/*
Part of the Shapes 3D library for Processing
http://www.lagers.org.uk
Copyright (c) 2010 Peter Lager
This library is free software; you can redistribute it and/or
modify it under the terms of the GNU Lesser General Public
License as published by the Free Software Foundation; either
version 2.1 of the License, or (at your option) any later version.
This library is distributed in the hope that it will be useful,
but WITHOUT ANY WARRANTY; without even the implied warranty of
MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU
Lesser General Public License for more details.
You should have received a copy of the GNU Lesser General
Public License along with this library; if not, write to the
Free Software Foundation, Inc., 59 Temple Place, Suite 330,
Boston, MA 02111-1307 USA
*/
/*
* The algorithms used in this class are based on the Rotation class
* found in the Apache Commons Math project at http://commons.apache.org/math/ <br>
*
* It has been rewritten to make use of the PVector class that is
* part of Processing.
*/
package shapes3d.utils;
import java.io.Serializable;
import processing.core.PVector;
/**
* 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
* any of these representations can be retrieved from a
* <code>Rotation</code> instance (see the various constructors and
* getters). In addition, a rotation can also be built implicitly
* 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
* following single line of code:</p>
* <pre>
* float[] angles = new Rotation(matrix, 1.0fe-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 implicitly 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
* methods: {@link #applyTo(Vector3D) applyTo(Vector3D)} and {@link
* #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(Rot) applyTo(Rotation)} and
* {@link #applyInverseTo(Rot) applyInverseTo(Rotation)}.</p>
*
* <p>Rotations are guaranteed to be immutable objects.</p>
*
* @version $Revision: 772119 $ $Date: 2009-05-06 05:43:28 -0400 (Wed, 06 May 2009) $
*
* @see PVector
* @see RotOrder
*/
public class Rot implements VectorConstants, Serializable {
/**
*
*/
private static final long serialVersionUID = 1077272288787175558L;
/** Identity rotation. */
public static final Rot IDENTITY = new Rot(1.0f, 0.0f, 0.0f, 0.0f, false);
/** Scalar coordinate of the quaternion. */
private final float q0;
/** First coordinate of the vectorial part of the quaternion. */
private final float q1;
/** Second coordinate of the vectorial part of the quaternion. */
private final float q2;
/** Third coordinate of the vectorial part of the quaternion. */
private final float q3;
/** 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 Rot(float q0, float q1, float q2, float q3,
boolean needsNormalization) {
if (needsNormalization) {
float inv = 1.0f / (float)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(PVector) apply} the instance to +i, we will get
* +j.</p>
* @param axis axis around which to rotate
* @param angle rotation angle.
*/
public Rot(PVector axis, float angle) {
float norm = axis.mag();
if (norm == 0) {
q0 = 1;
q1 = q2 = q3 = 0;
System.out.println(Messages.build("The axis vector {0} has no magnitude!!", axis));
return;
}
float halfAngle = -0.5f * angle;
float coeff = (float) Math.sin(halfAngle) / norm;
q0 = (float) Math.cos (halfAngle);
q1 = coeff * axis.x;
q2 = coeff * axis.y;
q3 = coeff * axis.z;
}
/** 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 Rot(float[][] m, float threshold){
// dimension check
if ((m.length != 3) || (m[0].length != 3) ||
