Figuring Out Affine Transformations: Frustrating But Doable

In summary, the conversation discusses determining the affine transformations used to produce a given image. The image is scaled by 1/3 and there are 3 linear transformations that put it at specific points. The challenge is finding the transformations for the vertical edges. The solution involves removing the y-coordinates, scaling by 1/3, and rotating the image. A suggestion is made to try a projection on the x-axis. The final transformation is described as f(x)= 0 if 0< x< 1/3, f(x)= 1 if 1/3< x< 2/3, and f(x)= 0 if 2/3< x< 1.
  • #1
srn
17
0
I need to determine the affine transformations used to produce the following image:

KIAsx.png


Been staring at it for an hour and it's frustrating me to no end because it's probably really easy. Clearly it gets scaled by 1/3 and there are 3 linear transformations that put it at (0,0), (1/3,1/3) and (2/3,0) but I don't know how to produce the vertical edges. It's not iterating anything, it's just a line that keeps getting bigger... Take the original image, remove the y-coordinates, scale by 1/3 and rotate, or something?
 
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  • #2
Yes, try some kind of projection on the x-axis.
 
  • #3
Got it. Thanks, guess I just needed a little push in the back. :)
 
  • #4
It's clear that the basic transformation is something like
f(x)= 0 if 0< x< 1/3
f(x)= 1 if 1/3< x< 2/3
f(x)= 0 if 2/3< x< 1.
 

What is an affine transformation?

An affine transformation is a type of transformation that preserves parallel lines and ratios of distances. It can include translations, rotations, reflections, and dilations, as long as it does not involve shearing or scaling.

Why is figuring out affine transformations frustrating?

Figuring out affine transformations can be frustrating because it requires a deep understanding of mathematical concepts such as linear algebra and matrix operations. It also involves a lot of trial and error and can be time-consuming.

How do I know if I have correctly figured out an affine transformation?

You can check if you have correctly figured out an affine transformation by applying it to a set of points and seeing if the resulting transformation preserves the properties of an affine transformation, such as parallel lines and ratios of distances.

What are some applications of affine transformations?

Affine transformations are commonly used in computer graphics, image processing, and computer vision. They can also be applied in areas such as robotics, physics, and engineering.

Are there any tools or resources that can help with figuring out affine transformations?

Yes, there are many online resources and software programs that can assist with figuring out affine transformations. Some examples include WolframAlpha, GeoGebra, and MATLAB. It can also be helpful to consult with a math tutor or professor for guidance and clarification.

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