Difference between perspective and parallel projection [closed]

Question

We know that in orthographic Projection Projectors (projection vectors) are perpendicular to the projection plane.

And in Perspective Projection Object positions are transformed to the view plane along lines that converge to the projection reference (center) point.

The Point $(x,y,z)$ is projected to position $(x_p,y_p,z_{vp})$ on the view plane.Since the view plane is placed at position $z_{vp}$ along the $z_v$ axis.

Then in parallel projection any point $(x,y,z)$ in viewing coordinates is transformed to projection coordinates as: $x_p=x,$$y_p=y$ and $z_{vp}=0.$

My question is why $(x,y,z)$ is projected to $(x_p,y_p)$, why not $(x_p,y_p,z_{vp}),$ I mean view plane is placed at position $z_{vp}$ ,so why $z_{vp}=0$ in $(x_p,y_p,z_{vp})?$ And another question in perspective projection a why $(x,y,z)$ is projected to $(x_p,y_p,z_{vp})?$

My final question is that in perspective projection projectors are perpendicular to the viewing plane?

N. B. - Reference

Answer 1

In OpenGL parlance, the projection matrix, whether orthographic or perspective, takes you from View Space (a.k.a. Camera Space) to Homogenous Clip Space. After clipping you arrive at Normalized Device Coordinates (also known as NDC) and these points are still 3D $(x_p, y_p, z_{vp})$ and finally it's the viewport transform that transforms NDC to Viewport coordinates (a.k.a. screen coordinates, window coordinates or (not normalized) device coordinates) which are 2D $(x_p, y_p)$.

May I suggest reading: OpenGL Transformation

And with respect to your final question: No, perspective "projectors" (projected rays) are not all perpendicular to the viewing plane - that would require a curved viewing surface. If all the rays projected from a point were perpendicular to the viewing surface, that surface would be a section of a sphere. There is one line that passes through the center of projection and meets the image plane at a right angle and that is the axis of projection arriving at the viewing plane at the principal point.

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More info:

Transformation just means change. With respect to 3D points it usually means the output bears some resemblance to the input. e.g.: Translation, Rotation, Scale, Skew, or even Projection is a kind of transformation. If the output can be transformed back to the input then the transformation is invertible. Projection is a specific kind of transformation that moves all the input points onto a plane. Either to the closest point (orthographic) or along a lines passing through a common projection point (perspective). The projection transformation is not invertible.

A transformation is said to be linear if it can be represented by a matrix multiply.

Here's the confusing part. To perform the perspective transformation, one must perform a non-linear step of dividing by Z. This can't be represented by a matrix. To get around this, we transform all the coordinates linearly into what is called clip space. And then the pipeline includes a non-linear transformation from clip space to normalized device coordinates by performing the division operation. The so-called PROJECTION MATRIX (in OpenGL) is actually just a linear transformation from camera/eye coordinates to clip coordinates. It's only the first of two steps required to perform an actual projection transformation. The actual projection transformation is completed in a second step by doing the divide, and arriving at NDC coordinates. This divide cannot be represented by a matrix multiply. The linear part of the transformation produces the numerator and the denominator of the quotient in separate coordinates. The non-linear part of the transformation divides.

Interestingly, the linear transformation from camera coordinates to clip coordinates (as given by the PROJECTION MATRIX) is invertible. And after dividing by Z it's still invertible (provided we retain the Z value). But after we discard the Z-value and go to just Screen coordinates, we can no longer invert the 2D screen-coordinates back into the original 3D coordinates. Similarly, if we actually transformed all the points onto the projection plane (still in 3D), we wouldn't be able to invert that transformation of points on the plane back into the original 3D points (information has been lost, because it's been scaled down to 0 in the "depth" direction).

Even after dividing, we're still in 3-dimensional coordinates. The third coordinate is still useful for writing to your depth buffer.

Only when we are ready to compute screen coordinates do we drop the Z value and use just 2-dimensional coordinates.

Other transformations such as perspective texture mapping (like to simulate a projector) follow a similar path and have a non-linear divide step to arrive at texture mapping coordinates (u,v).

So, to answer your question, in the pipeline we try to keep invertible 3D coordinates around as long as possible in the pipeline, deferring the divide as late as possible and discarding the Z information as late as possible so that things are linear for as much of the pipeline as possible (thanks to homogenous coordinates) and are invertible as long as possible (by retaining the depth coordinate in NDC space).

Answer 2

While it is often the case that a parallel projection such as an orthographic projection has all zero's in one row/column of a matrix, it is also easy to set a projection plane other than zero, such as the near plane which is often done in CG for things like shadow maps of infinitely distant point lights. (think cascade shadow maps)

Also, projection planes are very frequently perpendicular to the view plane but this is just the norm. It is possible and sometimes desirable to have an oblique projection plane. Oblique parallel projections are a specific subtype. See this paper for an example of an oblique perspective projection.

These terms are so overloaded in the industry that it is easy to start talking apples and oranges. So just be clear my terminology is that planar projections are the superset of all perspective and parallel projections. Under perspective projections, there are 1,2, and 3 point projections. Most people are referring to 3 point projections when they say "perspective projection". Under parallel projections, there are orthographic, axonometric, and oblique. So when you say parallel it sounds like you are referring to a large assortment of projections. Then again some folks refer to axonometric as parallel, it really depends on the industry their school, and where they work.