(m[1].length != 3) || (m[2].length != 3)) {
q0 = 1;
q1 = q2 = q3 = 0;
System.out.println(Messages.build("a {0}x{1} matrix cannot be a rotation matrix", m.length, m[0].length));
return;
}
// compute a "close" orthogonal matrix
float[][] ort = orthogonalizeMatrix(m, threshold);
if(ort == null){
q0 = 1;
q1 = q2 = q3 = 0;
System.out.println(Messages.build("unable to orthogonalize matrix in {0} iterations",10));
return;
}
// check the sign of the determinant
float 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 < 0.0f) {
q0 = 1;
q1 = q2 = q3 = 0;
System.out.println(Messages.build("the closest orthogonal matrix has a negative determinant {0}", det));
return;
}
// 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)
float s = ort[0][0] + ort[1][1] + ort[2][2];
if (s > -0.19) {
// compute q0 and deduce q1, q2 and q3
q0 = 0.5f * (float) Math.sqrt(s + 1.0f);
float inv = 0.25f / 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 = 0.5f * (float) Math.sqrt(s + 1.0f);
float inv = 0.25f / 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 = 0.5f * (float) Math.sqrt(s + 1.0f);
float inv = 0.25f / 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 = 0.5f * (float) Math.sqrt(s + 1.0f);
float inv = 0.25f / 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 Rot(PVector u1, PVector u2, PVector v1, PVector v2) {
// norms computation
float u1u1 = PVector.dot(u1, u1);
float u2u2 = PVector.dot(u2, u2);
float v1v1 = PVector.dot(v1, v1);
float v2v2 = PVector.dot(v2, v2);
if ((u1u1 == 0) || (u2u2 == 0) || (v1v1 == 0) || (v2v2 == 0)) {
q0 = 1;
q1 = q2 = q3 = 0;
System.out.println("zero norm for rotation defining vector");
return;
}
float u1x = u1.x;
float u1y = u1.y;
float u1z = u1.z;
float u2x = u2.x;
float u2y = u2.y;
float u2z = u2.z;
// normalize v1 in order to have (v1'|v1') = (u1|u1)
float coeff = (float)Math.sqrt (u1u1 / v1v1);
float v1x = coeff * v1.x;
float v1y = coeff * v1.y;
float v1z = coeff * v1.z;
v1 = new PVector(v1x, v1y, v1z);
// adjust v2 in order to have (u1|u2) = (v1|v2) and (v2'|v2') = (u2|u2)
float u1u2 = PVector.dot(u1, u2);
float v1v2 = PVector.dot(v1, v2);
float coeffU = u1u2 / u1u1;
float coeffV = v1v2 / u1u1;
float beta = (float) Math.sqrt((u2u2 - u1u2 * coeffU) / (v2v2 - v1v2 * coeffV));
float alpha = coeffU - beta * coeffV;
float v2x = alpha * v1x + beta * v2.x;
float v2y = alpha * v1y + beta * v2.y;
float v2z = alpha * v1z + beta * v2.z;
v2 = new PVector(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)
PVector uRef = u1;
PVector vRef = v1;
float dx1 = v1x - u1.x;
float dy1 = v1y - u1.y;
float dz1 = v1z - u1.z;
float dx2 = v2x - u2.x;
float dy2 = v2y - u2.y;
float dz2 = v2z - u2.z;
PVector k = new PVector(dy1 * dz2 - dz1 * dy2,
dz1 * dx2 - dx1 * dz2,
dx1 * dy2 - dy1 * dx2);
float c = k.x * (u1y * u2z - u1z * u2y) +
k.y * (u1z * u2x - u1x * u2z) +
k.z * (u1x * u2y - u1y * u2x);
if (c == 0) {
// the (q1, q2, q3) vector is in the (u1, u2) plane
// we try other vectors
PVector u3 = PVector.cross(u1, u2, null);
PVector v3 = PVector.cross(v1, v2, null);
float u3x = u3.x;
float u3y = u3.y;
float u3z = u3.z;
float v3x = v3.x;
float v3y = v3.y;
float v3z = v3.z;
float dx3 = v3x - u3x;
float dy3 = v3y - u3y;
float dz3 = v3z - u3z;
k = new PVector(dy1 * dz3 - dz1 * dy3,
dz1 * dx3 - dx1 * dz3,
dx1 * dy3 - dy1 * dx3);
c = k.x * (u1y * u3z - u1z * u3y) +
k.y * (u1z * u3x - u1x * u3z) +
k.z * (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 PVector(dy2 * dz3 - dz2 * dy3,
dz2 * dx3 - dx2 * dz3,
dx2 * dy3 - dy2 * dx3);
c = k.x * (u2y * u3z - u2z * u3y) +
