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The Apache Commons Math project is a library of lightweight, self-contained mathematics and statistics components addressing the most common practical problems not immediately available in the Java programming language or commons-lang.
/*
* 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.complex;
import java.io.Serializable;
import org.apache.commons.math3.util.FastMath;
import org.apache.commons.math3.util.MathUtils;
import org.apache.commons.math3.util.Precision;
import org.apache.commons.math3.exception.DimensionMismatchException;
import org.apache.commons.math3.exception.ZeroException;
import org.apache.commons.math3.exception.util.LocalizedFormats;
/**
* This class implements
* quaternions (Hamilton's hypercomplex numbers).
*
* Instance of this class are guaranteed to be immutable.
*
* @since 3.1
*/
public final class Quaternion implements Serializable {
/** Identity quaternion. */
public static final Quaternion IDENTITY = new Quaternion(1, 0, 0, 0);
/** Zero quaternion. */
public static final Quaternion ZERO = new Quaternion(0, 0, 0, 0);
/** i */
public static final Quaternion I = new Quaternion(0, 1, 0, 0);
/** j */
public static final Quaternion J = new Quaternion(0, 0, 1, 0);
/** k */
public static final Quaternion K = new Quaternion(0, 0, 0, 1);
/** Serializable version identifier. */
private static final long serialVersionUID = 20092012L;
/** First component (scalar part). */
private final double q0;
/** Second component (first vector part). */
private final double q1;
/** Third component (second vector part). */
private final double q2;
/** Fourth component (third vector part). */
private final double q3;
/**
* Builds a quaternion from its components.
*
* @param a Scalar component.
* @param b First vector component.
* @param c Second vector component.
* @param d Third vector component.
*/
public Quaternion(final double a,
final double b,
final double c,
final double d) {
this.q0 = a;
this.q1 = b;
this.q2 = c;
this.q3 = d;
}
/**
* Builds a quaternion from scalar and vector parts.
*
* @param scalar Scalar part of the quaternion.
* @param v Components of the vector part of the quaternion.
*
* @throws DimensionMismatchException if the array length is not 3.
*/
public Quaternion(final double scalar,
final double[] v)
throws DimensionMismatchException {
if (v.length != 3) {
throw new DimensionMismatchException(v.length, 3);
}
this.q0 = scalar;
this.q1 = v[0];
this.q2 = v[1];
this.q3 = v[2];
}
/**
* Builds a pure quaternion from a vector (assuming that the scalar
* part is zero).
*
* @param v Components of the vector part of the pure quaternion.
*/
public Quaternion(final double[] v) {
this(0, v);
}
/**
* Returns the conjugate quaternion of the instance.
*
* @return the conjugate quaternion
*/
public Quaternion getConjugate() {
return new Quaternion(q0, -q1, -q2, -q3);
}
/**
* Returns the Hamilton product of two quaternions.
*
* @param q1 First quaternion.
* @param q2 Second quaternion.
* @return the product {@code q1} and {@code q2}, in that order.
*/
public static Quaternion multiply(final Quaternion q1, final Quaternion q2) {
// Components of the first quaternion.
final double q1a = q1.getQ0();
final double q1b = q1.getQ1();
final double q1c = q1.getQ2();
final double q1d = q1.getQ3();
// Components of the second quaternion.
final double q2a = q2.getQ0();
final double q2b = q2.getQ1();
final double q2c = q2.getQ2();
final double q2d = q2.getQ3();
// Components of the product.
final double w = q1a * q2a - q1b * q2b - q1c * q2c - q1d * q2d;
final double x = q1a * q2b + q1b * q2a + q1c * q2d - q1d * q2c;
final double y = q1a * q2c - q1b * q2d + q1c * q2a + q1d * q2b;
final double z = q1a * q2d + q1b * q2c - q1c * q2b + q1d * q2a;
return new Quaternion(w, x, y, z);
}
/**
* Returns the Hamilton product of the instance by a quaternion.
*
* @param q Quaternion.
* @return the product of this instance with {@code q}, in that order.
*/
public Quaternion multiply(final Quaternion q) {
return multiply(this, q);
}
/**
* Computes the sum of two quaternions.
*
* @param q1 Quaternion.
* @param q2 Quaternion.
* @return the sum of {@code q1} and {@code q2}.
*/
public static Quaternion add(final Quaternion q1,
final Quaternion q2) {
return new Quaternion(q1.getQ0() + q2.getQ0(),
q1.getQ1() + q2.getQ1(),
q1.getQ2() + q2.getQ2(),
q1.getQ3() + q2.getQ3());
}
/**
* Computes the sum of the instance and another quaternion.
*
* @param q Quaternion.
* @return the sum of this instance and {@code q}
*/
public Quaternion add(final Quaternion q) {
return add(this, q);
}
/**
* Subtracts two quaternions.
*
* @param q1 First Quaternion.
* @param q2 Second quaternion.
* @return the difference between {@code q1} and {@code q2}.
*/
public static Quaternion subtract(final Quaternion q1,
final Quaternion q2) {
return new Quaternion(q1.getQ0() - q2.getQ0(),
q1.getQ1() - q2.getQ1(),
q1.getQ2() - q2.getQ2(),
q1.getQ3() - q2.getQ3());
}
/**
* Subtracts a quaternion from the instance.
*
* @param q Quaternion.
* @return the difference between this instance and {@code q}.
*/
public Quaternion subtract(final Quaternion q) {
return subtract(this, q);
}
/**
* Computes the dot-product of two quaternions.
*
* @param q1 Quaternion.
* @param q2 Quaternion.
* @return the dot product of {@code q1} and {@code q2}.
