Sin along a vector?

Question

I have a 2D vector that I currently use to plot a line in its direction.

I calculate the next point using:

p_new = p_old + v * i

Were i going from 0 to some upper limit where the line should end, say for example 100.

How would I go about to plot a sin curve in direction v?

Answer 1

Calculate the perpendicular $v^{\perp}$ to $v$ and normalize both vectors. Then plot $$A\sin(\omega t)v^{\perp} + tv$$ for some values of the constants $A$ and $\omega$, which control the amplitude and frequency of the sine wave respectively.