k.y * (u2z * u3x - u2x * u3z) +
k.z * (u2x * u3y - u2y * u3x);
if (c == 0) {
// the (q1, q2, q3) vector is aligned with everything
// this is really the identity rotation
q0 = 1.0f;
q1 = 0.0f;
q2 = 0.0f;
q3 = 0.0f;
return;
}
// we will have to use u2 and v2 to compute the scalar part
uRef = u2;
vRef = v2;
}
}
// compute the vectorial part
c = (float) Math.sqrt(c);
float inv = 1.0f / (c + c);
q1 = inv * k.x;
q2 = inv * k.y;
q3 = inv * k.z;
// compute the scalar part
k = new PVector(uRef.y * q3 - uRef.z * q2,
uRef.z * q1 - uRef.x * q3,
uRef.x * q2 - uRef.y * q1);
c = PVector.dot(k, k);
q0 = PVector.dot(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 Rot(PVector u, PVector v) {
float normProduct = u.mag() * v.mag();
if (normProduct == 0) {
q0 = 1;
q1 = q2 = q3 = 0;
System.out.println("zero norm for rotation defining vector");
return;
}
float dot = PVector.dot(u, v);
if (dot < ((2.0e-15 - 1.0f) * normProduct)) {
// special case u = -v: we select a PI angle rotation around
// an arbitrary vector orthogonal to u
PVector w = VectorUtil.orthogonal(u);
q0 = 0.0f;
q1 = (float) -w.x;
q2 = (float) -w.y;
q3 = (float) -w.z;
} else {
// general case: (u, v) defines a plane, we select
// the shortest possible rotation: axis orthogonal to this plane
q0 = (float) Math.sqrt(0.5 * (1.0f + dot / normProduct));
float coeff = 1.0f / (2.0f * q0 * normProduct);
q1 = (float) (coeff * (v.y * u.z - v.z * u.y));
q2 = (float) (coeff * (v.z * u.x - v.x * u.z));
q3 = (float) (coeff * (v.x * u.y - v.y * u.x));
}
}
/** 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 being 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 being 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 Rot(RotOrder order, float alpha1, float alpha2, float alpha3) {
Rot r1 = new Rot(order.getA1(), alpha1);
Rot r2 = new Rot(order.getA2(), alpha2);
Rot r3 = new Rot(order.getA3(), alpha3);
Rot 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 Rot revert() {
return new Rot(-q0, q1, q2, q3, false);
}
/** Get the scalar coordinate of the quaternion.
* @return scalar coordinate of the quaternion
*/
public float 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 float 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 float 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 float getQ3() {
return q3;
}
/** Get the normalized axis of the rotation.
* @return normalized axis of the rotation
*/
public PVector getAxis() {
float squaredSine = q1 * q1 + q2 * q2 + q3 * q3;
if (squaredSine == 0) {
return new PVector(1, 0, 0);
}
else if (q0 < 0) {
float inverse = 1 / (float) Math.sqrt(squaredSine);
return new PVector(q1 * inverse, q2 * inverse, q3 * inverse);
}
float inverse = -1 / (float) Math.sqrt(squaredSine);
return new PVector(q1 * inverse, q2 * inverse, q3 * inverse);
}
/** Get the angle of the rotation.
* @return angle of the rotation (between 0 and π)
*/
public float getAngle() {
if ((q0 < -0.1) || (q0 > 0.1)) {
return 2 * (float) Math.asin(Math.sqrt(q1 * q1 + q2 * q2 + q3 * q3));
}
else if (q0 < 0) {
return 2 * (float) Math.acos(-q0);
}
return 2 * (float) 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 π + a<sub>1</sub>, π
* - a<sub>2</sub> and π + 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 -π/2 and π/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 π (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
* -π/2 or +π/2, for Euler angle singularities occur when the
* second angle is close to 0 or π, 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
* or null if angle singularity found.
