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What is homogeneous space?

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Can someone explain me what purpose do 4x4 matrix have? What is homogeneous space and homogeneous coordinates?

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quote:
Can someone explain me what purpose do 4x4 matrix have?

They are used to speed up the necessary calculations to transform a vertex from world space to screen space. Why? Because you can bundle as many linear transformations (rotations, transforms, etc.) as you want into one matrix by multipling the matrices for the specific operations.

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quote:
Original post by g3d
Can someone explain me what purpose do 4x4 matrix have?

The 4x4 matrices that are used in 3D graphics represent affine transformations.

quote:
Original post by g3d
What is homogeneous space and homogeneous coordinates?

By "homogeneous space" you are probably refering to what I call "projective space". You start with a vector space, like R^4. Take the origin out and define the equivalence relationship "to be proportional". The set of classes of that equivalence is called "projective space", and the classes are usually called "points". You can express a point in projective space by the coordinates of one of the representatives of the class. If you multiply the coordinates by any non-zero constant, you get another element of the class. We call these coordinates where the scale doesn''t matter "homogeneous coordinates".

I took an entire year of Projective Geometry in college. There is really a lot you can do with this concept.

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As Alvaro implied, but didn''t say, there is no homogeneous space. There is a homogeneous representation of some spaces. And while you can capture projective space with it, it''s not the only thing you can do. It''s "homogeneous" because you can specify vector transformations (rotation, skew, scaling, etc.) as well as translations in a single representation (an nxn matrix of rank 1 greater than the vector space).

The homogeneous representation can model a few different spaces: affine, projective, and Grassman. The way things work generally in games, you can consider affine space to be the transformations you do before perspective projection. Projective space is what it sounds like. But it''s a very interesting space that I''m sure Alvaro can tell you all kind of interesting things about. (For those who don''t know, in projective space, a parabolas, hyperbolas, and ellipses are the same class of beast - one can be transformed into another.) But Grassman spaces are very interesting too. The space is similar to perspective and sometimes gets mixed up in it, but it has some nice qualities on its own.

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A 4x4 matrix can be used to solve a system of 4 equations with 4 variables I know that''s not what you meant though.

4x4 matrices are used in 3D graphics because it''s not possible to make a 3x3 matrix that will support certain transformations (well translations in particular) for any given vector. In other words, it''s possible to derive one 4x4 matrix that will translate n points each by the same amount, wheres you would need n m3x3 matrices to do the same thing.

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YoshiN: you can''t translate 3D points with a 3x3 matrix at all.

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Just today a lecturer of mine mentioned something about the homogenuity of space meaning that any point in space is as good as any other point: the underlying physics remains the same. At least thats how I understood it.

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That''s a different topic entirely. Homogeneous in this discussion refers to the fact that the same representation (namely a 4x4 matrix) may represent vector transformation, translation, and perspective transformation. It also refers to the homogeneous representations of vectors and points in a vector of 1 degree greater than the dimension of the space. The physics concept to which you refer is that of "inertial frames":

http://plato.stanford.edu/entries/spacetime-iframes/

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quote:
YoshiN: you can't translate 3D points with a 3x3 matrix at all.

Hell yeah you can AP.. in fact you can only transform 3d points with 3x3 matrices and i dare you to prove otherwise.

However, in 3d homogenous space, were actually in 4d (x,y,z,w), and thats why you need a 4x4

[edited by - psamty10 on March 9, 2004 7:21:41 PM]

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If I remember correctly, "homogeneous" comes from which polynomials describe algebraic manifolds in the projective space.

If we describe a curve in a projective plane using a polynomial in three variables, it is required that if the polynomial is zero for (x,y,z) it also be zero for (a*x,a*y,a*z), where a is an arbitrary number. It is easy to prove that the polynomial must be homogeneous (all terms should have the same degree) for that condition to be satisfied.

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