This is a really basic question, yet I can't remember my old geometry classes nor could I find an answer via google.

Given a circle "tilted" at angle a to the horizontal plane, and given angle b inside the circle, I want to calculate the vertically projected angle, c, on the horizontal plane.

It's obvious that the circle's vertical projection on the plane is an ellipse, and that cos(b) = cos(c) but I still can't work out the formula that gives me c given only the 2 first angles.

I'm assuming this can be worked out regardless of the dimensions of the circle, although temporary arbritary dimensions can be used to calculate the solution.

projected angle

Note: As an example, in the above sketchup drawing a = 70 degrees, b = 45 degrees, and (what I want to calculate mathematically) c = 18.8 degrees.


Imagine that the circle is centered at the origin of a (titled by angle $a$) cartesian $x$-$y$ plane, with the $x$-axis along the intersection of the circle’s diameter and the ellipse’s major axis. Projection of points from the tilted plane onto the horizontal plane leaves $x$-coordinates alone and scales $y$-coordinates by a factor of $\cos a$. So the point $(\cos b,\sin b)$ (on the circle at angle $b$) is projected onto the point $(\cos b,\cos a\sin b)$ on the horizontal plane. Then $\tan c=\frac{\cos a \sin b}{\cos b}$, or $c=\arctan(\cos a \tan b)$. Note that $\arctan(\cos 70^\circ \tan 45^\circ)\approx18.882^\circ$

  • $\begingroup$ Brilliant. Thank you! $\endgroup$ – dude Mar 18 '15 at 3:16

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.