# Rotations in a three-dimensional space : Matrix « 2D Graphics GUI « Java

Rotations in a three-dimensional space

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* Licensed to the Apache Software Foundation (ASF) under one or more
* contributor license agreements.  See the NOTICE file distributed with
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* (the "License"); you may not use this file except in compliance with
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*
*
* Unless required by applicable law or agreed to in writing, software
* WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.
* See the License for the specific language governing permissions and
*/

import java.io.Serializable;

/**
* This class implements rotations in a three-dimensional space.
*
* <p>Rotations can be represented by several different mathematical
* entities (matrices, axe and angle, Cardan or Euler angles,
* quaternions). This class presents an higher level abstraction, more
* user-oriented and hiding this implementation details. Well, for the
* curious, we use quaternions for the internal representation. The
* user can build a rotation from any of these representations, and
* <code>Rotation</code> instance (see the various constructors and
* getters). In addition, a rotation can also be built implicitely
* from a set of vectors and their image.</p>
* <p>This implies that this class can be used to convert from one
* representation to another one. For example, converting a rotation
* matrix into a set of Cardan angles from can be done using the
* followong single line of code:</p>
* <pre>
* double[] angles = new Rotation(matrix, 1.0e-10).getAngles(RotationOrder.XYZ);
* </pre>
* <p>Focus is oriented on what a rotation <em>do</em> rather than on its
* underlying representation. Once it has been built, and regardless of its
* internal representation, a rotation is an <em>operator</em> which basically
* transforms three dimensional {@link Vector3D vectors} into other three
* dimensional {@link Vector3D vectors}. Depending on the application, the
* meaning of these vectors may vary and the semantics of the rotation also.</p>
* <p>For example in an spacecraft attitude simulation tool, users will often
* consider the vectors are fixed (say the Earth direction for example) and the
* rotation transforms the coordinates coordinates of this vector in inertial
* frame into the coordinates of the same vector in satellite frame. In this
* case, the rotation implicitely defines the relation between the two frames.
* Another example could be a telescope control application, where the rotation
* would transform the sighting direction at rest into the desired observing
* direction when the telescope is pointed towards an object of interest. In this
* case the rotation transforms the directionf at rest in a topocentric frame
* into the sighting direction in the same topocentric frame. In many case, both
* approaches will be combined, in our telescope example, we will probably also
* need to transform the observing direction in the topocentric frame into the
* observing direction in inertial frame taking into account the observatory
* location and the Earth rotation.</p>
*
* <p>These examples show that a rotation is what the user wants it to be, so this
* class does not push the user towards one specific definition and hence does not
* provide methods like <code>projectVectorIntoDestinationFrame</code> or
* <code>computeTransformedDirection</code>. It provides simpler and more generic
* #applyInverseTo(Vector3D) applyInverseTo(Vector3D)}.</p>
*
* <p>Since a rotation is basically a vectorial operator, several rotations can be
* composed together and the composite operation <code>r = r<sub>1</sub> o
* r<sub>2</sub></code> (which means that for each vector <code>u</code>,
* <code>r(u) = r<sub>1</sub>(r<sub>2</sub>(u))</code>) is also a rotation. Hence
* we can consider that in addition to vectors, a rotation can be applied to other
* rotations as well (or to itself). With our previous notations, we would say we
* can apply <code>r<sub>1</sub></code> to <code>r<sub>2</sub></code> and the result
* we get is <code>r = r<sub>1</sub> o r<sub>2</sub></code>. For this purpose, the
* class provides the methods: {@link #applyTo(Rotation) applyTo(Rotation)} and
*
* <p>Rotations are guaranteed to be immutable objects.</p>
*
* @version \$Revision: 627994 \$ \$Date: 2008-02-15 03:16:05 -0700 (Fri, 15 Feb 2008) \$
* @see Vector3D
* @see RotationOrder
* @since 1.2
*/

public class Rotation implements Serializable {

/** Build the identity rotation.
*/
public Rotation() {
q0 = 1;
q1 = 0;
q2 = 0;
q3 = 0;
}

/** Build a rotation from the quaternion coordinates.
* <p>A rotation can be built from a <em>normalized</em> quaternion,
* i.e. a quaternion for which q<sub>0</sub><sup>2</sup> +
* q<sub>1</sub><sup>2</sup> + q<sub>2</sub><sup>2</sup> +
* q<sub>3</sub><sup>2</sup> = 1. If the quaternion is not normalized,
* the constructor can normalize it in a preprocessing step.</p>
* @param q0 scalar part of the quaternion
* @param q1 first coordinate of the vectorial part of the quaternion
* @param q2 second coordinate of the vectorial part of the quaternion
* @param q3 third coordinate of the vectorial part of the quaternion
* @param needsNormalization if true, the coordinates are considered
* not to be normalized, a normalization preprocessing step is performed
* before using them
*/
public Rotation(double q0, double q1, double q2, double q3,
boolean needsNormalization) {

if (needsNormalization) {
// normalization preprocessing
double inv = 1.0 / Math.sqrt(q0 * q0 + q1 * q1 + q2 * q2 + q3 * q3);
q0 *= inv;
q1 *= inv;
q2 *= inv;
q3 *= inv;
}

this.q0 = q0;
this.q1 = q1;
this.q2 = q2;
this.q3 = q3;

}

/** Build a rotation from an axis and an angle.
* <p>We use the convention that angles are oriented according to
* the effect of the rotation on vectors around the axis. That means
* that if (i, j, k) is a direct frame and if we first provide +k as
* the axis and PI/2 as the angle to this constructor, and then
* {@link #applyTo(Vector3D) apply} the instance to +i, we will get
* +j.</p>
* @param axis axis around which to rotate
* @param angle rotation angle.
* @exception ArithmeticException if the axis norm is zero
*/
public Rotation(Vector3D axis, double angle) {

double norm = axis.getNorm();
if (norm == 0) {
throw new ArithmeticException("zero norm for rotation axis");
}

double halfAngle = -0.5 * angle;
double coeff = Math.sin(halfAngle) / norm;

q0 = Math.cos (halfAngle);
q1 = coeff * axis.getX();
q2 = coeff * axis.getY();
q3 = coeff * axis.getZ();

}

/** Build a rotation from a 3X3 matrix.

