使用正弦波获取交叉点

Question

I have a particle at a fixed position X and fixed amplitude AMP that travels on a sin wave.

I'd like to be able to dynamically change the period of the target sine wave WITHOUT moving the particle. (When I change the period, its X position doesn't line up to the same Y position of the graph with a different period, so it jumps all over the place.

I have a solution for this - Find where the graph with the new period intersects with the graph of the old period and create an X-offset, lining these two figures up.

For any given y on a sin wave, there will be 2 corresponding X values per period. I'm trying to get these two values. I'll then compare the derivatives to choose which one I want.

However, I don't know how to get both of these values. Any ideas?

Here's an image of what I'm after: Sin Wave Logic

Answer 1

If you have a sine-wave defined by

y = sin(a*(x+c))+d

then you can reverse it by

x = (arcsin(y-d)-c)/a

for a given a.

This yields the solution in the range of [-pi;pi]

The other solution you are looking for is then 2*pi-x

Answer 2

If you allow a little cheating, you can use a cross-fading effect to get a smooth transition from f(t) to g(t) via

h(s,t)=(1-s)*f(t)+s*g(t)

while s as a function of t is between 0 and 1 and use something like

y = h( (t-ts)/d, t)

for the time period from ts to ts+d .