apatriarca

It is 1-4 fps for me.. I tested it using a Quadro K1000M (it is a laptop GPU..).

Yeah, AFAIK, the word "space" doesn't belong there. We're contrasting 4D homogeneous coordinates to 3D/2D Cartesian coordinates.

 

Personally, I call it post-projection space, and after the divide by w, I call it NDC-space.

 

We (programmers) generally use "blah-space" to refer to some particular "basis" (view-space, world-space, post-projection space, etc...), but "w" comes into play not just because we're using some particular basis/space, but because we've switched from one coordinate system to another.

 

i.e. When working in object-space, view-space or world-space, we generally use 3D cartesian coordinates to identify points. When we're dealing with post-projection-space we switch over to using 4D homogenous coordinates to represent our points. We then transform them back to 2D cartesian coordinates in screen/viewport-space to identify pixels.

 

The reason we switch over to using 4D homogenous coordinates in order to implement perspective is because all our linear algebra (matrix/vector math) is... well... linear... whereas perspective is a non-linear effect! i.e. you want to scale something with "1/distance", which when you plot it, it isn't a straight line (not linear).

By working with 4D coordinates, where we say "we will divide by w later on", then we can continue to use linear algebra (matrices) to operate in a "non-linear space" (laymens terms, not formal math terms ).

 

Yes, I know. But homogeneous space is not a particularly useful search query in any case. I've actually always used the terms clip(ping) coordinates (and thus clip(ping) space) and normalized device coordinates (NDC) (and thus NDC-space) to denote the coordinates post-projection and post W-division in the graphics pipeline. The homogeneous coordinates has always been a more general term for me.

I understand that the way we use the homogeneous coordinates in the graphics pipeline is often not mathematically correct. We don't really care about what a projective space really is and what operation we can do on these coordinates. Some more advanced theory is however in my opinion sometimes useful to understand, for example, what is preserved by a projective transformation.

I understand that "homogeneous space" is how your book call it, but if you search that on google you get something completely different. "projective space" is a much better search query. This is the main reason I have introduced that name in my previous post.

I think you should simply look for something like "homogeneous coordinates computer graphics" on google. There are hundreds of beginner articles about these things already and it is not possible to explain everything is a simple forum post. If you don't care about mathematical concepts, then you should probably just understand how perspective projections are derived and basically consider the W division step as a technical trick. Indeed, if you compute the perspective projection equations, you will see that it is necessary to divide by z at some point. You can't divide by a coordinate using matrices. The trick is thus to write the denominator in a fourth coordinate and then divide by it after the matrix multiplication.

The additional coordinate is actually also used to make it possible to use matrices to represent translations but this has nothing to do with the notion of homogeneous coordinates. 

First of all, the correct (or at least more common) name of this space is (real) projective space and not homogeneous space. An homogeneous space is usually something different in mathematics*. I've never seen the term homogeneous space used in this context.

The term projective space comes from the fact it is the smaller/more natural space in which (central) projections can be defined. You can visualize this space as the set of lines passing through the origin in the (real) vector space one dimension larger. So the projective plane (the projective space of dimension two) is for example the set of 3D lines in the form

P(t) = t * (X, Y, Z).

Each line can be identified by any of its non-zero points. The homogeneous coordinates of a point of the projective space are the coordinates of one of the non-zero points of the corresponding lines and are written between square brackets (and often separated by a colon) [X0 : X1 : ... : Xn]. These homogeneous coordinates are not unique: two coordinates which differ by a non-zero scalar multiple represent the same point. The division by W is simply a way to choose a unique representative of a point. You basically represent all lines by their intersection with the plane W=1. All the lines parallel to that plane cannot be represented in this way. They are called points at infinity and they usually represent directions.. Note that dividing by W is just a convention and it is possible to divide by any other coordinate or use any other general plane. 

So, why using a space like that for projections? The main reason is that a projection transform a point with coordinates [x : y : z : 1] to some point [X : Y : Z : W] where W is no longer equal to one.. It is thus necessary to divide by W to get back a 3D point.. This is the main reason homogeneous coordinates have been defined and used in computer graphics.

The projection matrices are usually chosen so that the W coordinates basically represents the depth of the transformed point and the view frustum is transformed to the cube with all three coordinates inside [-1,1] after W divide (there are actually different conventions here between DirectX and OpenGL here..). But this is just a useful convention.

* It is usually used to denote some kind of space with a transitive G-action.. 

May I ask what your roll values actually means? Each 3D curve has a well defined local frame but that's clearly not the frame you are trying to compute. The usual Frenet or parallel transport methods are thus not directly applicable here.

It looks like you are using a fixed up vector which is then rotated around the tangent by some amount. But in the case you have displayed, the up vector is parallel to the tangent vector and that may cause problems.