Coordinate system transforms - Change of bases The
Cartesian coordinates of the Sun in the
horizontal coordinate system can be determined by successive
changes of bases.
Expression as transformation matrices A
transformation matrix from a system B to a system B' allows for calculating the coordinates of a point or
vector in system B' when its coordinates are known is system B. For example, to change the system by rotating by an angle α around the Z axis, the coordinates in the new system can be calculated from those in the old system as: \begin{pmatrix}\mathrm{X}' \\ \mathrm{Y}'\\ \mathrm{Z}'\\ \end{pmatrix} = \begin{pmatrix} \cos \alpha & \sin \alpha & 0 \\ -\sin \alpha & \cos \alpha & 0 \\ 0 & 0 & 1 \\ \end{pmatrix} \cdot \begin{pmatrix} \mathrm{X}\\ \mathrm{Y}\\ \mathrm{Z}\\ \end{pmatrix} Similarly, for rotation of an angle α around the X axis: \begin{pmatrix}\mathrm{X}' \\ \mathrm{Y}'\\ \mathrm{Z}'\\ \end{pmatrix} = \begin{pmatrix} 1 & 0 & 0 \\ 0 & \cos \alpha & \sin \alpha \\ 0 & -\sin \alpha & \cos \alpha \\ \end{pmatrix} \cdot \begin{pmatrix} \mathrm{X}\\ \mathrm{Y}\\ \mathrm{Z}\\ \end{pmatrix} And for rotation by the angle α around the Y axis: \begin{pmatrix}\mathrm{X}' \\ \mathrm{Y}'\\ \mathrm{Z}'\\ \end{pmatrix} = \begin{pmatrix} \cos \alpha & 0 & -\sin \alpha \\ 0 & 1 & 0 \\ \sin \alpha & 0 & \cos \alpha \\ \end{pmatrix} \cdot \begin{pmatrix} \mathrm{X}\\ \mathrm{Y}\\ \mathrm{Z}\\ \end{pmatrix}
Model of the apparent movement of the Sun The Cartesian coordinates of the Sun in the horizontal system of coordinates can be calculated using
change of basis matrices: \begin{pmatrix}\mathrm{X}_h \\ \mathrm{Y}_h\\ \mathrm{Z}_h\\ \end{pmatrix} = \begin{pmatrix} \cos (\frac{\pi}{2}-\phi) & 0 & -\sin (\frac{\pi}{2}-\phi) \\ 0 & 1 & 0 \\ \sin (\frac{\pi}{2}-\phi) & 0 & \cos (\frac{\pi}{2}-\phi) \\ \end{pmatrix} \cdot \begin{pmatrix} \cos (LMST) & \sin (LMST) & 0 \\ -\sin (LMST) & \cos (LMST) & 0 \\ 0 & 0 & 1 \\ \end{pmatrix} \cdot \begin{pmatrix} 1 & 0 & 0 \\ 0 & \cos (-\epsilon) & \sin (-\epsilon) \\ 0 & -\sin (-\epsilon) & \cos (-\epsilon) \\ \end{pmatrix} \begin{pmatrix} \cos(l_\odot)\\ \sin(l_\odot)\\ 0\\ \end{pmatrix} where: \phi :
Latitude of the place of observation LMST : Local mean
sidereal time \epsilon :
Axial tilt l_\odot :
Ecliptic longitude of the Sun
Projection of the shadow of a vertical gnomon Let \begin{pmatrix} 0\\ 0\\ L\\ \end{pmatrix} be the Cartesian coordinates, in the local coordinate system, of the end of a vertical gnomon of length L . The coordinates of the extremity of the shadow in the horizontal plane can be obtained with an
affine transform parallel to the line by \begin{pmatrix}\mathrm{X}_h \\ \mathrm{Y}_h\\ \mathrm{Z}_h\\ \end{pmatrix} and \begin{pmatrix} 0\\ 0\\ L\\ \end{pmatrix} .
Inclined and declined sundial The Cartesian coordinates of the Sun in the system of coordinates bound to an inclined sundial of given declination are: • \begin{pmatrix}\mathrm{X}'_h \\ \mathrm{Y}'_h\\ \mathrm{Z}'_h\\ \end{pmatrix} = \begin{pmatrix} \cos i & 0 & -\sin i \\ 0 & 1 & 0 \\ \sin i & 0 & \cos i \\ \end{pmatrix} \cdot \begin{pmatrix} \cos (-D) & \sin (-D) & 0 \\ -\sin (-D) & \cos(-D) & 0 \\ 0 & 0 & 1 \\ \end{pmatrix} \cdot \begin{pmatrix} \mathrm{X}_h\\ \mathrm{Y}_h\\ \mathrm{Z}_h\\ \end{pmatrix} where: D :
declination of the plane of the sundial i : inclination of the sundial, that is, the angle of the normal with respect to the zenith ==Other uses==