# Why are vanishing point and centre of projection the same [closed]

We know that in perspective projection object positions are transformed to the view plane along lines that converge to the projection reference (center) point like this:

But many websites say that COP(Centre of projection) and vanishing point both are the same. But we know that parallel lines that are not parallel to the viewing plane converge to a vanishing point. In the above picture, if I draw the parallel line through the point $$A, B, C$$, then they intersect at infinity which is the vanishing point.

My question is: How are COP and vanishing point both the same?

• "But we know that parallel lines that are not parallel to the viewing plane converge to a vanishing point." This sentence confuses me and I do not know where you got it from or what you are trying to say. From Wikipedia: " two lines in three-dimensional space which do not meet must be in a common plane to be considered parallel". So there shouldn't be any COB even in infinity cause this requires that the lines intersect/meet. Oct 20, 2021 at 14:18
• @wychmaster See this in perspective projection section.
– user17390
Oct 20, 2021 at 14:29
• I will take a look later and report back tomorrow if nobody else answers. Oct 20, 2021 at 14:36
• @wychmaster COB full form?
– user17390
Oct 20, 2021 at 14:38
• The fact that many websites have misinformation is nothing new. Vanishing point is a 2D point on the image plane. Center of Projection is a 3D point in space.
– Wyck
Oct 21, 2021 at 16:26

The answer is: they are not the same. Here is a related question from another site where somebody asked what the difference between both of them is (found the sentence I asked about in the comments there too). This is the main reference my answer is based on.

From Wikipedia:

"A vanishing point is a point on the image plane of a perspective drawing where the two-dimensional perspective projections (or drawings) of mutually parallel lines in three-dimensional space appear to converge."

The centre of projection (COP) is not on the image plane as you can see on the image you posted. It is the focal point where all your projection lines converge. If you place it on the image plane you would create a singularity and the whole projection math would fail.

However, the location of the vanishing point on the 2d plane depends on the position of your COP. Take the drawn cube on the projection plane of your image as an example. The top plane has 2 vanishing points because it has two sets of parallel lines. Now if you rotate the COP left or right around the 3d cube until one set of lines of the top face is parallel to the projection plane you would observe the following: One vanishing point would move directly to the 12 o'clock position while the other would accelerate away from the plane until it disappears because as you wrote:

parallel lines that are not parallel to the viewing plane converge to a vanishing point

Lines parallel to the projection plane do not have a vanishing point.

So how does it come that the internet says otherwise? My educated guess here is, that like with many other things, people start mixing up terms of closely related things. Also, you provided this link here as a reference. Strictly, they are not saying both are the same. They simply just name them identically. But that doesn't make it any better.

UPDATE A noteworthy difference between COP and vanishing point (as mentioned by Wyck in the questions comments) is that the vanishing point is actually a 2d point while the COP is a 3d point. One can deduce it from my answer, but I felt like it should be mentioned explicitly.

• ♦ vanishing point isn't not actually 2d point, it is points in projective plane $P^2$
– user17390
Oct 21, 2021 at 17:21
• @N-Sat well, a plane is a 2 dimensional space. Any point in that 2 dimensional space is a 2 dimensional (2d) point ;) Oct 21, 2021 at 17:30