I am studying computer aided geometry and I have a background in mathematics. For me a (real) projective transformation is a map f:RP^n->RP^n induced by a linear isomorphism F:R^(n+1)->R^(n+1) since being injective it maps lines to lines (m-subspaces to m-subspaces). In the context of graphic design they usually never propey define projective maps, they usually project something onto a plane or use the following construction: Given (x,y,z)inR^3 we consider the point (x,y,z,1)inR^4 (i.e. a copy of R^3 on the affine hyperplane {w=1}, then we apply a bijective linear map on R^4 and project the image back on the hyperplane (which causes some trouble if the image has forth coordinate equal to zero). This process on R^3 is called a projective map. What is the link with my definition? I am sure i

Emma Hobbs

Emma Hobbs

Answered question

2022-11-08

I am studying computer aided geometry and I have a background in mathematics. For me a (real) projective transformation is a map f : R P n R P n induced by a linear isomorphism F : R n + 1 R n + 1 since being injective it maps lines to lines (m-subspaces to m-subspaces). In the context of graphic design they usually never propey define projective maps, they usually project something onto a plane or use the following construction:
Given ( x , y , z ) R 3 we consider the point ( x , y , z , 1 ) R 4 (i.e. a copy of R 3 on the affine hyperplane { w = 1 }, then we apply a bijective linear map on R 4 and project the image back on the hyperplane (which causes some trouble if the image has forth coordinate equal to zero). This process on R 3 is called a projective map. What is the link with my definition? I am sure it might involve considering homogeneous coordinates ( x , y , z ) [ x : y : z : 1 ], anyhow I don't get why the confused notation, they seem different things.

Answer & Explanation

Izabella Henson

Izabella Henson

Beginner2022-11-09Added 20 answers

The 1 is an artifact of homogenous coordinates and makes sure that your vectors are normalized with respect to that coordinate. The 1 should never be zero, or anything else for that matter. This is because the augmented matrix form of an affine transformation takes advantage of this for computational reasons and should be normalized at every step to ensure that the 1 is always there. The OpenGL transformation object is a 4 × 4 matrix for this reason. Even if you're not working in OpenGL directly your API probably has this transform object implemented.
One advantage is that you may now do vector addition with matrix multiplication permitting you to encode the matrix and the vector it acts on within the identical matrix. every other advantage is for the attitude projection which calls for homogenous coordinates to do efficiently. it's really the angle projection that prompted the examine of projective areas, and gave them their call. you could vaguely see how vanishing factors have some thing to do with traces via the beginning however you can make this concept extra unique by using reading the viewing frustum.
Sooner or later the OpenGL remodel permits you to encode transformation and positional records in a single matrix. this is truly useful while drawing objects relative to some different object, including an arm with joints. You need each joint to act independently but stay linked so one element will need position and transform records about the other to attract successfully. The affine remodel matrix representing where every a part of the arm is with regards to the joints will clearly lessen to this matrix multiplication.

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