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ALine

question about matrix transformations

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Is it possible to determine a transformation matrix from just a x and y matrix and the resulted transformed matrix?

Say for example you have a few points on an x and y graph. You transform these points into a matrix.

Lets also say that you have another graph with x and y points and you change that into a matrix.

Lets finally say that a transformation matrix exist between the first and second matrix, however you have no idea what it is.

Is there a possible means of solving for that said transformation matrix.

Also apologies for the slight vagueness, have some linear algebra knowledge however I need to study up on it some more in order to give more concrete terms.

 

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A matrix M can be a function that "transforms" points into new ones in the sense of y = M*x, where x is the vector to the original point, y the vector to the point x gets transformed to, * the multiplication of matrix and vector, and M the transformation matrix. Given enough pairs of original points and transformed points, it is possible to reconstruct the transformation matrix.

Please note the x and y in this case are not what they were in your post: Here, they are two vectors, while your term "x and y graph" suggests you refer to the x- and y-coordinates. If you have a single point (x,y) or a set of points (x1, y1), (x2, y2), ... then it is not clear what "transforming these points into a matrix" means. In other words: There is no such thing as transforming points into a matrix.

 

Bottom line: It is possible to reconstruct a transformation matrix when enough pairs or original and transformed vectors are given. At minimum, for an N-by-N square matrix at least N pairs of N-dimensional vectors must be given (it does not guarantee success and will fail e.g. if two pairs are identical). The reconstruction is done by solving systems of N equations with N unknowns for N times. [And if I had the time right now I would add the explicit equations for the 2D case - that should be pretty straightforward].

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In 3D computer graphics, we have couple typical transformations that can be performed using a matrix: translation, rotation, scaling, and projection from 3D to 2D, and eventually inversed projection from 2D to 3D. Conversion from object-space to world-space and reverse operation.

How to construct transformation matrix you can find in e.g. OpenGL and Direct3D tutorials and manuals. There is often explained entire math used to get said matrices. e.g.

https://docs.microsoft.com/en-us/windows/win32/direct3d9/transforms

https://docs.microsoft.com/en-us/windows/win32/direct3d9/viewports-and-clipping

How to make example 3D to 2D projection matrix in OpenGL:

http://www.songho.ca/opengl/gl_projectionmatrix.html

(OpenGL/Direct3D use 4x4 matrices with 4d vectors)

Even if someone is not interested in 3D computer graphics, but is a novice in matrices, should read these articles, because they are better (more clearly) written than by mathematicians.

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Posted (edited)
On 5/25/2020 at 6:20 AM, ALine said:

 

Also apologies for the slight vagueness, have some linear algebra knowledge however I need to study up on it some more in order to give more concrete terms.

 

I think you can find some contexts in Schaum's Outlines / Linear Algebra. (book series)

if you would like to further improve yourself,then I can recommend Algebra for you under the same source series.

Edited by ahmet

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On 6/11/2020 at 4:57 PM, ahmet said:

I think you can find some contexts in Schaum's Outlines / Linear Algebra. (book series)

if you would like to further improve yourself,then I can recommend Algebra for you under the same source series

Sorry for the long reply, thank you for the assistance. I will check them out.

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