Ellipse construction in perspective
2yr
Philipp Meyer
Hi,
lately I had some doubts about the construction of ellipses in cylinders. In the art books dealing with perspective (e.g. Scott Robertson - How to draw, p.72) it is said that the minor axes of the ellipses will vanish to the same vanishing point as the middle axis of the cylinder and its sides. I was not fully convinced and therefore did some research on the topic. I found a math forum where people were discussing just that topic. The mathematicians seem to be of the opinion that the minor axes do have a different vanishing point than the sides of the cylinder without actually giving a proof. Check the thread here: https://math.stackexchange.com/questions/3823048/is-the-line-created-by-the-minor-axis-of-an-ellipse-concurrent-to-the-lines-runn
Does anybody have an actual formal proof for the theory of the minor axes sharing the vanishing point with the sides of the cylinder?
I did some drawings and tried to be accurate with the construction. In my drawings the vanishing points of the minor axes and the sides of the cylinder aligned. My drawing process was first constructing a cube in perspective, then transferring the cube and the vanishing points to another sheet of paper and constructing the ellipses inside of the squares in perspective. After inserting the minor axes I checked whether they were in the correct spot or not with a small mirror.
Actually that rule is valid (also very approximately) in very few cases. In most of the cases center of the ellipse will be located somewhere else: sometime on one of the diagonals, sometimes on the middle line of the square.
The rule mentioned by Robertson and others is a rule of thumb. It is not mathematically correct. However, it is the only practical guideline known to mankind, so let's stick to it...
This rule has a very easy explanation. Even if the information following your link is correct and true, the human eye is capable to see in focus only limited portion of the whole picture. So in order to draw a cylinder, one would look at it in a way, that the object (in our case ellipse) is in the middle of our sight. And in that case the rule with minor axis is 100% valid. To draw an ellipse on the other side of the cylinder one would again move eyes to see it clearly. And the rule will work again. So in fact people, when they create a picture, they fake the perspective. And only if you follow strictly mathematical approach you would get that confusing situation. And that also happens if you create mathematically correct drawing with a very wide angle of view (bigger that 30 degrees), which human eye is not even capable to catch. I hope that explanation was clear enough.
Can you give an explanation or a proof that it is just a rule of thumb and not mathematically correct?
