How can I rotate a parabola.

In summary, to rotate a parabola, you can use parametric equations to translate the pixels along the curve. By picking an angle to rotate by and using general equations for new coordinates, you can easily rotate the parabola in different directions. This method can also be applied to rotating other objects.
  • #1
RusselMorty
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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[itex]^{2}[/itex] 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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  • #2


Write the parabola in parametric form:

[tex]y = t^2, \quad x = t[/tex]

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

[tex]x'(t) = x(t) \cos \theta - y(t) \sin \theta \\
y'(t) = y(t) \cos \theta + x(t) \sin \theta[/tex]

(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 [itex]y(x)[/itex]).
 
  • #3


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.
 

What is a parabola?

A parabola is a U-shaped curve that is formed by the graph of a quadratic equation. It is characterized by its vertex, a point where the curve changes direction, and its symmetry, which is the line that divides the parabola into two mirror images.

How can I rotate a parabola?

To rotate a parabola, you can use the general equation for a rotated parabola, which is (x-h)^2 = 4p(y-k), where (h,k) is the vertex of the original parabola and p is the distance between the focus and the vertex. You can also use transformation matrices to rotate the coordinates of the parabola.

What is the effect of rotating a parabola?

Rotating a parabola changes the orientation of the curve, but it does not change its shape. The vertex and focus of the parabola will remain the same, but the axis of symmetry will be rotated to a new position.

Can I rotate a parabola by any angle?

Yes, you can rotate a parabola by any angle as long as you know the coordinates of the original vertex. You can use trigonometric functions to determine the angle of rotation and then use the appropriate equations or matrices to rotate the parabola.

What are some real-life applications of rotating a parabola?

Rotating a parabola can be useful in engineering and architecture, where parabolic shapes are used to design structures such as arches and bridges. It is also used in physics to study the trajectory of projectiles and in astronomy to track the movement of celestial bodies.

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