector
* @param v2 second vector
* @return the distance between v1 and v2 according to the L2 norm
*/
public static double distance(Vector3D v1, Vector3D v2) {
final double dx = v2.x - v1.x;
final double dy = v2.y - v1.y;
final double dz = v2.z - v1.z;
return FastMath.sqrt(dx * dx + dy * dy + dz * dz);
}
/** Compute the distance between two vectors according to the L∞ norm.
* Calling this method is equivalent to calling:
* v1.subtract(v2).getNormInf() except that no intermediate
* vector is built
* @param v1 first vector
* @param v2 second vector
* @return the distance between v1 and v2 according to the L∞ norm
*/
public static double distanceInf(Vector3D v1, Vector3D v2) {
final double dx = FastMath.abs(v2.x - v1.x);
final double dy = FastMath.abs(v2.y - v1.y);
final double dz = FastMath.abs(v2.z - v1.z);
return FastMath.max(FastMath.max(dx, dy), dz);
}
/** Compute the square of the distance between two vectors.
* Calling this method is equivalent to calling:
* v1.subtract(v2).getNormSq() except that no intermediate
* vector is built
* @param v1 first vector
* @param v2 second vector
* @return the square of the distance between v1 and v2
*/
public static double distanceSq(Vector3D v1, Vector3D v2) {
final double dx = v2.x - v1.x;
final double dy = v2.y - v1.y;
final double dz = v2.z - v1.z;
return dx * dx + dy * dy + dz * dz;
}
/** Get a string representation of this vector.
* @return a string representation of this vector
*/
@Override
public String toString() {
return DEFAULT_FORMAT.format(this);
}
}