How to interpret angle of Ellipse in drawing guide?

Question

Im a designer who, in my work, often uses drawing guides for drawing ellipses. A few nights ago I attempted to design some guides of my own but I was baffled as a I could not understand how to define my ellipses in the same way they are defined on the standardized drawing guides. On these guides the ellipses are described by an angle that ranges from 90 (circular) to 0+ (0 is just a line)

In the picture below, you can see a drawing guide with ellipses described as "45 degrees". I drew some lines of top of them and you can clearly see that neither A, B or C is 45 degrees (blue line is 45).

Im sure this is very simple for you guys but I cannot figure out how it works, thanks!

Answer 1

It's probably something related to the angular eccentricity $\alpha$ of the ellipse. If your extremes of 0 and 90° are correct, it would be $90^\circ-\alpha$ rather than $\alpha$ itself.

This would correspond to the intersection between your blue 45° line and the major axis being the focus of the ellipse, and the angle is then the angle between the major axis and the line that connects the focus to the end of the minor axis.

This also matches n00b's description of the angle between the line-of-sight and the plane of a circle that is foreshortened by perspective to have the aspect ratio of the ellipse.

Answer 2

$30 *\sin(\pi/4) = 21.213$, it's a circle viewed from an angle of 45 degrees

Answer 3

If you think of a straight line parallel to the ground coming out from your eye and hitting a perpendicular piece of paper with a circle on it, you get a 90° angle. As you tilt the paper away from you, there's a bigger angle on the side closest to you, and a smaller angle on the other side of the paper. That's the angle of the ellipse, and when the paper is parallel with you sight line the angle has reached 0°. Diagram

It's kind of hard to see it but there's a perfect circle on the paper on the desk that from this angle (15°) looks the same as the 15° ellipse drawn on the post-it note which is perpendicular to the camera.

Not the most mathematical explanation, but probably the most practical for any artists/designers who end up here from google wondering the answer to this like me.