• #### homography of 4 point polygon by interpolation or shape functions

From thomas.plehn@gmail.com@21:1/5 to All on Fri Jun 16 10:29:06 2017
from sci.math

The rectangle is big, the return values are small

the following is all the same

(i) projectivity on rectangle

(ii) bilinear interpolation on a rectangle

(iii) functions f_x(x,y)=1*g+x*a+y*b+xy*c, f_y(x,y)=1*h+x*d+y*e+xy*f

(iiii) all of them are linear on all rays between two points

For some applied context, I need to know if that is correct.

and even:
does the scanline algorithm on a 4 point polygon define a projectivity?
(if there are two return values)

Am Donnerstag, 15. Juni 2017 19:49:26 UTC+2 schrieb David Petry:
On Thursday, June 15, 2017 at 8:00:31 AM UTC-7, Thomas Plehn wrote:

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

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.

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.

The desired word is perspectivity

The scanline algorithm should only suggest interpolating a 4 point
polygon line by line. First on the edges, the on a horizontal line
connecting the edges.
This is a generalization of bilinear interpolation.
What I want to ask, is, if that is a perspectivity in 2d space, even the perspectivity defined by the mappings of the 4 points.

The second question is, can any such 2d perspectivity of 4 points be
described as
f_x(x,y)=g+a*x+b*y+c*xy
f_y(x,y)=h+d*x+e*y+f*xy

and if such is true, can we calculate this coefficients from the
perspectivity matrix

The second question is, can any such 2d perspectivity of 4 points be described as
f_x(x,y)=g+a*x+b*y+c*xy
f_y(x,y)=h+d*x+e*y+f*xy

as you can see, such pair of functions can also be described by 4
definition points. Is it the same as the perspectivity defined by such 4 points.

and is the generalized bilinear interpolation sheme of a 4 point polygon describing the perspectivity defined by such 4 points?

and if such is true, can we calculate this coefficients from the perspectivity matrix

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