Interpolating rotation

• Problem: Given 2 rotations \(r_1, r_2\)
• • - Find rotation \(r(t)\) st \(r(0)=r_1\), \(r(1)=r_2\), and varies smoothly along \(t\)

• Matrix representation
• • Componentwise interpolation not adapted
  • • ex. \(t\,R_1 + (1-t)\,R_2\) is not a rotation: introduce scaling/shearing
  • The correct formulation to interpolate on the manifold would be \(R1 \,exp(t \log(R_1^T\,R_2))\)
  • • But matrix exponential is complex to compute

• Euler angle
• • Can interpolate the 3 angles separately
  • (+) Leads to a rotation
  • (-) Doesn't necessarily follows simplest trajectory

• Axis-Angle
• • No trivial interpolation scheme with different axes

• Quaternion
• • Well adapted (see later)