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I've seen some algorithms with the following transformation matrix:

enter image description here

In such an algorithm, this matrix is used to draw some parallel circumferences on a sphere, i.e. the blue circumferences in the following figure: enter image description here

Each curve is approximated as a set of points (so each circumference is approximated with a polygon). Each point is defined with the spherical coordinates $\rho, \phi, \theta$.

enter image description here

I want to find out some informations about this matrix. It seems to contain the classical transformation matrix from cartesian to spherical coordinates:

enter image description here

but it as another coloumn at right. Why has this been inserted in my matrix?

$X_c, Y_c, Z_c$ express the centre of the sphere.

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  • $\begingroup$ Cartesian, after the Philosopher and Mathematician René Descartes. $\endgroup$ – Bazin Mar 20 at 16:59
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Too long for a comment. The mapping $\mathbb R_+\times (-π, π)\times (0,π)\ni(r,\theta, \phi)\mapsto (x,y,z)$ is given by\begin{cases} x=r \cos \theta\sin\phi\\ y=r \sin \theta\sin\phi\\ z=r\cos \phi \end{cases} and its differential is $$ \begin{pmatrix} \cos \theta\sin\phi& -r \sin \theta\sin\phi&r \cos \theta\cos\phi\\ \sin \theta\sin\phi&r \cos \theta\sin\phi&r \sin \theta\cos\phi\\ \cos \phi&0&-r \sin\phi \end{pmatrix}. $$ Permuting the second and third row, making $r=1$ you get $$ \begin{pmatrix} \cos \theta\sin\phi& \cos \theta\cos\phi& - \sin \theta\sin\phi\\ \sin \theta\sin\phi& \sin \theta\cos\phi& \cos \theta\sin\phi\\ \cos \phi&- \sin\phi&0 \end{pmatrix}. $$ The longitude lines are defined by $\theta=\theta_0$ and the latitude lines by $\phi=\phi_0$ (the latitude is $\frac π2-\phi_0$, positive on the northern hemisphere, negative on the southern). What was the use of your $4\times 4$ matrix?

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