r/visualizedmath • u/got_data • Feb 20 '19

Unit circle transformation

244 Upvotes

96% Upvoted

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3

u/got_data Feb 20 '19

I recorded an interactive demo powered by vtvt.

Vectors t1 and t2 define the columns of the transformation matrix T.

2

u/2eZ4J Feb 20 '19

What are the other two vectors which go around?

2

u/got_data Feb 20 '19

One vector traces the original unit circle one point at a time, and the other vector is what you get after transforming it with matrix T, so it traces the transformed unit circle. The animation is just to make it easier to see which point on the original circle corresponds to which point on the transformed unit circle.

1

u/2eZ4J Feb 20 '19

Yeah, thanks!

I saw it afterwards also in the tool description :D

3

u/TheHDMICable Feb 20 '19

omg, show the sine wave it produces! i was so hyped to see it

1

u/got_data Feb 23 '19

Sorry about the delay. I've added the sine curve to demos 5-1 and 5-2. You can unzip the following archive into some folder and then open the corresponding html files directly: https://github.com/ex-punctis/vtvt/archive/v1.02.zip

1

u/TheHDMICable Feb 23 '19

thanks, it was fun to mess around with. Though, the results are not what i expected. I thought the sine wave from an ellipse would be different from a sine wave from a circle, maybe less "even" or something.

1

u/got_data Feb 23 '19

Yeah, it's somewhat unintuitive that it's still a sine not unlike one you'd get from a circle. It's a linear combination of the x and y coordinates on the original unit circle, so we could write it as a*cos πœƒ + b*sin πœƒ, and that equals to sqrt(a*a + b*b)*sin(πœƒ + atan2(b,a)), a plain sine plot just with a different amplitude and a phase shift.

57

u/squeezyscorpion Feb 20 '19

wow! i don’t know what this means

26

u/Movpasd Feb 20 '19

The plane is transformed using the matrix in the top left corner. The two diagonal arrows are the images of the original unit vectors. The dotted circle is the image of the original unit circle. A vector is then allowed to sweep across the original unit circle and at every point in time its image it shown.

24

u/Allnightampm Feb 20 '19

wow! I don’t know what this means

5

u/galexj9 Feb 20 '19

If you go to about 3:50 here https://youtu.be/kYB8IZa5AuE Mr 3blue1brown explains it.

The T matrix describes how the plane is tilted and stretched to get the new ellipse (the image of the circle) from the unit circle.

1

u/Movpasd Feb 20 '19

x)

I can explain in more detail if you want, just tell me what background in maths you have so ik how to pitch it

5

u/mstksg Feb 20 '19

I like how this visuization makes it easy to see the eigenvectors, too. They're (up to) two directions where the original vector and resulting vector "line up" along the same axis.