Given an ellipse centred at the origin with major and minor axes and slope of the major axis specified:
How can I convert those three parameters into parameters that would express a scaled and sheared ellipse at the origin?
What I need is the width and height of the green parallelogram and the slope of the blue line when the red line is lying flat.
I might also be interested in versions with angles rather than slopes. I'm assuming it's easier with slopes but I could be wrong.
(I only have maybe highschool level maths so please excuse my ignorance of proper terminology. The images are not mine, just "close enough" ones I found on the net, the angles should actually match. I hope they are clear enough.)
Proposed rewording by @Blue.
A rotated ellipse can be interpreted as a (horizontally-)sheared ellipse. For instance, "an ellipse with radii $a$ and $b$, transformed by rotation through angle $\theta$" is just as well described as "an ellipse with radii $p$ and $q$, transformed by (horizontal) shear of angle $\phi$".
I want to know how to convert from one set of parameters to the other. That is,
$$\text{Given $a$, $b$, $\theta$, what are $p$, $q$, $\phi$?}$$
- It may be easier to express the amount of rotation and shear as slopes rather than angles.
- For the shear, I'm more interested the height of the bounding parallelogram ($q$ in the figure) not the transformed radius ($q^\prime$). (Of course, these are related by $q = q^\prime \sin\phi$.)