*/
public float[] getAngles(RotOrder order){
if (order == RotOrder.XYZ) {
// r (plusK) coordinates are :
// sin (theta), -cos (theta) sin (phi), cos (theta) cos (phi)
// (-r) (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]
PVector v1 = applyToNew(PLUS_K);
PVector v2 = applyInverseToNew(PLUS_I);
if ((v2.z < -0.9999999999) || (v2.z > 0.9999999999)) {
return null;
}
return new float[] {
(float) Math.atan2(-(v1.y), v1.z),
(float) Math.asin(v2.z),
(float) Math.atan2(-(v2.y), v2.x)
};
} else if (order == RotOrder.XZY) {
// r (plusJ) coordinates are :
// -sin (psi), cos (psi) cos (phi), cos (psi) sin (phi)
// (-r) (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]
PVector v1 = applyToNew(PLUS_J);
PVector v2 = applyInverseToNew(PLUS_I);
if ((v2.y < -0.9999999999) || (v2.y > 0.9999999999)) {
return null;
}
return new float[] {
(float) Math.atan2(v1.z, v1.y),
-(float) Math.asin(v2.y),
(float) Math.atan2(v2.z, v2.x)
};
} else if (order == RotOrder.YXZ) {
// r (plusK) coordinates are :
// cos (phi) sin (theta), -sin (phi), cos (phi) cos (theta)
// (-r) (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]
PVector v1 = applyToNew(PLUS_K);
PVector v2 = applyInverseToNew(PLUS_J);
if ((v2.z < -0.9999999999) || (v2.z > 0.9999999999)) {
return null;
}
return new float[] {
(float) Math.atan2(v1.x, v1.z),
-(float) Math.asin(v2.z),
(float) Math.atan2(v2.x, v2.y)
};
} else if (order == RotOrder.YZX) {
// r (plusI) coordinates are :
// cos (psi) cos (theta), sin (psi), -cos (psi) sin (theta)
// (-r) (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]
PVector v1 = applyToNew(PLUS_I);
PVector v2 = applyInverseToNew(PLUS_J);
if ((v2.x < -0.9999999999) || (v2.x > 0.9999999999)) {
return null;
}
return new float[] {
(float) Math.atan2(-(v1.z), v1.x),
(float) Math.asin(v2.x),
(float) Math.atan2(-(v2.z), v2.y)
};
} else if (order == RotOrder.ZXY) {
// r (plusJ) coordinates are :
// -cos (phi) sin (psi), cos (phi) cos (psi), sin (phi)
// (-r) (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]
PVector v1 = applyToNew(PLUS_J);
PVector v2 = applyInverseToNew(PLUS_K);
if ((v2.y < -0.9999999999) || (v2.y > 0.9999999999)) {
return null;
}
return new float[] {
(float) Math.atan2(-(v1.x), v1.y),
(float) Math.asin(v2.y),
(float) Math.atan2(-(v2.x), v2.z)
};
} else if (order == RotOrder.ZYX) {
// r (plusI) coordinates are :
// cos (theta) cos (psi), cos (theta) sin (psi), -sin (theta)
// (-r) (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]
PVector v1 = applyToNew(PLUS_I);
PVector v2 = applyInverseToNew(PLUS_K);
if ((v2.x < -0.9999999999) || (v2.x > 0.9999999999)) {
return null;
}
return new float[] {
(float) Math.atan2(v1.y, v1.x),
-(float) Math.asin(v2.x),
(float) Math.atan2(v2.y, v2.z)
};
} else if (order == RotOrder.XYX) {
// r (plusI) coordinates are :
// cos (theta), sin (phi1) sin (theta), -cos (phi1) sin (theta)
// (-r) (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]
