Move object at angle like a sine wave

Question

I currently have this code, which moves an object in a straight line at an angle, where "dir" is the angle.

velX = (float)(Math.cos(Math.toRadians(dir))) * speed;

velY = (float)(Math.sin(Math.toRadians(dir))) * speed;

x += velX;

y += velY;

What I need now is to have the object move like a sine wave, do I have to change the way I calculate velX and velY or is there some way of changing dir that would make it work?

I tried this but it didn't work, my objects would just got bunched up near 0, 0.
Move a sprite in a sine wave along different angles

Answer 1

Let's take a unit vector \$\vec u\$ in the direction we want to travel, and a vector perpendicular to it, \$\vec v\$:

$$\vec u = \begin{bmatrix}cos \Theta\\sin \Theta\end{bmatrix}, \vec v = \begin{bmatrix}-sin\Theta\\cos \Theta\end{bmatrix}$$

Then we can define a point on a sinusoidal curve through starting point \$\vec p_0\$, travelling in the direction \$\vec u\$, with a given speed, frequency, and amplitude, after time \$t\$ as:

$$\vec p(t) = \vec p_0 + \vec u \cdot speed \cdot t + \vec v \cdot amplitude \cdot sin(frequency \cdot t)$$

Taking the derivative, we get:

$$\frac{\delta \vec p(t)}{\delta t} = \vec u \cdot speed + \vec v \cdot amplitude \cdot cos(frequency \cdot t) \cdot frequency$$

So, we can rewrite your velocity expressions as...

c = (float)(Math.cos(Math.toRadians(dir)));
s = (float)(Math.sin(Math.toRadians(dir)));

wobble = amplitude * (float)Math.cos(frequency * t) * frequency;

velX = c * speed - s * wobble;
velY = s * speed + c * wobble;

Note that steering the object purely by velocity, you can accumulate rounding and integration errors. While the path will have the overall shape, it may stray from the parametric equation \$\vec p(t)\$ above. To fix this, you could instead compute your next position directly from the parametric equation, then compute a velocity to reach it.