* <p>Rotation matrices are orthogonal matrices, i.e. unit matrices
* (which are matrices for which m.m<sup>T</sup> = I) with real
* coefficients. The module of the determinant of unit matrices is
* 1, among the orthogonal 3X3 matrices, only the ones having a
* positive determinant (+1) are rotation matrices.</p>

* <p>When a rotation is defined by a matrix with truncated values
* (typically when it is extracted from a technical sheet where only
* four to five significant digits are available), the matrix is not
* orthogonal anymore. This constructor handles this case
* transparently by using a copy of the given matrix and applying a
* correction to the copy in order to perfect its orthogonality. If
* the Frobenius norm of the correction needed is above the given
* threshold, then the matrix is considered to be too far from a
* true rotation matrix and an exception is thrown.<p>

* @param m rotation matrix
* @param threshold convergence threshold for the iterative
* orthogonality correction (convergence is reached when the
* difference between two steps of the Frobenius norm of the
* correction is below this threshold)

* @exception NotARotationMatrixException if the matrix is not a 3X3
* matrix, or if it cannot be transformed into an orthogonal matrix
* with the given threshold, or if the determinant of the resulting
* orthogonal matrix is negative

*/
public Rotation(double[][] m, double threshold)
{

// dimension check
if ((m.length != 3) || (m[0].length != 3) ||
(m[1].length != 3) || (m[2].length != 3)) {
System.out.println("a {0}x{1} matrix" +
" cannot be a rotation matrix");
}

// compute a "close" orthogonal matrix
double[][] ort = orthogonalizeMatrix(m, threshold);

// check the sign of the determinant
double det = ort[0][0] * (ort[1][1] * ort[2][2] - ort[2][1] * ort[1][2]) -
ort[1][0] * (ort[0][1] * ort[2][2] - ort[2][1] * ort[0][2]) +
ort[2][0] * (ort[0][1] * ort[1][2] - ort[1][1] * ort[0][2]);
if (det < 0.0) {
System.out.println("the closest orthogonal matrix" +
" has a negative determinant {0}");
}

// There are different ways to compute the quaternions elements
// from the matrix. They all involve computing one element from
// the diagonal of the matrix, and computing the three other ones
// using a formula involving a division by the first element,
// which unfortunately can be zero. Since the norm of the
// quaternion is 1, we know at least one element has an absolute
// value greater or equal to 0.5, so it is always possible to
// select the right formula and avoid division by zero and even
// numerical inaccuracy. Checking the elements in turn and using
// the first one greater than 0.45 is safe (this leads to a simple
// test since qi = 0.45 implies 4 qi^2 - 1 = -0.19)
double s = ort[0][0] + ort[1][1] + ort[2][2];
if (s > -0.19) {
// compute q0 and deduce q1, q2 and q3
q0 = 0.5 * Math.sqrt(s + 1.0);
double inv = 0.25 / q0;
q1 = inv * (ort[1][2] - ort[2][1]);
q2 = inv * (ort[2][0] - ort[0][2]);
q3 = inv * (ort[0][1] - ort[1][0]);
} else {
s = ort[0][0] - ort[1][1] - ort[2][2];
if (s > -0.19) {
// compute q1 and deduce q0, q2 and q3
q1 = 0.5 * Math.sqrt(s + 1.0);
double inv = 0.25 / q1;
q0 = inv * (ort[1][2] - ort[2][1]);
q2 = inv * (ort[0][1] + ort[1][0]);
q3 = inv * (ort[0][2] + ort[2][0]);
} else {
s = ort[1][1] - ort[0][0] - ort[2][2];
if (s > -0.19) {
// compute q2 and deduce q0, q1 and q3
q2 = 0.5 * Math.sqrt(s + 1.0);
double inv = 0.25 / q2;
q0 = inv * (ort[2][0] - ort[0][2]);
q1 = inv * (ort[0][1] + ort[1][0]);
q3 = inv * (ort[2][1] + ort[1][2]);
} else {
// compute q3 and deduce q0, q1 and q2
s = ort[2][2] - ort[0][0] - ort[1][1];
q3 = 0.5 * Math.sqrt(s + 1.0);
double inv = 0.25 / q3;
q0 = inv * (ort[0][1] - ort[1][0]);
q1 = inv * (ort[0][2] + ort[2][0]);
q2 = inv * (ort[2][1] + ort[1][2]);
}
}
}

}

/** Build the rotation that transforms a pair of vector into another pair.

* <p>Except for possible scale factors, if the instance were applied to
* the pair (u<sub>1</sub>, u<sub>2</sub>) it will produce the pair
* (v<sub>1</sub>, v<sub>2</sub>).</p>

* <p>If the angular separation between u<sub>1</sub> and u<sub>2</sub> is
* not the same as the angular separation between v<sub>1</sub> and
* v<sub>2</sub>, then a corrected v'<sub>2</sub> will be used rather than
* v<sub>2</sub>, the corrected vector will be in the (v<sub>1</sub>,
* v<sub>2</sub>) plane.</p>

* @param u1 first vector of the origin pair
* @param u2 second vector of the origin pair
* @param v1 desired image of u1 by the rotation
* @param v2 desired image of u2 by the rotation
* @exception IllegalArgumentException if the norm of one of the vectors is zero
*/
public Rotation(Vector3D u1, Vector3D u2, Vector3D v1, Vector3D v2) {

// norms computation
double u1u1 = Vector3D.dotProduct(u1, u1);
double u2u2 = Vector3D.dotProduct(u2, u2);
double v1v1 = Vector3D.dotProduct(v1, v1);
double v2v2 = Vector3D.dotProduct(v2, v2);
if ((u1u1 == 0) || (u2u2 == 0) || (v1v1 == 0) || (v2v2 == 0)) {
throw new IllegalArgumentException("zero norm for rotation defining vector");
}

double u1x = u1.getX();
double u1y = u1.getY();
double u1z = u1.getZ();

double u2x = u2.getX();
double u2y = u2.getY();
double u2z = u2.getZ();

// normalize v1 in order to have (v1'|v1') = (u1|u1)
double coeff = Math.sqrt (u1u1 / v1v1);
double v1x   = coeff * v1.getX();
double v1y   = coeff * v1.getY();
double v1z   = coeff * v1.getZ();
v1 = new Vector3D(v1x, v1y, v1z);

