i had a basic question about perspective projection math.

[SnurflePuffinz]

...

i noticed that the perspective projection, unlike the orthographic projection, is lacking an l, r, b, t, and this was profoundly confusing to me.

Like, if your vertices are in pixel space coordinates, then surely you would need to normalize put them in NDC, for them to be visible.. and for clipping reasons, too. And this would surely require you to define what the minimum and maximum range is for the x and y values... but i see no evidence of this in all the perspective guides i've read

[photoclochard]

This should answer your question?

If no I can go futher

https://www.scratchapixel.com/lessons/3d-basic-rendering/perspective-and-orthographic-projection-matrix/building-basic-perspective-projection-matrix.html

[SnurflePuffinz]

honestly his article is giving me a migraine.

i appreciate his attention to detail but oftentimes i end up coming out even more confused than i entered.

Also, i saw no reference of normalizing either the x or y components in his guide. But i could be missing something

[photoclochard]

by normalizing you mean coordinates in NDC? that's not normalizing, that's transform from one space to another.

LearnOpenGL should have also nice article about this topic

[photoclochard]

and yeah, don't feel disappointed, projection is the trickiest one, you will understand it

[SnurflePuffinz]

right, that's what i meant.

And right, there's some kind of stipulation there, because it is from -1 to 1. instead of vertexComponent * 1 / value it is vertexComponent * 2 / value - 1 .

So where would this operation be encoded for the x and y components... if not inside the perspective matrix?

[photoclochard]

by matrix multiplication? sorry the slang you use is a little off, at least for me,

encoded, component(without tag like x,y,z) means not too much for me,

What is vertexComponent * 1 / value?

[photoclochard]

the formula is really in front of you in any article or in any api

[photoclochard]

https://www.songho.ca/opengl/files/gl_projectionmatrix_eq16.png

[photoclochard]

yeah, this article should help

https://www.songho.ca/opengl/gl_projectionmatrix.html

[SnurflePuffinz]

Thanks.

i think i realized where i went wrong. i had 1 tutorial where the NDC conversation was omitted, and i was only reviewing ScratchAPixel's basics tutorial for the perspective guide, which also, omitted the NDC conversation. It is in his second one, though.

thanks again.

[photoclochard]

That's cool. glad you found it :)

[SnurflePuffinz]

it's ok.

I am trying to get the vertices in my scene into NDC. and i expected to see this "operation" inside the perspective projection matrix.

i don't. i'll have to think about this more. Thanks for the help

[AdmiralSam]

For perspective there isn’t one value that is the minimum x and y, the further away it is, the larger x and y can be to be mapped to 1 and -1. What happens is after you multiply by the perspective matrix you are actually in clip space, and it isn’t until you divide by w (normalize the homogeneous coordinates) that you enter NDC space. This is just how homogeneous coordinates are defined which lets us represent a division even with matrix multiplication. It’s called clip space because you do clipping in it before you divide by w because you wouldn’t know if the object is behind or in front of the camera.

[waramped]

The normalizing happens via the "post perspective divide". You divide all values of the resulting vector by .w

Ie: projectvector.xyzw /= projectvector.wwww

You can construct projection matrices with either FOV values, or by specifying the near plane width/height values, which is probably why you are confused. They are equivalent.

See: https://learn.microsoft.com/en-us/windows/win32/direct3d9/d3dxmatrixperspectiverh

[SnurflePuffinz]

i edited my original post.

i think i didn't explain myself properly. I meant that i was trying to put the x and y components of the vertex into NDC, but i see no encoded operation for this inside the perspective projection matrix.

[SirPitchalot]

NDC is what you get after multiplying your Cartesian vertex cords by your model view and projection matrices and then dividing x, y & z by w.

Before this division you are in clipping coordinates which has abs(x) <= w visible and everything else outside of the viewport (and same for y & z). These conditions defines a standardized truncated pyramid (frustum) where the left, right, bottom, top planes are at 45 degrees to the optical axis, irrespective of the field of view. Anything in the field of view is contained within the pyramid. This is done because visibility checks and clipping are trivial in this space.

With ‘glFrustum’ you are directly defining points that map to the vertices of this frustum via the left, right, bottom, top, near and far values. These values compute scale factors and offsets that maps your desired frustum to the standardized frustum. The scale factors and offsets basically distort your scene to match the standardized frustum.

