Rotation of a 3D ball on a plane

Question

This used to be on the physics stack exchange but I was told to put it here:

I'm making a game (with physics) and I have a 3D ball rolling down a flat plane. It moves perfectly fine, but the rotation is messed up. It seems way too fast, and the axis seems to be all wrong. Here's the code that runs every tick (20 times per second). (normal is the normal to the plane, acceleration is gravity vector projected onto the plane, in m/tick², and velocity is the velocity in m/tick.)

vec3 rotationAxis = normalize(cross(normal, acceleration));
quat newRotation = rotate(oldRotation, length(velocity) / sphereRadius, rotationAxis); // rotate by v/r radians about the rotationAxis

I'm not sure why this doesn't work, since according to angular mechanics, v=rω and ∆θ=v∆t/r (and ∆t=1 since velocity is in meters/tick). Am I doing something wrong? Why is the rotation so much faster than it should be? And is my axis of rotation correct?

Rephrased in a more physics way, is this correct in 3D:

$$\hat{x} = \frac{\hat{N} \times \vec{a}}{|\hat{N} \times \vec{a}|}$$

$$\Delta \theta \textrm{ about } \hat{x} = \frac{v}{r} \Delta t$$

And if it is correct, then what’s wrong with my code?

Answer 1

Your rotationAxis should be the cross product between your velocity vector and the surface normal vector as velocity contains the direction of movement whereas acceleration is not necessarily in the direction of movement. (E.g. Projectile motion where acceleration is always down.)

i.e. vec3 rotationAxis = normalize(cross(normal, velocity));

∆θ = length(velocity) / sphereRadius seems right since

$$ 2\pi Radius \ meters/tick = 1 \ (full) \ rotations/tick = 2\pi \ radians/tick $$

So, $$ x \ meters/tick = \frac{x}{Radius} \ radians / tick $$

You say velocity is per tick so length(velocity) is the speed in meters/tick. Hence, $$|Velocity| \ meters/tick = \frac{|Velocity|}{Radius} \ radians / tick $$

Therefore, every new tick, you should rotate by $$\frac{|Velocity|}{Radius} \ radians$$ which is what you're doing.

Make sure your sphereRadius is also in meters as length(velocity) is in meters/tick.