// adjust v2 in order to have (u1|u2) = (v1|v2) and (v2'|v2') = (u2|u2)
double u1u2   = Vector3D.dotProduct(u1, u2);
double v1v2   = Vector3D.dotProduct(v1, v2);
double coeffU = u1u2 / u1u1;
double coeffV = v1v2 / u1u1;
double beta   = Math.sqrt((u2u2 - u1u2 * coeffU) / (v2v2 - v1v2 * coeffV));
double alpha  = coeffU - beta * coeffV;
double v2x    = alpha * v1x + beta * v2.getX();
double v2y    = alpha * v1y + beta * v2.getY();
double v2z    = alpha * v1z + beta * v2.getZ();
v2 = new Vector3D(v2x, v2y, v2z);

// preliminary computation (we use explicit formulation instead
// of relying on the Vector3D class in order to avoid building lots
// of temporary objects)
Vector3D uRef = u1;
Vector3D vRef = v1;
double dx1 = v1x - u1.getX();
double dy1 = v1y - u1.getY();
double dz1 = v1z - u1.getZ();
double dx2 = v2x - u2.getX();
double dy2 = v2y - u2.getY();
double dz2 = v2z - u2.getZ();
Vector3D k = new Vector3D(dy1 * dz2 - dz1 * dy2,
dz1 * dx2 - dx1 * dz2,
dx1 * dy2 - dy1 * dx2);
double c = k.getX() * (u1y * u2z - u1z * u2y) +
k.getY() * (u1z * u2x - u1x * u2z) +
k.getZ() * (u1x * u2y - u1y * u2x);

if (c == 0) {
// the (q1, q2, q3) vector is in the (u1, u2) plane
// we try other vectors
Vector3D u3 = Vector3D.crossProduct(u1, u2);
Vector3D v3 = Vector3D.crossProduct(v1, v2);
double u3x  = u3.getX();
double u3y  = u3.getY();
double u3z  = u3.getZ();
double v3x  = v3.getX();
double v3y  = v3.getY();
double v3z  = v3.getZ();

double dx3 = v3x - u3x;
double dy3 = v3y - u3y;
double dz3 = v3z - u3z;
k = new Vector3D(dy1 * dz3 - dz1 * dy3,
dz1 * dx3 - dx1 * dz3,
dx1 * dy3 - dy1 * dx3);
c = k.getX() * (u1y * u3z - u1z * u3y) +
k.getY() * (u1z * u3x - u1x * u3z) +
k.getZ() * (u1x * u3y - u1y * u3x);

if (c == 0) {
// the (q1, q2, q3) vector is aligned with u1:
// we try (u2, u3) and (v2, v3)
k = new Vector3D(dy2 * dz3 - dz2 * dy3,
dz2 * dx3 - dx2 * dz3,
dx2 * dy3 - dy2 * dx3);
c = k.getX() * (u2y * u3z - u2z * u3y) +
k.getY() * (u2z * u3x - u2x * u3z) +
k.getZ() * (u2x * u3y - u2y * u3x);

if (c == 0) {
// the (q1, q2, q3) vector is aligned with everything
// this is really the identity rotation
q0 = 1.0;
q1 = 0.0;
q2 = 0.0;
q3 = 0.0;
return;
}

// we will have to use u2 and v2 to compute the scalar part
uRef = u2;
vRef = v2;

}

}

// compute the vectorial part
c = Math.sqrt(c);
double inv = 1.0 / (c + c);
q1 = inv * k.getX();
q2 = inv * k.getY();
q3 = inv * k.getZ();

// compute the scalar part
k = new Vector3D(uRef.getY() * q3 - uRef.getZ() * q2,
uRef.getZ() * q1 - uRef.getX() * q3,
uRef.getX() * q2 - uRef.getY() * q1);
c = Vector3D.dotProduct(k, k);
q0 = Vector3D.dotProduct(vRef, k) / (c + c);

}

/** Build one of the rotations that transform one vector into another one.

* <p>Except for a possible scale factor, if the instance were
* applied to the vector u it will produce the vector v. There is an
* infinite number of such rotations, this constructor choose the
* one with the smallest associated angle (i.e. the one whose axis
* is orthogonal to the (u, v) plane). If u and v are colinear, an
* arbitrary rotation axis is chosen.</p>

* @param u origin vector
* @param v desired image of u by the rotation
* @exception IllegalArgumentException if the norm of one of the vectors is zero
*/
public Rotation(Vector3D u, Vector3D v) {

double normProduct = u.getNorm() * v.getNorm();
if (normProduct == 0) {
throw new IllegalArgumentException("zero norm for rotation defining vector");
}

double dot = Vector3D.dotProduct(u, v);

if (dot < ((2.0e-15 - 1.0) * normProduct)) {
// special case u = -v: we select a PI angle rotation around
// an arbitrary vector orthogonal to u
Vector3D w = u.orthogonal();
q0 = 0.0;
q1 = -w.getX();
q2 = -w.getY();
q3 = -w.getZ();
} else {
// general case: (u, v) defines a plane, we select
// the shortest possible rotation: axis orthogonal to this plane
q0 = Math.sqrt(0.5 * (1.0 + dot / normProduct));
double coeff = 1.0 / (2.0 * q0 * normProduct);
q1 = coeff * (v.getY() * u.getZ() - v.getZ() * u.getY());
q2 = coeff * (v.getZ() * u.getX() - v.getX() * u.getZ());
q3 = coeff * (v.getX() * u.getY() - v.getY() * u.getX());
}

}

/** Build a rotation from three Cardan or Euler elementary rotations.

* <p>Cardan rotations are three successive rotations around the
* canonical axes X, Y and Z, each axis beeing used once. There are
* 6 such sets of rotations (XYZ, XZY, YXZ, YZX, ZXY and ZYX). Euler
* rotations are three successive rotations around the canonical
* axes X, Y and Z, the first and last rotations beeing around the
* same axis. There are 6 such sets of rotations (XYX, XZX, YXY,
* YZY, ZXZ and ZYZ), the most popular one being ZXZ.</p>
* <p>Beware that many people routinely use the term Euler angles even
* for what really are Cardan angles (this confusion is especially
* widespread in the aerospace business where Roll, Pitch and Yaw angles
* are often wrongly tagged as Euler angles).</p>

* @param order order of rotations to use
* @param alpha1 angle of the first elementary rotation
* @param alpha2 angle of the second elementary rotation
* @param alpha3 angle of the third elementary rotation
*/
public Rotation(RotationOrder order,
double alpha1, double alpha2, double alpha3) {
Rotation r1 = new Rotation(order.getA1(), alpha1);
Rotation r2 = new Rotation(order.getA2(), alpha2);
Rotation r3 = new Rotation(order.getA3(), alpha3);
Rotation composed = r1.applyTo(r2.applyTo(r3));
q0 = composed.q0;
q1 = composed.q1;
q2 = composed.q2;
q3 = composed.q3;
}

