Discussion:
Projectivity in 2D on a Rectangle
Thomas Plehn
2017-06-15 15:00:10 UTC
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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)
David Petry
2017-06-15 17:49:18 UTC
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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.
Thomas Plehn
2017-06-15 18:33:50 UTC
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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.
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.
Thomas Plehn
2017-06-16 11:00:05 UTC
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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.
https://en.wikipedia.org/wiki/Perspectivity
Thomas Plehn
2017-06-16 11:09:50 UTC
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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.
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
Thomas Plehn
2017-06-16 11:35:40 UTC
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Post by Thomas Plehn
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
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?
Post by Thomas Plehn
and if such is true, can we calculate this coefficients from the
perspectivity matrix
Tim Golden BandTech.com
2017-06-16 19:12:32 UTC
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Post by Thomas Plehn
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.
does the scanline algorithm on a 4 point polygon define a projectivity?
(if there are two return values)
It is my sincere opinion that projection as defined in mathematics is a flawed definition; as I recall they insist upon a P P^-1 = I or some such nonsense.
The fact is that a projection down to two dimensions from three dimensional space does not carry any reverse function that returns the original object. The proper projection from 3D to 2D is simply via the multiplication of a 3D point position series by a 3x2 matrix to yield a 2D point series. In this way the 3x2 matrix literally is the projection and variations upon it yield different images.
Thomas Plehn
2017-06-17 12:36:15 UTC
Raw Message
Post by Tim Golden BandTech.com
Post by Thomas Plehn
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.
does the scanline algorithm on a 4 point polygon define a projectivity?
(if there are two return values)
It is my sincere opinion that projection as defined in mathematics is a flawed definition; as I recall they insist upon a P P^-1 = I or some such nonsense.
The fact is that a projection down to two dimensions from three dimensional space does not carry any reverse function that returns the original object. The proper projection from 3D to 2D is simply via the multiplication of a 3D point position series by a 3x2 matrix to yield a 2D point series. In this way the 3x2 matrix literally is the projection and variations upon it yield different images.
a perspectivity is a special geometric map. It is a collineation, which
preserves ratios.
That mean, all points on a line are mapped to points on another line.
And all ratios of that points on the original line are preserved on the
other line.
Thomas Plehn
2017-06-17 12:58:26 UTC
Raw Message
Post by Thomas Plehn
Post by Tim Golden BandTech.com
Post by Thomas Plehn
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.
does the scanline algorithm on a 4 point polygon define a projectivity?
(if there are two return values)
It is my sincere opinion that projection as defined in mathematics is
a flawed definition; as I recall they insist upon a P P^-1 = I or some
such nonsense.
The fact is that a projection down to two dimensions from three
dimensional space does not carry any reverse function that returns the
original object. The proper projection from 3D to 2D is simply via the
multiplication of a 3D point position series by a 3x2 matrix to yield
a 2D point series. In this way the 3x2 matrix literally is the
projection and variations upon it yield different images.
a perspectivity is a special geometric map. It is a collineation, which
preserves ratios.
That mean, all points on a line are mapped to points on another line.
And all ratios of that points on the original line are preserved on the
other line.
https://en.wikipedia.org/wiki/Homography