calculate the spherical rotation from change in position

Question

I'm trying to understand how to work the rotation out of the sphere in the gif linked below. Given the spheres Radius, currentMatrix, and a nextPosition Vector, how do I work out the sphere's newMatrix?

Does any one have any ideas? I've tried googling a bunch of stuff but I'm not really sure if I've found the right equation of formula to work this out. Any guidance would be really appreciated

Input: Sphere Radius

Sphere currentMatrix

Move around: nextPosition Vector

Output: Sphere newMatrix

Answer 1

I assume by sphereMatrix you mean the 3×3 orientation matrix describing the local to global transformation matrix.

In general a body with rotational velocity ω=(ωx,ωy,ωz) and orientation matrix R will update the orientation matrix after some time Δt by converting the rotational velocity into an axis of rotation and a small angle of rotation.

So you have R' = Rot(k, Δθ)*R

What is the rotation axis k? Take the direction of ω with k=ω/mag(ω) and the small angle Δθ=mag(ω)*Δt.

Lastly look up the Rodrigues' rotation formula to conver the axis angle into a 3×3 rotation matrix.

In the video the rotation matrix is found by the no-slip condition on the contact of the sphere with the floor. From rigid body mechanics it is found that

ω = [vy/R,0,-vx/R]

if R is the radius of the sphere, +y is the "up" direction and vx, vy are the velocity components of the center of the sphere along the x and y directions.