/** Revert a rotation.
* Build a rotation which reverse the effect of another
* rotation. This means that if r(u) = v, then r.revert(v) = u. The
* instance is not changed.
* @return a new rotation whose effect is the reverse of the effect
* of the instance
*/
public Rotation revert() {
return new Rotation(-q0, q1, q2, q3, false);
}

/** Get the scalar coordinate of the quaternion.
* @return scalar coordinate of the quaternion
*/
public double getQ0() {
return q0;
}

/** Get the first coordinate of the vectorial part of the quaternion.
* @return first coordinate of the vectorial part of the quaternion
*/
public double getQ1() {
return q1;
}

/** Get the second coordinate of the vectorial part of the quaternion.
* @return second coordinate of the vectorial part of the quaternion
*/
public double getQ2() {
return q2;
}

/** Get the third coordinate of the vectorial part of the quaternion.
* @return third coordinate of the vectorial part of the quaternion
*/
public double getQ3() {
return q3;
}

/** Get the normalized axis of the rotation.
* @return normalized axis of the rotation
*/
public Vector3D getAxis() {
double squaredSine = q1 * q1 + q2 * q2 + q3 * q3;
if (squaredSine == 0) {
return new Vector3D(1, 0, 0);
} else if (q0 < 0) {
double inverse = 1 / Math.sqrt(squaredSine);
return new Vector3D(q1 * inverse, q2 * inverse, q3 * inverse);
}
double inverse = -1 / Math.sqrt(squaredSine);
return new Vector3D(q1 * inverse, q2 * inverse, q3 * inverse);
}

/** Get the angle of the rotation.
* @return angle of the rotation (between 0 and &pi;)
*/
public double getAngle() {
if ((q0 < -0.1) || (q0 > 0.1)) {
return 2 * Math.asin(Math.sqrt(q1 * q1 + q2 * q2 + q3 * q3));
} else if (q0 < 0) {
return 2 * Math.acos(-q0);
}
return 2 * Math.acos(q0);
}

/** Get the Cardan or Euler angles corresponding to the instance.

* <p>The equations show that each rotation can be defined by two
* different values of the Cardan or Euler angles set. For example
* if Cardan angles are used, the rotation defined by the angles
* a<sub>1</sub>, a<sub>2</sub> and a<sub>3</sub> is the same as
* the rotation defined by the angles &pi; + a<sub>1</sub>, &pi;
* - a<sub>2</sub> and &pi; + a<sub>3</sub>. This method implements
* the following arbitrary choices:</p>
* <ul>
*   <li>for Cardan angles, the chosen set is the one for which the
*   second angle is between -&pi;/2 and &pi;/2 (i.e its cosine is
*   positive),</li>
*   <li>for Euler angles, the chosen set is the one for which the
*   second angle is between 0 and &pi; (i.e its sine is positive).</li>
* </ul>

* <p>Cardan and Euler angle have a very disappointing drawback: all
* of them have singularities. This means that if the instance is
* too close to the singularities corresponding to the given
* rotation order, it will be impossible to retrieve the angles. For
* Cardan angles, this is often called gimbal lock. There is
* <em>nothing</em> to do to prevent this, it is an intrinsic problem
* with Cardan and Euler representation (but not a problem with the
* rotation itself, which is perfectly well defined). For Cardan
* angles, singularities occur when the second angle is close to
* -&pi;/2 or +&pi;/2, for Euler angle singularities occur when the
* second angle is close to 0 or &pi;, this implies that the identity
* rotation is always singular for Euler angles!</p>

* @param order rotation order to use
* @return an array of three angles, in the order specified by the set
* @exception CardanEulerSingularityException if the rotation is
* singular with respect to the angles set specified
*/
public double[] getAngles(RotationOrder order)
{

if (order == RotationOrder.XYZ) {

// r (Vector3D.plusK) coordinates are :
//  sin (theta), -cos (theta) sin (phi), cos (theta) cos (phi)
// (-r) (Vector3D.plusI) coordinates are :
// cos (psi) cos (theta), -sin (psi) cos (theta), sin (theta)
// and we can choose to have theta in the interval [-PI/2 ; +PI/2]
Vector3D v1 = applyTo(Vector3D.plusK);
Vector3D v2 = applyInverseTo(Vector3D.plusI);
if  ((v2.getZ() < -0.9999999999) || (v2.getZ() > 0.9999999999)) {
System.out.println("CardanEulerSingularityException");
}
return new double[] {
Math.atan2(-(v1.getY()), v1.getZ()),
Math.asin(v2.getZ()),
Math.atan2(-(v2.getY()), v2.getX())
};

} else if (order == RotationOrder.XZY) {

// r (Vector3D.plusJ) coordinates are :
// -sin (psi), cos (psi) cos (phi), cos (psi) sin (phi)
// (-r) (Vector3D.plusI) coordinates are :
// cos (theta) cos (psi), -sin (psi), sin (theta) cos (psi)
// and we can choose to have psi in the interval [-PI/2 ; +PI/2]
Vector3D v1 = applyTo(Vector3D.plusJ);
Vector3D v2 = applyInverseTo(Vector3D.plusI);
if ((v2.getY() < -0.9999999999) || (v2.getY() > 0.9999999999)) {
System.out.println("CardanEulerSingularityException");
}
return new double[] {
Math.atan2(v1.getZ(), v1.getY()),
-Math.asin(v2.getY()),
Math.atan2(v2.getZ(), v2.getX())
};

} else if (order == RotationOrder.YXZ) {

// r (Vector3D.plusK) coordinates are :
//  cos (phi) sin (theta), -sin (phi), cos (phi) cos (theta)
// (-r) (Vector3D.plusJ) coordinates are :
// sin (psi) cos (phi), cos (psi) cos (phi), -sin (phi)
// and we can choose to have phi in the interval [-PI/2 ; +PI/2]
Vector3D v1 = applyTo(Vector3D.plusK);
Vector3D v2 = applyInverseTo(Vector3D.plusJ);
if ((v2.getZ() < -0.9999999999) || (v2.getZ() > 0.9999999999)) {
System.out.println("CardanEulerSingularityException");
}
return new double[] {
Math.atan2(v1.getX(), v1.getZ()),
-Math.asin(v2.getZ()),
Math.atan2(v2.getX(), v2.getY())
};

