Perspective in Mathematics

[tame kernel]

Is there a mathematical approach to perspective projection?
I would like to study one-point, two-point, and three-point perspective from a geometric standpoint.

I want to understand mathematically:

How vanishing points represent a 3D object,
How the Station Point (SP) can be represented on a 2D plane,
Where to draw the Measuring Line (ML),
How to determine the angle of the Picture Plane (PP),
How to express Euler angles,
etc....

The reason I want to learn this is that I want to draw anime characters and architectural structures realistically using perspective
(this is not about computer graphics)
I want to approach these concepts through mathematics.

Are there any books or articles about perspective in mathematics?

[tame kernel]

I am curious about why it is drawn this way.

[proud sigil]

What an interesting topic

[proud sigil]

I draw too and i think this is a great idea and in anime prespective is so important since its a 2D artstyle

[tame kernel]

Exactly!

[tame kernel]

Of course, drawing is done by intuition.
However, I want to at least know approximately what the Euler angles are, and how far objects are,
through mathematics(?).

It's delusional, but I want to draw something similar to this using perspective:

[noble scaffold]

Optics/perspective is a really interesting field of geometry. The basic concept is this; model your eye as a point and the vertex of a cone, representing your field of view. The "apparent size" of an object, the amount of your visual field it takes up, is then not dependent only on its "actual size", but also its distance from your eye. You can see this in the classic example of how the Moon is the same apparent size as a quarter held at arm's length. This is because an object's "apparent size" is determined by the angle the rays from its edges make with the point of your eye.

Hence this concept is known as "angular size".

[tame kernel]

Would it be correct to say that perspective is fundamentally based on a cone?

[noble scaffold]

Possibly. Not sure about the phrasing.

[tame kernel]

How to find the vanishing point in 3D space

Derived based on

However, I don't understand why the three-point perspective is related to the orthocenter.

[tame kernel]

.

In my opinion, 90% of perspective seems to be VP

[noble scaffold]

There's no such thing as a vanishing point in 3D space.

Remember what I said, about your field of view being a cone with your eye at the vertex?

The "vanishing point" is then the center of the base of the cone. Except the cone has infinite height.

So there isn't actually a base.

[tame kernel]

https://tenor.com/view/maxwell-cat-cat-meme-silly-cat-silly-gif-4508364476527350133