Lets say I have a XYZ coordinate system, with a secondary coordinate system made with rotations of 15 degrees on the X axis and 5 degrees on the Z axis. On a sketch made on this new coordinate system in the X-Z plane, what kind of eclipse would I need to make in order to see a perfect circle when looking at the sketch from the ORIGINAL X-Z plane? As in, what major and minor radii would I need, as well as any other transformations afterwards?
I've been thinking about this for a while and it seems really simple when there's a rotation on only one axis, lets say the X axis. The major radius would change on the eclipse (in the Z axis direction) and would be equal to the following equation, while the minor radius would just be the original radius of the circle since there's no "shift" in that direction.
major radius = original radius / cosine (angle of rotation along X axis)
I just can't wrap my head around what changes when you rotate the coordinate system around two angles. Originally I thought the second angle would only affect the minor diameter and the two components only affected their respective radii on the eclipse, but the circle doesn't match perfectly when trying this on a CAD system. Is there a subsequent rotation to the eclipse that I need to make after or some combination of trigonometry?