} else if (order == RotationOrder.YZX) {

// r (Vector3D.plusI) coordinates are :
// cos (psi) cos (theta), sin (psi), -cos (psi) sin (theta)
// (-r) (Vector3D.plusJ) coordinates are :
// sin (psi), cos (phi) cos (psi), -sin (phi) cos (psi)
// and we can choose to have psi in the interval [-PI/2 ; +PI/2]
Vector3D v1 = applyTo(Vector3D.plusI);
Vector3D v2 = applyInverseTo(Vector3D.plusJ);
if ((v2.getX() < -0.9999999999) || (v2.getX() > 0.9999999999)) {
System.out.println("CardanEulerSingularityException");
}
return new double[] {
Math.atan2(-(v1.getZ()), v1.getX()),
Math.asin(v2.getX()),
Math.atan2(-(v2.getZ()), v2.getY())
};

} else if (order == RotationOrder.ZXY) {

// r (Vector3D.plusJ) coordinates are :
// -cos (phi) sin (psi), cos (phi) cos (psi), sin (phi)
// (-r) (Vector3D.plusK) coordinates are :
// -sin (theta) cos (phi), sin (phi), cos (theta) cos (phi)
// and we can choose to have phi in the interval [-PI/2 ; +PI/2]
Vector3D v1 = applyTo(Vector3D.plusJ);
Vector3D v2 = applyInverseTo(Vector3D.plusK);
if ((v2.getY() < -0.9999999999) || (v2.getY() > 0.9999999999)) {
System.out.println("CardanEulerSingularityException");
}
return new double[] {
Math.atan2(-(v1.getX()), v1.getY()),
Math.asin(v2.getY()),
Math.atan2(-(v2.getX()), v2.getZ())
};

} else if (order == RotationOrder.ZYX) {

// r (Vector3D.plusI) coordinates are :
//  cos (theta) cos (psi), cos (theta) sin (psi), -sin (theta)
// (-r) (Vector3D.plusK) coordinates are :
// -sin (theta), sin (phi) cos (theta), cos (phi) cos (theta)
// and we can choose to have theta in the interval [-PI/2 ; +PI/2]
Vector3D v1 = applyTo(Vector3D.plusI);
Vector3D v2 = applyInverseTo(Vector3D.plusK);
if ((v2.getX() < -0.9999999999) || (v2.getX() > 0.9999999999)) {
System.out.println("CardanEulerSingularityException");
}
return new double[] {
Math.atan2(v1.getY(), v1.getX()),
-Math.asin(v2.getX()),
Math.atan2(v2.getY(), v2.getZ())
};

} else if (order == RotationOrder.XYX) {

// r (Vector3D.plusI) coordinates are :
//  cos (theta), sin (phi1) sin (theta), -cos (phi1) sin (theta)
// (-r) (Vector3D.plusI) coordinates are :
// cos (theta), sin (theta) sin (phi2), sin (theta) cos (phi2)
// and we can choose to have theta in the interval [0 ; PI]
Vector3D v1 = applyTo(Vector3D.plusI);
Vector3D v2 = applyInverseTo(Vector3D.plusI);
if ((v2.getX() < -0.9999999999) || (v2.getX() > 0.9999999999)) {
System.out.println("CardanEulerSingularityException");
}
return new double[] {
Math.atan2(v1.getY(), -v1.getZ()),
Math.acos(v2.getX()),
Math.atan2(v2.getY(), v2.getZ())
};

} else if (order == RotationOrder.XZX) {

// r (Vector3D.plusI) coordinates are :
//  cos (psi), cos (phi1) sin (psi), sin (phi1) sin (psi)
// (-r) (Vector3D.plusI) coordinates are :
// cos (psi), -sin (psi) cos (phi2), sin (psi) sin (phi2)
// and we can choose to have psi in the interval [0 ; PI]
Vector3D v1 = applyTo(Vector3D.plusI);
Vector3D v2 = applyInverseTo(Vector3D.plusI);
if ((v2.getX() < -0.9999999999) || (v2.getX() > 0.9999999999)) {
System.out.println("CardanEulerSingularityException");
}
return new double[] {
Math.atan2(v1.getZ(), v1.getY()),
Math.acos(v2.getX()),
Math.atan2(v2.getZ(), -v2.getY())
};

} else if (order == RotationOrder.YXY) {

// r (Vector3D.plusJ) coordinates are :
//  sin (theta1) sin (phi), cos (phi), cos (theta1) sin (phi)
// (-r) (Vector3D.plusJ) coordinates are :
// sin (phi) sin (theta2), cos (phi), -sin (phi) cos (theta2)
// and we can choose to have phi in the interval [0 ; PI]
Vector3D v1 = applyTo(Vector3D.plusJ);
Vector3D v2 = applyInverseTo(Vector3D.plusJ);
if ((v2.getY() < -0.9999999999) || (v2.getY() > 0.9999999999)) {
System.out.println("CardanEulerSingularityException");
}
return new double[] {
Math.atan2(v1.getX(), v1.getZ()),
Math.acos(v2.getY()),
Math.atan2(v2.getX(), -v2.getZ())
};

} else if (order == RotationOrder.YZY) {

// r (Vector3D.plusJ) coordinates are :
//  -cos (theta1) sin (psi), cos (psi), sin (theta1) sin (psi)
// (-r) (Vector3D.plusJ) coordinates are :
// sin (psi) cos (theta2), cos (psi), sin (psi) sin (theta2)
// and we can choose to have psi in the interval [0 ; PI]
Vector3D v1 = applyTo(Vector3D.plusJ);
Vector3D v2 = applyInverseTo(Vector3D.plusJ);
if ((v2.getY() < -0.9999999999) || (v2.getY() > 0.9999999999)) {
System.out.println("CardanEulerSingularityException");
}
return new double[] {
Math.atan2(v1.getZ(), -v1.getX()),
Math.acos(v2.getY()),
Math.atan2(v2.getZ(), v2.getX())
};

} else if (order == RotationOrder.ZXZ) {

// r (Vector3D.plusK) coordinates are :
//  sin (psi1) sin (phi), -cos (psi1) sin (phi), cos (phi)
// (-r) (Vector3D.plusK) coordinates are :
// sin (phi) sin (psi2), sin (phi) cos (psi2), cos (phi)
// and we can choose to have phi in the interval [0 ; PI]
Vector3D v1 = applyTo(Vector3D.plusK);
Vector3D v2 = applyInverseTo(Vector3D.plusK);
if ((v2.getZ() < -0.9999999999) || (v2.getZ() > 0.9999999999)) {
System.out.println("CardanEulerSingularityException");
}
return new double[] {
Math.atan2(v1.getX(), -v1.getY()),
Math.acos(v2.getZ()),
Math.atan2(v2.getX(), v2.getY())
};

