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#Actualmax343

Posted 06 November 2012 - 05:34 PM

Scaling by [1,1,1,0] is just multiplying element-wise, or in matrix notation: diag(1,1,1,0)*[x,y,z,w]^T. Since there's a zero in one of the elements, the scale is singular.
A full fledged scaling will look something like this: [|d|, |d|, |d|, sqrt(d^2+||n||^2), 0], with 'n' being the normal of the projection plane and 'd' is the plane parameter.

I should also note, that this is only one interpretation of RP^3. There are more colorful, so to speak, interpretations of it. Though generally they're harder to grasp. However, the advantage of some of those interpretations is that they relate to objects which have a direct connection to R^3.

Anywho, textbooks. The kicker with projective geometry is that all textbooks (that I know of) that talk about RP^3 assume at least late undergrad (or mostly grad) level of math knowledge.
However, there's one notable book that describes these topics very well for an audience other than math students. For projective geometry they use a simpler model than RP^3, namely RP^2. But it shouldn't get in your way of understanding how projections work, since RP^2 is applicable as well to project 3D objects onto a plane.
The book's name is "Geometry, by David A. Brannan, Matthew F. Esplen, Jeremy J. Gray".

#1max343

Posted 06 November 2012 - 05:33 PM

Scaling by [1,1,1,0] is just multiplying element-wise, or in matrix notation: diag(1,1,1,0)*[x,y,z,w]^T. Since there's a zero in one of the elements, the scale is singular.
A full fledged scaling will look something like this: [|d|, |d|, |d|, sqrt(d^2+||n||^2), 0], with 'n' being the normal of the projection plane and 'd' is the plane parameter.

I should also note, that this is only one interpretation of RP^3. There are more colorful, so to speak, interpretations of it. Though generally they're harder to grasp. However, the advantage of some of those interpretations is that they relate to objects which have a direct connection to R^3.

Anywho, textbooks. The kicker with projective geometry is that all textbooks (that I know of) that talk about RP^3 assume at least late undergrad (or mostly grad) level of math knowledge.
However, there's one notable book that describes these topics very well for an audience other than math students. For projective geometry It uses a simpler model than RP^3, namely RP^2. But it shouldn't get in your way of understanding how projections work, since RP^2 is applicable as well to project 3D objects onto a plane.
The book's name is "Geometry, by David A. Brannan, Matthew F. Esplen, Jeremy J. Gray".

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