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:

${\displaystyle {\begin{pmatrix}\mathrm {X} '\\\mathrm {Y} '\\\mathrm {Z} '\\\end{pmatrix}}={\begin{pmatrix}\cos \alpha &\sin \alpha &0\\-\sin \alpha &\cos \alpha &0\\0&0&1\\\end{pmatrix}}\cdot {\begin{pmatrix}\mathrm {X} \\\mathrm {Y} \\\mathrm {Z} \\\end{pmatrix}}}$

Similarly, for rotation of an angle α around the X axis:

${\displaystyle {\begin{pmatrix}\mathrm {X} '\\\mathrm {Y} '\\\mathrm {Z} '\\\end{pmatrix}}={\begin{pmatrix}1&0&0\\0&\cos \alpha &\sin \alpha \\0&-\sin \alpha &\cos \alpha \\\end{pmatrix}}\cdot {\begin{pmatrix}\mathrm {X} \\\mathrm {Y} \\\mathrm {Z} \\\end{pmatrix}}}$

And for rotation by the angle α around the Y axis:

${\displaystyle {\begin{pmatrix}\mathrm {X} '\\\mathrm {Y} '\\\mathrm {Z} '\\\end{pmatrix}}={\begin{pmatrix}\cos \alpha &0&-\sin \alpha \\0&1&0\\\sin \alpha &0&\cos \alpha \\\end{pmatrix}}\cdot {\begin{pmatrix}\mathrm {X} \\\mathrm {Y} \\\mathrm {Z} \\\end{pmatrix}}}$

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:

${\displaystyle {\begin{pmatrix}\mathrm {X} _{h}\\\mathrm {Y} _{h}\\\mathrm {Z} _{h}\\\end{pmatrix}}={\begin{pmatrix}\cos({\frac {\pi }{2}}-\phi )&0&-\sin({\frac {\pi }{2}}-\phi )\\0&1&0\\\sin({\frac {\pi }{2}}-\phi )&0&\cos({\frac {\pi }{2}}-\phi )\\\end{pmatrix}}\cdot {\begin{pmatrix}\cos(LMST)&\sin(LMST)&0\\-\sin(LMST)&\cos(LMST)&0\\0&0&1\\\end{pmatrix}}\cdot {\begin{pmatrix}1&0&0\\0&\cos(-\epsilon )&\sin(-\epsilon )\\0&-\sin(-\epsilon )&\cos(-\epsilon )\\\end{pmatrix}}{\begin{pmatrix}\cos(l_{\odot })\\\sin(l_{\odot })\\0\\\end{pmatrix}}}$

where:

${\displaystyle \phi }$: Latitude of the place of observation

${\displaystyle LMST}$: Local mean sidereal time

${\displaystyle \epsilon }$: Axial tilt

${\displaystyle l_{\odot }}$: Ecliptic longitude of the Sun

Projection of the shadow of a vertical gnomon

Let ${\displaystyle {\begin{pmatrix}0\\0\\L\\\end{pmatrix}}}$ be the Cartesian coordinates, in the local coordinate system, of the end of a vertical gnomon of length ${\displaystyle L}$.

The coordinates of the extremity of the shadow in the horizontal plane can be obtained with an affine transform parallel to the line by ${\displaystyle {\begin{pmatrix}\mathrm {X} _{h}\\\mathrm {Y} _{h}\\\mathrm {Z} _{h}\\\end{pmatrix}}}$ and ${\displaystyle {\begin{pmatrix}0\\0\\L\\\end{pmatrix}}}$.

Inclined and declined sundial

The Cartesian coordinates of the Sun in the system of coordinates bound to an inclined sundial of given declination are:

• ${\displaystyle {\begin{pmatrix}\mathrm {X} '_{h}\\\mathrm {Y} '_{h}\\\mathrm {Z} '_{h}\\\end{pmatrix}}={\begin{pmatrix}\cos i&0&-\sin i\\0&1&0\\\sin i&0&\cos i\\\end{pmatrix}}\cdot {\begin{pmatrix}\cos(-D)&\sin(-D)&0\\-\sin(-D)&\cos(-D)&0\\0&0&1\\\end{pmatrix}}\cdot {\begin{pmatrix}\mathrm {X} _{h}\\\mathrm {Y} _{h}\\\mathrm {Z} _{h}\\\end{pmatrix}}}$

where:

${\displaystyle D}$: declination of the plane of the sundial

${\displaystyle i}$: inclination of the sundial, that is, the angle of the normal with respect to the zenith