Warning, file /education/kstars/kstars/ekos/align/rotations.cpp was not indexed or was modified since last indexation (in which case cross-reference links may be missing, inaccurate or erroneous).

 /*
     SPDX-FileCopyrightText: 2021 Hy Murveit <hy@murveit.com>
 
     SPDX-License-Identifier: GPL-2.0-or-later
 */
 
 #include "rotations.h"
 
 #include <cmath>
 
 // In the coordinate system used in this file:
 // x points forward
 // y points to the left
 // z points up.
 // In this system, theta is the angle between x & y and is related
 // to our azimuth: theta = 90-azimuth.
 // Phi is the angle away from z, and is related to our altitude as 90-altitude.
 
 namespace Rotations
 {
 
 double d2r(double degrees)
 {
     return 2 * M_PI * degrees / 360.0;
 }
 
 double r2d(double radians)
 {
     return 360.0 * radians / (2 * M_PI);
 }
 
 V3 V3::normal(const V3 &v1, const V3 &v2, const V3 &v3)
 {
     // First subtract d21 = V2-V1; d32 = V3-V2
     const V3 d21 = V3(v2.x() - v1.x(), v2.y() - v1.y(), v2.z() - v1.z());
     const V3 d32 = V3(v3.x() - v2.x(), v3.y() - v2.y(), v3.z() - v2.z());
     // Now take the cross-product of d21 and d31
     const V3 cross = V3(d21.y() * d32.z() - d21.z() * d32.y(),
                         d21.z() * d32.x() - d21.x() * d32.z(),
                         d21.x() * d32.y() - d21.y() * d32.x());
     // Finally normalize cross so that it is a unit vector.
     const double lenSq = cross.x() * cross.x() + cross.y() * cross.y() + cross.z() * cross.z();
     if (lenSq == 0.0) return V3();
     const double len = sqrt(lenSq);
     // Should we also fail if len < e.g. 5e-8 ??
     return V3(cross.x() / len, cross.y() / len, cross.z() / len);
 }
 
 double V3::length()
 {
     return sqrt(X * X + Y * Y + Z * Z);
 }
 
 V3 azAlt2xyz(const QPointF &azAlt)
 {
     // Convert the new point to xyz
     // See https://mathworld.wolfram.com/SphericalCoordinates.html
     const double azRadians = d2r(azAlt.x());
     const double altRadians = d2r(azAlt.y());
 
     const double theta = -azRadians;
     const double phi = (M_PI / 2.0) - altRadians;
     const double x = cos(theta) * sin(phi);
     const double y = sin(theta) * sin (phi);
     const double z = cos(phi);
     return V3(x, y, z);
 
 }
 
 QPointF xyz2azAlt(const V3 &xyz)
 {
     // Deal with degenerate values for the atan along the meridian (y == 0).
     if (xyz.y() == 0.0 && xyz.x() == 0.0)
     {
         // Straight overhead
         return QPointF(0.0, 90.0);
     }
 
     const double azRadians = (xyz.y() == 0) ? 0.0 : -atan2(xyz.y(), xyz.x());
     const double altRadians = (M_PI / 2.0) - acos(xyz.z());
 
     return QPointF(r2d(azRadians), r2d(altRadians));
 }
 
 V3 haDec2xyz(const QPointF &haDec, double latitude)
 {
     const double haRadians = d2r(haDec.x());
     // Since dec=90 points at the pole.
     const double decRadians = d2r(haDec.y() - 90);
 
     const double x = cos(decRadians);
     const double y = sin(haRadians) * sin(decRadians);
     const double z = cos(haRadians) * sin(decRadians);
     return rotateAroundY(V3(x, y, z), -fabs(latitude));
 }
 
 QPointF xyz2haDec(const V3 &xyz, double latitude)
 {
 
     const V3 pt = rotateAroundY(xyz, latitude);
     const double ha = r2d(atan2(pt.y(), pt.z()));
 
     V3 pole = Rotations::azAlt2xyz(QPointF(0, fabs(latitude)));
     const double dec = 90 - getAngle(xyz, pole);
     return QPointF(ha, dec);
 }
 
 V3 getAxis(const V3 &p1, const V3 &p2, const V3 &p3)
 {
     return V3::normal(p1, p2, p3);
 }
 
 // Returns the angle between two points on a sphere using the spherical law of
 // cosines: https://en.wikipedia.org/wiki/Great-circle_distance
 double getAngle(const V3 &p1, const V3 &p2)
 {
     QPointF a1 = xyz2azAlt(p1);
     QPointF a2 = xyz2azAlt(p2);
     return r2d(acos(sin(d2r(a1.y())) * sin(d2r(a2.y())) +
                     cos(d2r(a1.y())) * cos(d2r(a2.y())) * cos(d2r(a1.x()) - d2r(a2.x()))));
 }
 
 // Using the equations in
 // https://sites.google.com/site/glennmurray/Home/rotation-matrices-and-formulas/rotation-about-an-arbitrary-axis-in-3-dimensions
 // This rotates point around axis (which should be a unit vector) by degrees.
 V3 rotateAroundAxis(const V3 &point, const V3 &axis, double degrees)
 {
     const double cosAngle = cos(d2r(degrees));
     const double sinAngle = sin(d2r(degrees));
     const double pointDotAxis = (point.x() * axis.x() + point.y() * axis.y() + point.z() * axis.z());
     const double x = axis.x() * pointDotAxis * (1.0 - cosAngle) + point.x() * cosAngle + (-axis.z() * point.y() + axis.y() *
                      point.z()) * sinAngle;
     const double y = axis.y() * pointDotAxis * (1.0 - cosAngle) + point.y() * cosAngle + (axis.z() * point.x() + -axis.x() *
                      point.z()) * sinAngle;
     const double z = axis.z() * pointDotAxis * (1.0 - cosAngle) + point.z() * cosAngle + (-axis.y() * point.x() + axis.x() *
                      point.y()) * sinAngle;
     return V3(x, y, z);
 }
 
 // Simpler version of above for altitude rotations, our main case.
 // Multiply [x,y,z] by the rotate-Y by "angle" rotation matrix
 // as in https://en.wikipedia.org/wiki/Rotation_matrix
 //   cos(angle)  0   sin(angle)
 //       0       1       0
 //  -sin(angle)  0   cos(angle)
 V3 rotateAroundY(const V3 &point, double degrees)
 {
     const double radians = d2r(degrees);
     const double cosAngle = cos(radians);
     const double sinAngle = sin(radians);
     return V3( point.x() * cosAngle + point.z() * sinAngle,
                point.y(),
                -point.x() * sinAngle + point.z() * cosAngle);
 }
 
 V3 rotateAroundZ(const V3 &point, double degrees)
 {
     const double radians = d2r(degrees);
     const double cosAngle = cos(radians);
     const double sinAngle = sin(radians);
     return V3( point.x() * cosAngle - point.y() * sinAngle,
                point.x() * sinAngle + point.y() * cosAngle,
                point.z());
 }
 
 // Rotates in altitude then azimuth, as is done to correct for polar alignment.
 // Note, NOT a single rotation along a great circle, but rather two separate
 // rotations.
 QPointF rotateRaAxis(const QPointF &azAltPoint, const QPointF &azAltRotation)
 {
     const V3 point = azAlt2xyz(azAltPoint);
     const V3 altRotatedPoint = rotateAroundY(point, -azAltRotation.y());
 
     // Az rotation is simply adding in the az angle.
     const QPointF altAz = xyz2azAlt(altRotatedPoint);
     return QPointF(altAz.x() + azAltRotation.x(), altAz.y());
 }
 
 }  // namespace rotations