Given a rectangle $ABCD,$ how do I calculate points $A, B, C, \; \text{and}\; D\;$ if I place the rectangle tangent to a sphere, centered at a given Latitude and Longitude, and given a "Rotation" which would be degrees clockwise from "North" on the sphere?
You can assume that the sphere is a Unit Sphere centered at the Origin.
I'm trying include this in a software application, so I would appreciate solutions suitable for programming, i.e. algebra or trigonometry rather than calculus.
Thank you!
If you are aware of matrices, you can construct your rotation matrix from simple rotation matrices. Let's have rectangle parallel to plane $yz$ (with $z$ being the axis of spherical coordinates), sides parallel to $y$ and $z$ and center on the sphere (you can derive the coordinates of the vertices yourself).
We first rotate rectangle around its central point (axis $x$) by angle $\varphi$: $$ \Phi=\begin{pmatrix} 0 & 0 & 0 \\ 0 & \cos\varphi& -\sin\varphi\\ 0 & \sin\varphi& \phantom{-}\cos\varphi \end{pmatrix}. $$ Then we rotate the rectangle by given latitude angle (around axis $y$): $$ \Theta=\begin{pmatrix} \phantom{-}\cos\theta& 0& \sin\theta\\ 0 & 0 & 0 \\ -\sin\theta& 0& \cos\varphi \end{pmatrix}. $$ Finally, we will rotate around axis $z$ by longitude angle $\lambda$: $$ \Lambda=\begin{pmatrix} \cos\lambda& -\sin\lambda&0\\ \sin\lambda& \phantom{-}\cos\lambda & 0\\ 0 & 0 & 0 \\ \end{pmatrix}. $$
The total rotation is the product of the matrices. Since we apply matrices to vector from left to right, the order is like this: $$ R = \Lambda \Theta \Phi $$
From programming point of view, you can either derive the final equation for the corners on paper and then write it in code. Or preferably, add multiplication by matrix as a function your code and apply matrix multiplication on demand. The latter case will give you more flexibility and is probably less prone to bugs.