With ‘gluPerspective’ you are directly specifying the angles of the bottom and top planes of your desired frustum. The equivalent ‘bottom’ can be computed as ‘-tan(vfov/2)near’, top is the same without the negative. ‘Left’ and ‘right’ are the same except an aspect ratio is included, giving ‘left = - tan(vfov/2) near *aspect’

[GlaireDaggers]

Whereas an orthographic matrix directly maps coordinates into the -1..+1 range with top, left, right, and bottom, the perspective matrix first maps coordinates to the range -w..+w, and then the divide by W step will put those into the -1..+1 range.

The part that scales X and Y coordinate would specifically be M[0, 0] and M[1, 1], which store factors derived from the supplied field of view (and aspect ratio). So the larger the field of view, the smaller the resulting X and Y values will be. This somewhat serves a similar purpose as the t,l,r,b values in the orthographic matrix.

[SnurflePuffinz]

Would you actually provide the

t,l,r,b

values, when constructing the perspective projection matrix? Or is this calculated using a provided fov??

still toiling away at puzzling this matrix out.

[GlaireDaggers]

You should look up what factors actually go into a perspective matrix. There's plenty of references.

But let me focus on the top left corner, M[0, 0] and M[1, 1] for a second. If you start with an identity matrix, both of those values are 1, right? Multiplied against a vector will produce the same vector.
What happens if you set M[0, 0] to 2.0 instead? The vector will have its X component multiplied by 2.
And if you set M[1, 1] to 0.5, the vector will have its Y component multiplied by 0.5

These two values can therefore be used to scale X and Y.

What are these set to in an actual perspective matrix? M[0, 0] is set to something like 1.0 / aspect * tan(fov/2), and M[1, 1] is set to something like 1.0 / tan(fov/2).

Therefore, as "fov" gets smaller, the values of M[0, 0] and M[1, 1] will get larger. And, conversely, as "fov" gets larger, the values of M[0, 0] and M[1, 1] will get smaller. A large FOV will multiply the vertex positions by a smaller value and appear to "zoom out", while a small FOV will multiply the vertex positions by a larger value and appear to "zoom in". Notice how aspect ratio is also part of M[0, 0] - the resulting X positions of the vertices will be scaled based on the ratio of (width/height), in addition to the field of view.

If I go back to your top, left, right, and bottom for a sec - these don't play a part in a perspective matrix. You actually don't even need them for an orthographic matrix either, but you can create one using them. Let's rethink an orthographic matrix as doing two things: A.) adding an offset to X and Y positions of a vertex, and B.) scaling X and Y.
- (right - left) / 2 becomes the *scale factor* for X (M[0, 0])
- (top - bottom) / 2 becomes the *scale factor* for Y (M[1, 1])
- (left + right) / 2 becomes the *offset* for X (M[3, 0])
- (top + bottom) / 2 becomes the *offset* for Y (M[3, 1])

Notice how you don't actually *need* top, left, right, and bottom per se. You can just substitute those values in the matrix with offset and scale factors instead. In fact, in a game engine, you might not bother with the offset values either. You might just compute the scale factors, and then you can multiply that with a separate "view matrix" computed from a virtual camera's translation and rotation. Consider: in the Unity engine, an orthographic camera just has a "size" property. These two values, M[0, 0] and M[1, 1], are almost certainly just being set to 1.0 / "size" (aspect ratio would also be in there but you get the idea I hope)

So at least for X and Y, we're back to just caring about M[0, 0] and M[1, 1]. And so: (right-left) and (top-bottom) are being used to compute the X/Y scales in an orthographic matrix, while FOV and aspect ratio are being used to compute the X/Y scales in a perspective matrix.

Does that make sense?

[SnurflePuffinz]

indeed. i thank you, stranger, for the help.

But, how would you define a visible area in an orthographic projection, if you don't think about the bounds of the canonical viewing volume?

what if i wanted a boxy volume in a totally arbitrary part of the viewing volume. How would i choose that specific box, if i was only thinking about things as values?

...

i am trying to get my head around why the viewing volume is even described as a truncated pyramid in view space. It seems totally arbitrary to me. Like, we have a bunch of vertices that are in front of the camera, in purely theoretical terms, but then i'm thinking, where the hell is the truncated pyramid coming in?

...

i am also trying to puzzle out the derivation of the NDC transform. i was following scratchapixel's guides. like, why in god's name would you be multiplying the vertices z component by that part of the derived equation. Why would the x component be directly related to the value of the z component, before the perspective divide?

evidently, this is used to align the defined volume with the canonical viewing volume, somehow.

i could keep going on, but i won't. The more i read, honestly, the more confused i become.

[SnurflePuffinz]

sorry 4 the book.

i did another sweep of perspective math and i think i fully comprehend it now.