Transformation of unit sqaure to circle

AI Thread Summary
Transforming a unit square into a circle can be achieved using polar coordinates, where the points are mapped to the circle using the equations x' = rcos(θ) and y' = rsin(θ). The radius r varies from 0 to an upper limit, while the angle θ ranges from 0 to 2π. This transformation involves mapping the square onto a rectangle in the (r, θ)-plane. However, it's important to note that this transformation is not linear, as it maps straight lines of the square into arcs of the circle, making it unsuitable for representation as a simple matrix multiplication. Understanding these concepts is crucial for successfully executing the transformation.
unique_pavadrin
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I was wondering if anybody knew how to transform a unit square (point A (0,0) point B (1,0) point C (1,1) point D (0,1)) into a circle using a transformation matrix, or if there is a special name for this case.
Many thanks,
unique_pavadrin
 
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Well, look closely on the standard polar coordinates transformation.
This takes a rectangle onto a circle.

So, all you need to do is to map the unit square onto that rectangle, and you're done.
 
hmm...so the x' and y' points would be rcos(theta) and rsin(theta) respectivly?
thanks
 
Yes, with r varying between 0 and some upper limit, and the angle between 0 and 2pi.

This is a rectangle in the (r,theta)-plane.
 
thank you, however I am am still a little unsure...sorry...
 
Since this transformation maps a line (side of the square) into an arc of a circle, it is not "linear" and cannot be written as a matrix multiplication.
 
thank you HallsofIvy
 
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