*/
public static double dotProduct(final Quaternion q1,
final Quaternion q2) {
return q1.getQ0() * q2.getQ0() +
q1.getQ1() * q2.getQ1() +
q1.getQ2() * q2.getQ2() +
q1.getQ3() * q2.getQ3();
}
/**
* Computes the dot-product of the instance by a quaternion.
*
* @param q Quaternion.
* @return the dot product of this instance and {@code q}.
*/
public double dotProduct(final Quaternion q) {
return dotProduct(this, q);
}
/**
* Computes the norm of the quaternion.
*
* @return the norm.
*/
public double getNorm() {
return FastMath.sqrt(q0 * q0 +
q1 * q1 +
q2 * q2 +
q3 * q3);
}
/**
* Computes the normalized quaternion (the versor of the instance).
* The norm of the quaternion must not be zero.
*
* @return a normalized quaternion.
* @throws ZeroException if the norm of the quaternion is zero.
*/
public Quaternion normalize() {
final double norm = getNorm();
if (norm < Precision.SAFE_MIN) {
throw new ZeroException(LocalizedFormats.NORM, norm);
}
return new Quaternion(q0 / norm,
q1 / norm,
q2 / norm,
q3 / norm);
}
/**
* {@inheritDoc}
*/
@Override
public boolean equals(Object other) {
if (this == other) {
return true;
}
if (other instanceof Quaternion) {
final Quaternion q = (Quaternion) other;
return q0 == q.getQ0() &&
q1 == q.getQ1() &&
q2 == q.getQ2() &&
q3 == q.getQ3();
}
return false;
}
/**
* {@inheritDoc}
*/
@Override
public int hashCode() {
// "Effective Java" (second edition, p. 47).
int result = 17;
for (double comp : new double[] { q0, q1, q2, q3 }) {
final int c = MathUtils.hash(comp);
result = 31 * result + c;
}
return result;
}
/**
* Checks whether this instance is equal to another quaternion
* within a given tolerance.
*
* @param q Quaternion with which to compare the current quaternion.
* @param eps Tolerance.
* @return {@code true} if the each of the components are equal
* within the allowed absolute error.
*/
public boolean equals(final Quaternion q,
final double eps) {
return Precision.equals(q0, q.getQ0(), eps) &&
Precision.equals(q1, q.getQ1(), eps) &&
Precision.equals(q2, q.getQ2(), eps) &&
Precision.equals(q3, q.getQ3(), eps);
}
/**
* Checks whether the instance is a unit quaternion within a given
* tolerance.
*
* @param eps Tolerance (absolute error).
* @return {@code true} if the norm is 1 within the given tolerance,
* {@code false} otherwise
*/
public boolean isUnitQuaternion(double eps) {
return Precision.equals(getNorm(), 1d, eps);
}
/**
* Checks whether the instance is a pure quaternion within a given
* tolerance.
*
* @param eps Tolerance (absolute error).
* @return {@code true} if the scalar part of the quaternion is zero.
*/
public boolean isPureQuaternion(double eps) {
return FastMath.abs(getQ0()) <= eps;
}
/**
* Returns the polar form of the quaternion.
*
* @return the unit quaternion with positive scalar part.
*/
public Quaternion getPositivePolarForm() {
if (getQ0() < 0) {
final Quaternion unitQ = normalize();
// The quaternion of rotation (normalized quaternion) q and -q
// are equivalent (i.e. represent the same rotation).
return new Quaternion(-unitQ.getQ0(),
-unitQ.getQ1(),
-unitQ.getQ2(),
-unitQ.getQ3());
} else {
return this.normalize();
}
}
/**
* Returns the inverse of this instance.
* The norm of the quaternion must not be zero.
*
* @return the inverse.
* @throws ZeroException if the norm (squared) of the quaternion is zero.
*/
public Quaternion getInverse() {
final double squareNorm = q0 * q0 + q1 * q1 + q2 * q2 + q3 * q3;
if (squareNorm < Precision.SAFE_MIN) {
throw new ZeroException(LocalizedFormats.NORM, squareNorm);
}
return new Quaternion(q0 / squareNorm,
-q1 / squareNorm,
-q2 / squareNorm,
-q3 / squareNorm);
}
/**
* Gets the first component of the quaternion (scalar part).
*
* @return the scalar part.
*/
public double getQ0() {
return q0;
}
/**
* Gets the second component of the quaternion (first component
* of the vector part).
*
* @return the first component of the vector part.
*/
public double getQ1() {
return q1;
}
/**
* Gets the third component of the quaternion (second component
* of the vector part).
*
* @return the second component of the vector part.
*/
public double getQ2() {
return q2;
}
/**
* Gets the fourth component of the quaternion (third component
* of the vector part).
*
* @return the third component of the vector part.
*/
public double getQ3() {
return q3;
}
/**
* Gets the scalar part of the quaternion.
*
* @return the scalar part.
* @see #getQ0()
*/
public double getScalarPart() {
return getQ0();
}
/**
* Gets the three components of the vector part of the quaternion.
*
* @return the vector part.
* @see #getQ1()
* @see #getQ2()
* @see #getQ3()
*/
public double[] getVectorPart() {
return new double[] { getQ1(), getQ2(), getQ3() };
}
/**
* Multiplies the instance by a scalar.
*
* @param alpha Scalar factor.
* @return a scaled quaternion.
*/
public Quaternion multiply(final double alpha) {
return new Quaternion(alpha * q0,
alpha * q1,
alpha * q2,
alpha * q3);
}
/**
* {@inheritDoc}
*/
@Override
public String toString() {
final String sp = " ";
final StringBuilder s = new StringBuilder();
s.append("[")
.append(q0).append(sp)
.append(q1).append(sp)
.append(q2).append(sp)
.append(q3)
.append("]");
return s.toString();
}
}