} else { // last possibility is ZYZ

// r (Vector3D.plusK) coordinates are :
//  cos (psi1) sin (theta), sin (psi1) sin (theta), cos (theta)
// (-r) (Vector3D.plusK) coordinates are :
// -sin (theta) cos (psi2), sin (theta) sin (psi2), cos (theta)
// and we can choose to have theta in the interval [0 ; PI]
Vector3D v1 = applyTo(Vector3D.plusK);
Vector3D v2 = applyInverseTo(Vector3D.plusK);
if ((v2.getZ() < -0.9999999999) || (v2.getZ() > 0.9999999999)) {
throw new RuntimeException("false");
}
return new double[] {
Math.atan2(v1.getY(), v1.getX()),
Math.acos(v2.getZ()),
Math.atan2(v2.getY(), -v2.getX())
};

}

}

/** Get the 3X3 matrix corresponding to the instance
* @return the matrix corresponding to the instance
*/
public double[][] getMatrix() {

// products
double q0q0  = q0 * q0;
double q0q1  = q0 * q1;
double q0q2  = q0 * q2;
double q0q3  = q0 * q3;
double q1q1  = q1 * q1;
double q1q2  = q1 * q2;
double q1q3  = q1 * q3;
double q2q2  = q2 * q2;
double q2q3  = q2 * q3;
double q3q3  = q3 * q3;

// create the matrix
double[][] m = new double[3][];
m[0] = new double[3];
m[1] = new double[3];
m[2] = new double[3];

m [0][0] = 2.0 * (q0q0 + q1q1) - 1.0;
m [1][0] = 2.0 * (q1q2 - q0q3);
m [2][0] = 2.0 * (q1q3 + q0q2);

m [0][1] = 2.0 * (q1q2 + q0q3);
m [1][1] = 2.0 * (q0q0 + q2q2) - 1.0;
m [2][1] = 2.0 * (q2q3 - q0q1);

m [0][2] = 2.0 * (q1q3 - q0q2);
m [1][2] = 2.0 * (q2q3 + q0q1);
m [2][2] = 2.0 * (q0q0 + q3q3) - 1.0;

return m;

}

/** Apply the rotation to a vector.
* @param u vector to apply the rotation to
* @return a new vector which is the image of u by the rotation
*/
public Vector3D applyTo(Vector3D u) {

double x = u.getX();
double y = u.getY();
double z = u.getZ();

double s = q1 * x + q2 * y + q3 * z;

return new Vector3D(2 * (q0 * (x * q0 - (q2 * z - q3 * y)) + s * q1) - x,
2 * (q0 * (y * q0 - (q3 * x - q1 * z)) + s * q2) - y,
2 * (q0 * (z * q0 - (q1 * y - q2 * x)) + s * q3) - z);

}

/** Apply the inverse of the rotation to a vector.
* @param u vector to apply the inverse of the rotation to
* @return a new vector which such that u is its image by the rotation
*/
public Vector3D applyInverseTo(Vector3D u) {

double x = u.getX();
double y = u.getY();
double z = u.getZ();

double s = q1 * x + q2 * y + q3 * z;
double m0 = -q0;

return new Vector3D(2 * (m0 * (x * m0 - (q2 * z - q3 * y)) + s * q1) - x,
2 * (m0 * (y * m0 - (q3 * x - q1 * z)) + s * q2) - y,
2 * (m0 * (z * m0 - (q1 * y - q2 * x)) + s * q3) - z);

}

/** Apply the instance to another rotation.
* Applying the instance to a rotation is computing the composition
* in an order compliant with the following rule : let u be any
* vector and v its image by r (i.e. r.applyTo(u) = v), let w be the image
* of v by the instance (i.e. applyTo(v) = w), then w = comp.applyTo(u),
* where comp = applyTo(r).
* @param r rotation to apply the rotation to
* @return a new rotation which is the composition of r by the instance
*/
public Rotation applyTo(Rotation r) {
return new Rotation(r.q0 * q0 - (r.q1 * q1 + r.q2 * q2 + r.q3 * q3),
r.q1 * q0 + r.q0 * q1 + (r.q2 * q3 - r.q3 * q2),
r.q2 * q0 + r.q0 * q2 + (r.q3 * q1 - r.q1 * q3),
r.q3 * q0 + r.q0 * q3 + (r.q1 * q2 - r.q2 * q1),
false);
}

/** Apply the inverse of the instance to another rotation.
* Applying the inverse of the instance to a rotation is computing
* the composition in an order compliant with the following rule :
* let u be any vector and v its image by r (i.e. r.applyTo(u) = v),
* let w be the inverse image of v by the instance
* (i.e. applyInverseTo(v) = w), then w = comp.applyTo(u), where
* comp = applyInverseTo(r).
* @param r rotation to apply the rotation to
* @return a new rotation which is the composition of r by the inverse
* of the instance
*/
public Rotation applyInverseTo(Rotation r) {
return new Rotation(-r.q0 * q0 - (r.q1 * q1 + r.q2 * q2 + r.q3 * q3),
-r.q1 * q0 + r.q0 * q1 + (r.q2 * q3 - r.q3 * q2),
-r.q2 * q0 + r.q0 * q2 + (r.q3 * q1 - r.q1 * q3),
-r.q3 * q0 + r.q0 * q3 + (r.q1 * q2 - r.q2 * q1),
false);
}

/** Perfect orthogonality on a 3X3 matrix.
* @param m initial matrix (not exactly orthogonal)
* @param threshold convergence threshold for the iterative
* orthogonality correction (convergence is reached when the
* difference between two steps of the Frobenius norm of the
* correction is below this threshold)
* @return an orthogonal matrix close to m
* @exception NotARotationMatrixException if the matrix cannot be
* orthogonalized with the given threshold after 10 iterations
*/
private double[][] orthogonalizeMatrix(double[][] m, double threshold)
{
double[] m0 = m[0];
double[] m1 = m[1];
double[] m2 = m[2];
double x00 = m0[0];
double x01 = m0[1];
double x02 = m0[2];
double x10 = m1[0];
double x11 = m1[1];
double x12 = m1[2];
double x20 = m2[0];
double x21 = m2[1];
double x22 = m2[2];
double fn = 0;
double fn1;

double[][] o = new double[3][3];
double[] o0 = o[0];
double[] o1 = o[1];
double[] o2 = o[2];