PVector v1 = applyToNew(PLUS_I);
PVector v2 = applyInverseToNew(PLUS_I);
if ((v2.x < -0.9999999999) || (v2.x > 0.9999999999)) {
return null;
}
return new float[] {
(float) Math.atan2(v1.y, -v1.z),
(float) Math.acos(v2.x),
(float) Math.atan2(v2.y, v2.z)
};
} else if (order == RotOrder.XZX) {
// r (plusI) coordinates are :
// cos (psi), cos (phi1) sin (psi), sin (phi1) sin (psi)
// (-r) (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]
PVector v1 = applyToNew(PLUS_I);
PVector v2 = applyInverseToNew(PLUS_I);
if ((v2.x < -0.9999999999) || (v2.x > 0.9999999999)) {
return null;
}
return new float[] {
(float) Math.atan2(v1.z, v1.y),
(float) Math.acos(v2.x),
(float) Math.atan2(v2.z, -v2.y)
};
} else if (order == RotOrder.YXY) {
// r (plusJ) coordinates are :
// sin (theta1) sin (phi), cos (phi), cos (theta1) sin (phi)
// (-r) (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]
PVector v1 = applyToNew(PLUS_J);
PVector v2 = applyInverseToNew(PLUS_J);
if ((v2.y < -0.9999999999) || (v2.y > 0.9999999999)) {
return null;
}
return new float[] {
(float) Math.atan2(v1.x, v1.z),
(float) Math.acos(v2.y),
(float) Math.atan2(v2.x, -v2.z)
};
} else if (order == RotOrder.YZY) {
// r (plusJ) coordinates are :
// -cos (theta1) sin (psi), cos (psi), sin (theta1) sin (psi)
// (-r) (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]
PVector v1 = applyToNew(PLUS_J);
PVector v2 = applyInverseToNew(PLUS_J);
if ((v2.y < -0.9999999999) || (v2.y > 0.9999999999)) {
return null;
}
return new float[] {
(float) Math.atan2(v1.z, -v1.x),
(float) Math.acos(v2.y),
(float) Math.atan2(v2.z, v2.x)
};
} else if (order == RotOrder.ZXZ) {
// r (plusK) coordinates are :
// sin (psi1) sin (phi), -cos (psi1) sin (phi), cos (phi)
// (-r) (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]
PVector v1 = applyToNew(PLUS_K);
PVector v2 = applyInverseToNew(PLUS_K);
if ((v2.z < -0.9999999999) || (v2.z > 0.9999999999)) {
return null;
}
return new float[] {
(float) Math.atan2(v1.x, -v1.y),
(float) Math.acos(v2.z),
(float) Math.atan2(v2.x, v2.y)
};
} else { // last possibility is ZYZ
// r (plusK) coordinates are :
// cos (psi1) sin (theta), sin (psi1) sin (theta), cos (theta)
// (-r) (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]
PVector v1 = applyToNew(PLUS_K);
PVector v2 = applyInverseToNew(PLUS_K);
if ((v2.z < -0.9999999999) || (v2.z > 0.9999999999)) {
return null;
}
return new float[] {
(float) Math.atan2(v1.y, v1.x),
(float) Math.acos(v2.z),
(float) Math.atan2(v2.y, -v2.x)
};
}
}
/** Get the 3X3 matrix corresponding to the instance
* @return the matrix corresponding to the instance
*/
public float[][] getMatrix() {
// products
float q0q0 = q0 * q0;
float q0q1 = q0 * q1;
float q0q2 = q0 * q2;
float q0q3 = q0 * q3;
float q1q1 = q1 * q1;
float q1q2 = q1 * q2;
float q1q3 = q1 * q3;
float q2q2 = q2 * q2;
float q2q3 = q2 * q3;
float q3q3 = q3 * q3;
// create the matrix
float[][] m = new float[3][];
m[0] = new float[3];
