Post by David Petry Post by Thomas Plehn
does the scanline algorithm on a 4 point polygon define a projectivity?
(if there are two return values)
Even after doing a considerable amount of google searching, I can't figure out what you're asking.
If you want an answer from people in this newsgroup, you should probably define the word "projectivity". But even then, I doubt many people here are familiar with scanline algorithms. I don't know what to suggest.
Thanks for all your efforts.
Because I am german, projectivity is a false translation. Perhaps, I should use the word homography. Homography arises in many image processing areas.
The 2d homography is defined by the 3x3 homography matrix (note the increase of one dimension). We even do this for example in image warping.
The displacement of the 4 corners describes a homography over the image domain. As I rememeber the displacement at any point of the image can calculated by means of bilinear interpolation from the corner displacements. However the function defined by this procedure is the homography defined by the corner displacements.
As I rememeber, every mapping of coordinates, which preserves the ratio how the distances between two points are devided by a third point on the line, is by definition a projectivity, even in higher spaces.
So I wonder if the following things are equal:
(i) it is a projectivity of a 4 point polygon, not necessaryly a rectangle
(ii) displacements are defined by the functions f_x(, f_y(
(iii) displacements of rectangles, can be calculated by bilinear interpolation
(iiii) diaplacements of other 4 point polygons can be calculated by interpolating linear first on the edges then between the new points
since every projectivity is defined by preserving the division of distances between points and we only have to prove for a 2d projectivity,
this property can be used to prove (ii), (iii), and (iiii)
we then know, how to ahndele a projectivity by means of linear algebray,
it seems a little bit confusing, that we calculate displacements instead of the absolute position. however by means of linear algebra, this can be achieved by adding the identity. Since displacements are small, the resulting homography is almost identity.