// iterative correction: Xn+1 = Xn - 0.5 * (Xn.Mt.Xn - M)
int i = 0;
while (++i < 11) {

// Mt.Xn
double mx00 = m0[0] * x00 + m1[0] * x10 + m2[0] * x20;
double mx10 = m0[1] * x00 + m1[1] * x10 + m2[1] * x20;
double mx20 = m0[2] * x00 + m1[2] * x10 + m2[2] * x20;
double mx01 = m0[0] * x01 + m1[0] * x11 + m2[0] * x21;
double mx11 = m0[1] * x01 + m1[1] * x11 + m2[1] * x21;
double mx21 = m0[2] * x01 + m1[2] * x11 + m2[2] * x21;
double mx02 = m0[0] * x02 + m1[0] * x12 + m2[0] * x22;
double mx12 = m0[1] * x02 + m1[1] * x12 + m2[1] * x22;
double mx22 = m0[2] * x02 + m1[2] * x12 + m2[2] * x22;

// Xn+1
o0[0] = x00 - 0.5 * (x00 * mx00 + x01 * mx10 + x02 * mx20 - m0[0]);
o0[1] = x01 - 0.5 * (x00 * mx01 + x01 * mx11 + x02 * mx21 - m0[1]);
o0[2] = x02 - 0.5 * (x00 * mx02 + x01 * mx12 + x02 * mx22 - m0[2]);
o1[0] = x10 - 0.5 * (x10 * mx00 + x11 * mx10 + x12 * mx20 - m1[0]);
o1[1] = x11 - 0.5 * (x10 * mx01 + x11 * mx11 + x12 * mx21 - m1[1]);
o1[2] = x12 - 0.5 * (x10 * mx02 + x11 * mx12 + x12 * mx22 - m1[2]);
o2[0] = x20 - 0.5 * (x20 * mx00 + x21 * mx10 + x22 * mx20 - m2[0]);
o2[1] = x21 - 0.5 * (x20 * mx01 + x21 * mx11 + x22 * mx21 - m2[1]);
o2[2] = x22 - 0.5 * (x20 * mx02 + x21 * mx12 + x22 * mx22 - m2[2]);

// correction on each elements
double corr00 = o0[0] - m0[0];
double corr01 = o0[1] - m0[1];
double corr02 = o0[2] - m0[2];
double corr10 = o1[0] - m1[0];
double corr11 = o1[1] - m1[1];
double corr12 = o1[2] - m1[2];
double corr20 = o2[0] - m2[0];
double corr21 = o2[1] - m2[1];
double corr22 = o2[2] - m2[2];

// Frobenius norm of the correction
fn1 = corr00 * corr00 + corr01 * corr01 + corr02 * corr02 +
corr10 * corr10 + corr11 * corr11 + corr12 * corr12 +
corr20 * corr20 + corr21 * corr21 + corr22 * corr22;

// convergence test
if (Math.abs(fn1 - fn) <= threshold)
return o;

// prepare next iteration
x00 = o0[0];
x01 = o0[1];
x02 = o0[2];
x10 = o1[0];
x11 = o1[1];
x12 = o1[2];
x20 = o2[0];
x21 = o2[1];
x22 = o2[2];
fn  = fn1;

}
return null;
// the algorithm did not converge after 10 iterations
//System.out.println("unable to orthogonalize matrix" +
//" in {0} iterations");
}

/** Scalar coordinate of the quaternion. */
private final double q0;

/** First coordinate of the vectorial part of the quaternion. */
private final double q1;

/** Second coordinate of the vectorial part of the quaternion. */
private final double q2;

/** Third coordinate of the vectorial part of the quaternion. */
private final double q3;

/** Serializable version identifier */
private static final long serialVersionUID = 8225864499430109352L;

}

/**
* This class is a utility representing a rotation order specification
* for Cardan or Euler angles specification.
*
* This class cannot be instanciated by the user. He can only use one
* of the twelve predefined supported orders as an argument to either
* constructor or the {@link Rotation#getAngles} method.
*
* @version \$Revision: 620312 \$ \$Date: 2008-02-10 12:28:59 -0700 (Sun, 10 Feb 2008) \$
* @since 1.2
*/
final class RotationOrder {

/** Private constructor.
* This is a utility class that cannot be instantiated by the user,
* so its only constructor is private.
* @param name name of the rotation order
* @param a1 axis of the first rotation
* @param a2 axis of the second rotation
* @param a3 axis of the third rotation
*/
private RotationOrder(String name,
Vector3D a1, Vector3D a2, Vector3D a3) {
this.name = name;
this.a1   = a1;
this.a2   = a2;
this.a3   = a3;
}

/** Get a string representation of the instance.
* @return a string representation of the instance (in fact, its name)
*/
public String toString() {
return name;
}

/** Get the axis of the first rotation.
* @return axis of the first rotation
*/
public Vector3D getA1() {
return a1;
}

/** Get the axis of the second rotation.
* @return axis of the second rotation
*/
public Vector3D getA2() {
return a2;
}

/** Get the axis of the second rotation.
* @return axis of the second rotation
*/
public Vector3D getA3() {
return a3;
}

/** Set of Cardan angles.
* this ordered set of rotations is around X, then around Y, then
* around Z
*/
public static final RotationOrder XYZ =
new RotationOrder("XYZ", Vector3D.plusI, Vector3D.plusJ, Vector3D.plusK);

/** Set of Cardan angles.
* this ordered set of rotations is around X, then around Z, then
* around Y
*/
public static final RotationOrder XZY =
new RotationOrder("XZY", Vector3D.plusI, Vector3D.plusK, Vector3D.plusJ);

/** Set of Cardan angles.
* this ordered set of rotations is around Y, then around X, then
* around Z
*/
public static final RotationOrder YXZ =
new RotationOrder("YXZ", Vector3D.plusJ, Vector3D.plusI, Vector3D.plusK);

/** Set of Cardan angles.
* this ordered set of rotations is around Y, then around Z, then
* around X
*/
public static final RotationOrder YZX =
new RotationOrder("YZX", Vector3D.plusJ, Vector3D.plusK, Vector3D.plusI);

