Is perspective a homothetic transformation?

[swampwiz]

I was reading about the homothetic transformation, and it seems that the perspective transform is a type of this.

 

[fresh_42]

Isn't this exactly the definition?
https://en.wikipedia.org/wiki/Homothetic_transformation

 

[swampwiz]

It doesn't seem to say that a perspective transform is a type of homothetic, although it sure looks like it.

 

[fresh_42]

It doesn't seem to say that a perspective transform is a type of homothetic, although it sure looks like it.

It does say it. They define a homothetic transformation as $M \mapsto \lambda \cdot \stackrel{\longrightarrow}{SM}$. That's exactly what a perspective does to a point $M$ as seen from $S$.

 

[swampwiz]

It does say it. They define a homothetic transformation as $M \mapsto \lambda \cdot \stackrel{\longrightarrow}{SM}$. That's exactly what a perspective does to a point $M$ as seen from $S$.

That's what I thought.

Now, what about affine transformations? Is the set of all Homothetic transformations also affine transformations? or vice-versa? It seems that since lines are preserved in a homothetic transformation, it is also an affine transformation.

 

[swampwiz]

On a side note, I have a question about a perspective projection in general. It seems that if I look at a line that is infinitely long, to see such at "infinity", I will need to look at it along the line in the exact same direction of the line, so there should never be a vanishing point, and essentially that a proper perspective projection is not linear. That said, if the differential of lines immediately in front of the camera being viewed is used, then there is a vanishing point, even though the real view would not vanish. This seems to be similar to the idea of using the paraxial approximation in geometrical optics, even though that breaks down for any system with any non-differential size. Is this accurate?

 

[fresh_42]

That's what I thought.

Now, what about affine transformations? Is the set of all Homothetic transformations also affine transformations? or vice-versa? It seems that since lines are preserved in a homothetic transformation, it is also an affine transformation.

They are affine transformations. The difference to linear transformations is only whether $S=0$ or $S\neq 0$.

 

[fresh_42]

I'm afraid I haven't understood your last post. This could be due to the fact that I haven't expertise in optics. What came to my mind while reading:

I will need to look at it along the line in the exact same direction of the line, so there should never be a vanishing point, ...

In reality there is something like resolution. Although it might not exist theoretically, there is a real margin below which we have indistinguishability.

... and essentially that a proper perspective projection is not linear.

No, it's affine linear, i.e. the origin is the center of projection and not the origin of the coordinate system.

I got the impression, that a perspective in your view is one whose center is at infinity, the other way around so to say. In this case the center does indeed not exist as part of the screen. A concept which deals with those infinite points is projective geometry where the horizon has a coordinate representation.