#linear perspective

The shapes of circles are intuitive, the angles of them are less so.

The angle of an ellipse should match the angle of the surface that it is drawn on. However, the angle of a surface is not as simple as vertical or horizontal when the camera is at an angle. What makes things worse is that the lines surrounding a surface do a really bad job of telling you the angle of it. In perspective, we would look at the angle of a surface’s horizon to figure out its angle. Unfortunately, we don't have horizons in orthographic projection. Our best option is to look at the lines that are perpendicular to that surface instead.

To figure out what angle you should draw an ellipse on a specific surface, try to figure out what lines would be perpendicular to that surface, and draw the ellipse perpendicular to those. I’ve heard a lot of people phrase this like “when drawing a cylinder, the minor axis of the ellipses should align with the center of the cylinder”. I do not like this phrasing. I think that it focuses too much on cylinders specifically. I believe that phrasing it as finding the angle of a surface makes it easier to intuit how all 3d shapes should be drawn. Because if you want a drawing to feel like it has volume, then you’re going to use this rule constantly.

I’ve already talked a bit about the shapes of circles in perspective in ‘the basics part one’. However, just like in orthographic projection, figuring out what angle to draw them in is a bit more complicated. Many people seem to have adopted the orthographic method into perspective and will tell you that the minor axis of the ellipses should point towards the vanishing point that is perpendicular to it.

This is incorrect.

The method that I have found that seems most accurate is to have their major axis align with a hyperbola that gets steeper the further away from the horizon it gets.

I have unfortunately not been able to find a satisfying explanation as to why it works (a proper explanation would require you to calculate the intersections of cones at different angles, and I’m not a mathematician) so I will unfortunately not be able to provide one either. What I can do is prove that the other method is wrong.

When a horizontal surface is at eye level, it should perfectly align with the horizon. When using the first method, the ellipses will always be on a slight curve,  meaning that they can never align with a straight horizon.