m[1] = new float[3];
m[2] = new float[3];
m [0][0] = 2.0f * (q0q0 + q1q1) - 1.0f;
m [1][0] = 2.0f * (q1q2 - q0q3);
m [2][0] = 2.0f * (q1q3 + q0q2);
m [0][1] = 2.0f * (q1q2 + q0q3);
m [1][1] = 2.0f * (q0q0 + q2q2) - 1.0f;
m [2][1] = 2.0f * (q2q3 - q0q1);
m [0][2] = 2.0f * (q1q3 - q0q2);
m [1][2] = 2.0f * (q2q3 + q0q1);
m [2][2] = 2.0f * (q0q0 + q3q3) - 1.0f;
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 PVector applyTo(PVector u) {
float x = u.x;
float y = u.y;
float z = u.z;
float s = q1 * x + q2 * y + q3 * z;
float nx = 2 * (q0 * (x * q0 - (q2 * z - q3 * y)) + s * q1) - x;
float ny = 2 * (q0 * (y * q0 - (q3 * x - q1 * z)) + s * q2) - y;
float nz = 2 * (q0 * (z * q0 - (q1 * y - q2 * x)) + s * q3) - z;
u.x = nx;
u.y = ny;
u.z = nz;
return u;
}
/**
* Same as applyTo but u is unchanged and a new PVector is returned.
* @param u
*/
public PVector applyToNew(final PVector u) {
float x = u.x;
float y = u.y;
float z = u.z;
float s = q1 * x + q2 * y + q3 * z;
float nx = 2 * (q0 * (x * q0 - (q2 * z - q3 * y)) + s * q1) - x;
float ny = 2 * (q0 * (y * q0 - (q3 * x - q1 * z)) + s * q2) - y;
float nz = 2 * (q0 * (z * q0 - (q1 * y - q2 * x)) + s * q3) - z;
return new PVector(nx,ny,nz);
}
/** 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 PVector applyInverseTo(PVector u) {
float x = u.x;
float y = u.y;
float z = u.z;
float s = q1 * x + q2 * y + q3 * z;
float m0 = -q0;
float nx = 2 * (m0 * (x * m0 - (q2 * z - q3 * y)) + s * q1) - x;
float ny = 2 * (m0 * (y * m0 - (q3 * x - q1 * z)) + s * q2) - y;
float nz = 2 * (m0 * (z * m0 - (q1 * y - q2 * x)) + s * q3) - z;
u.x = nx;
u.y = ny;
u.z = nz;
return u;
}
/**
* Same as applyInverseTo but u is unchanged and a new PVector is returned.
* @param u
*/
public PVector applyInverseToNew(PVector u) {
float x = u.x;
float y = u.y;
float z = u.z;
float s = q1 * x + q2 * y + q3 * z;
float m0 = -q0;
float nx = 2 * (m0 * (x * m0 - (q2 * z - q3 * y)) + s * q1) - x;
float ny = 2 * (m0 * (y * m0 - (q3 * x - q1 * z)) + s * q2) - y;
float nz = 2 * (m0 * (z * m0 - (q1 * y - q2 * x)) + s * q3) - z;
return new PVector(nx, ny, nz);
}
/** 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 Rot applyTo(Rot r) {
return new Rot(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 Rot applyInverseTo(Rot r) {
return new Rot(-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 float[][] orthogonalizeMatrix(float[][] m, float threshold){
float[] m0 = m[0];
float[] m1 = m[1];
float[] m2 = m[2];
float x00 = m0[0];
float x01 = m0[1];
float x02 = m0[2];
float x10 = m1[0];
float x11 = m1[1];
float x12 = m1[2];
float x20 = m2[0];
float x21 = m2[1];
float x22 = m2[2];
float fn = 0;
float fn1;
float[][] orth = new float[3][3];
float[] o0 = orth[0];
float[] o1 = orth[1];
float[] o2 = orth[2];
// iterative correction: Xn+1 = Xn - 0.5 * (Xn.Mt.Xn - M)
int i = 0;
while (++i < 11) {
// Mt.Xn
float mx00 = m0[0] * x00 + m1[0] * x10 + m2[0] * x20;
