How can I rotate a parabola.

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To rotate a parabola for image distortion, start by expressing the parabola in parametric form: y = t^2 and x = t. To achieve rotation, select an angle θ and apply the transformation equations: x'(t) = x(t) cos θ - y(t) sin θ and y'(t) = y(t) cos θ + x(t) sin θ. This method allows for the manipulation of the parabola's orientation, enabling the desired wavy effect for images. Understanding this rotation technique can also aid in rotating other shapes.
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How can I "rotate" a parabola.

I want to take an image and make it kind of wavy to distort it (like in a capcha form the letters are distorted sometimes with a wave) . So i figure that I could translate the pixels along a parabola. So the function where y=X^{2} would make the image curve downwards. If i want to curve it in other directions I would need to somehow rotate that parabola. I am not an expert at math so I have no idea how to do this. I am sure it is a simple thing but It would really help me alot. Thanks for any advice.
 
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Write the parabola in parametric form:

y = t^2, \quad x = t

Pick an angle \theta to rotate by. The general equations governing the new coordinates are

x&#039;(t) = x(t) \cos \theta - y(t) \sin \theta \\<br /> y&#039;(t) = y(t) \cos \theta + x(t) \sin \theta

(It's not strictly necessary to write in parametric, but it's usually easy to plot that way rather than go back and resolve for y(x)).
 


Thank you very much for your answer. It also gives me some more insight on how to rotate other things too. It will be very helpful.
 
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