/** Set of Cardan angles.
* this ordered set of rotations is around Z, then around X, then
* around Y
*/
public static final RotationOrder ZXY =
new RotationOrder("ZXY", Vector3D.plusK, Vector3D.plusI, Vector3D.plusJ);

/** Set of Cardan angles.
* this ordered set of rotations is around Z, then around Y, then
* around X
*/
public static final RotationOrder ZYX =
new RotationOrder("ZYX", Vector3D.plusK, Vector3D.plusJ, Vector3D.plusI);

/** Set of Euler angles.
* this ordered set of rotations is around X, then around Y, then
* around X
*/
public static final RotationOrder XYX =
new RotationOrder("XYX", Vector3D.plusI, Vector3D.plusJ, Vector3D.plusI);

/** Set of Euler angles.
* this ordered set of rotations is around X, then around Z, then
* around X
*/
public static final RotationOrder XZX =
new RotationOrder("XZX", Vector3D.plusI, Vector3D.plusK, Vector3D.plusI);

/** Set of Euler angles.
* this ordered set of rotations is around Y, then around X, then
* around Y
*/
public static final RotationOrder YXY =
new RotationOrder("YXY", Vector3D.plusJ, Vector3D.plusI, Vector3D.plusJ);

/** Set of Euler angles.
* this ordered set of rotations is around Y, then around Z, then
* around Y
*/
public static final RotationOrder YZY =
new RotationOrder("YZY", Vector3D.plusJ, Vector3D.plusK, Vector3D.plusJ);

/** Set of Euler angles.
* this ordered set of rotations is around Z, then around X, then
* around Z
*/
public static final RotationOrder ZXZ =
new RotationOrder("ZXZ", Vector3D.plusK, Vector3D.plusI, Vector3D.plusK);

/** Set of Euler angles.
* this ordered set of rotations is around Z, then around Y, then
* around Z
*/
public static final RotationOrder ZYZ =
new RotationOrder("ZYZ", Vector3D.plusK, Vector3D.plusJ, Vector3D.plusK);

/** Name of the rotations order. */
private final String name;

/** Axis of the first rotation. */
private final Vector3D a1;

/** Axis of the second rotation. */
private final Vector3D a2;

/** Axis of the third rotation. */
private final Vector3D a3;

}

/**
* This class implements vectors in a three-dimensional space.
* <p>Instance of this class are guaranteed to be immutable.</p>
* @version \$Revision: 627998 \$ \$Date: 2008-02-15 03:24:50 -0700 (Fri, 15 Feb 2008) \$
* @since 1.2
*/

class Vector3D
implements Serializable {

/** First canonical vector (coordinates: 1, 0, 0). */
public static final Vector3D plusI = new Vector3D(1, 0, 0);

/** Opposite of the first canonical vector (coordinates: -1, 0, 0). */
public static final Vector3D minusI = new Vector3D(-1, 0, 0);

/** Second canonical vector (coordinates: 0, 1, 0). */
public static final Vector3D plusJ = new Vector3D(0, 1, 0);

/** Opposite of the second canonical vector (coordinates: 0, -1, 0). */
public static final Vector3D minusJ = new Vector3D(0, -1, 0);

/** Third canonical vector (coordinates: 0, 0, 1). */
public static final Vector3D plusK = new Vector3D(0, 0, 1);

/** Opposite of the third canonical vector (coordinates: 0, 0, -1).  */
public static final Vector3D minusK = new Vector3D(0, 0, -1);

/** Null vector (coordinates: 0, 0, 0). */
public static final Vector3D zero   = new Vector3D(0, 0, 0);

/** Simple constructor.
* Build a null vector.
*/
public Vector3D() {
x = 0;
y = 0;
z = 0;
}

/** 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 (&alpha;) around Z
*              (0 is +X, &pi;/2 is +Y, &pi; is -X and 3&pi;/2 is -Y)
* @param delta elevation (&delta;) above (XY) plane, from -&pi;/2 to +&pi;/2
* @see #getAlpha()
* @see #getDelta()
*/
public Vector3D(double alpha, double delta) {
double cosDelta = Math.cos(delta);
this.x = Math.cos(alpha) * cosDelta;
this.y = Math.sin(alpha) * cosDelta;
this.z = Math.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 norm for the vector.
* @return euclidian norm for the vector
*/
public double getNorm() {
return Math.sqrt (x * x + y * y + z * z);
}

/** Get the azimuth of the vector.
* @return azimuth (&alpha;) of the vector, between -&pi; and +&pi;
* @see #Vector3D(double, double)
*/
public double getAlpha() {
return Math.atan2(y, x);
}

/** Get the elevation of the vector.
* @return elevation (&delta;) of the vector, between -&pi;/2 and +&pi;/2
* @see #Vector3D(double, double)
*/
public double getDelta() {
return Math.asin(z / getNorm());
}

/** Add a vector to the instance.
* @param v vector to add
* @return a new vector
*/
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 ArithmeticException("cannot normalize a zero norm vector");
}
return scalarMultiply(1 / s);
}

/** Get a vector orthogonal to the instance.
* <p>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 :
* <pre><code>
*   Vector3D k = u.normalize();
*   Vector3D i = k.orthogonal();
*   Vector3D j = Vector3D.crossProduct(k, i);
* </code></pre></p>
* @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 ArithmeticException("null norm");
}

if ((x >= -threshold) && (x <= threshold)) {
double inverse  = 1 / Math.sqrt(y * y + z * z);
return new Vector3D(0, inverse * z, -inverse * y);
} else if ((y >= -threshold) && (y <= threshold)) {
double inverse  = 1 / Math.sqrt(x * x + z * z);
return new Vector3D(-inverse * z, 0, inverse * x);
}
double inverse  = 1 / Math.sqrt(x * x + y * y);
return new Vector3D(inverse * y, -inverse * x, 0);

}

/** Compute the angular separation between two vectors.
* <p>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 allow to have a
* good accuracy in all cases, even for vectors very close to each
* other.</p>
* @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 ArithmeticException("null 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 Math.asin(v3.getNorm() / normProduct);
}
return Math.PI - Math.asin(v3.getNorm() / normProduct);
}

// the vectors are sufficiently separated to use the cosine
return Math.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);
}

/** 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(Vector3D v1, Vector3D 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);
}

/** Abscissa. */
private final double x;

/** Ordinate. */
private final double y;

/** Height. */
private final double z;

/** Serializable version identifier */
private static final long serialVersionUID = -5721105387745193385L;

}

```

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