float mx10 = m0[1] * x00 + m1[1] * x10 + m2[1] * x20;
float mx20 = m0[2] * x00 + m1[2] * x10 + m2[2] * x20;
float mx01 = m0[0] * x01 + m1[0] * x11 + m2[0] * x21;
float mx11 = m0[1] * x01 + m1[1] * x11 + m2[1] * x21;
float mx21 = m0[2] * x01 + m1[2] * x11 + m2[2] * x21;
float mx02 = m0[0] * x02 + m1[0] * x12 + m2[0] * x22;
float mx12 = m0[1] * x02 + m1[1] * x12 + m2[1] * x22;
float mx22 = m0[2] * x02 + m1[2] * x12 + m2[2] * x22;
// Xn+1
o0[0] = x00 - 0.5f * (x00 * mx00 + x01 * mx10 + x02 * mx20 - m0[0]);
o0[1] = x01 - 0.5f * (x00 * mx01 + x01 * mx11 + x02 * mx21 - m0[1]);
o0[2] = x02 - 0.5f * (x00 * mx02 + x01 * mx12 + x02 * mx22 - m0[2]);
o1[0] = x10 - 0.5f * (x10 * mx00 + x11 * mx10 + x12 * mx20 - m1[0]);
o1[1] = x11 - 0.5f * (x10 * mx01 + x11 * mx11 + x12 * mx21 - m1[1]);
o1[2] = x12 - 0.5f * (x10 * mx02 + x11 * mx12 + x12 * mx22 - m1[2]);
o2[0] = x20 - 0.5f * (x20 * mx00 + x21 * mx10 + x22 * mx20 - m2[0]);
o2[1] = x21 - 0.5f * (x20 * mx01 + x21 * mx11 + x22 * mx21 - m2[1]);
o2[2] = x22 - 0.5f * (x20 * mx02 + x21 * mx12 + x22 * mx22 - m2[2]);
// correction on each elements
float corr00 = o0[0] - m0[0];
float corr01 = o0[1] - m0[1];
float corr02 = o0[2] - m0[2];
float corr10 = o1[0] - m1[0];
float corr11 = o1[1] - m1[1];
float corr12 = o1[2] - m1[2];
float corr20 = o2[0] - m2[0];
float corr21 = o2[1] - m2[1];
float 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 orth;
// 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;
}
/** Compute the <i>distance</i> between two rotations.
* <p>The <i>distance</i> is intended here as a way to check if two
* rotations are almost similar (i.e. they transform vectors the same way)
* or very different. It is mathematically defined as the angle of
* the rotation r that prepended to one of the rotations gives the other
* one:</p>
* <pre>
* r<sub>1</sub>(r) = r<sub>2</sub>
* </pre>
* <p>This distance is an angle between 0 and π. Its value is the smallest
* possible upper bound of the angle in radians between r<sub>1</sub>(v)
* and r<sub>2</sub>(v) for all possible vectors v. This upper bound is
* reached for some v. The distance is equal to 0 if and only if the two
* rotations are identical.</p>
* <p>Comparing two rotations should always be done using this value rather
* than for example comparing the components of the quaternions. It is much
* more stable, and has a geometric meaning. Also comparing quaternions
* components is error prone since for example quaternions (0.36, 0.48, -0.48, -0.64)
* and (-0.36, -0.48, 0.48, 0.64) represent exactly the same rotation despite
* their components are different (they are exact opposites).</p>
* @param r1 first rotation
* @param r2 second rotation
* @return <i>distance</i> between r1 and r2
*/
public static float distance(Rot r1, Rot r2) {
return r1.applyInverseTo(r2).getAngle();
}
public String toString(){
return "Q["+q0+" "+q1+" "+q2+" "+q3+"] ";
}
}
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