Question

This may be ridiculously obvious, but math wasn’t my strong suit in school. I’ve been banging my head against the wall long enough that I finally figured I’d ask.

I’m trying to animate a sprite moving along a 2D parabolic path, from point A to point B. Both points are at the same y-coordinate. The desired height of the parabola from the starting/ending y-coordinate is also given (or, if you prefer, a desired velocity). Currently in my code I have a timer firing at a high frequency. I would like to calculate the new location of the ball based on the amount of time that has passed. So a parametric parabola equation should work nicely.

I found this answer from GameDev adequate, until my requirements grew (although I’m not sure its really a parabolic path… I can’t follow the derivation of the final equations there provided). Now I would like to squish/stretch the sprite at different points along the parabolic path. But to get the effect to work right, I’ll need to rotate the sprite so that it’s primary axis is tangential to the path. So I need to be able to derive the angle of the tangent at any given location/time.

I can find all sorts of equations for each of these requirements (parametric parabola, tangent at a point, etc.), but I just can’t figure out how to unify them all. Could someone with more math skills help a fellow coder out and provide a set of equations that will work? Thanks ever so much in advance.

Answer 1

Let X be the distance from A to B, Y the desired height of the parabola, V the horizontal speed.

x = Vt

y = Y - (4Y/X^2) (X/2-Vt)^2

tangent dy/dx = (8Y/X^2) (X/2-Vt)