2.4 Interpolation on the Manifold

The exponential map and logarithm provides an intuitive method for interpolation or blending of transformations. Consider transformation $X,Y\in\mathrm{G}$ and an interpolation coefficient $t\in\left[0,1\right]\subset\mathbb{R}$ . The function

blends the two transformations by moving steadily along the geodesic between them:

$\displaystyle f:\mathrm{G}\times\mathrm{G}\times\mathbb{R}$	$\displaystyle \rightarrow$	$\displaystyle \mathrm{G}$	(6)
$\displaystyle f\left(X,Y,t\right)$	$\displaystyle =$	$\displaystyle \exp\left(t\cdot\log\left(Y\cdot X^{-1}\right)\right)\cdot X$	(7)
$\displaystyle \implies f\left(X,Y,0\right)$	$\displaystyle =$	$\displaystyle X$	(8)
$\displaystyle \implies f\left(X,Y,1\right)$	$\displaystyle =$	$\displaystyle Y$	(9)
$\displaystyle \implies f\left(X,Y,\frac{1}{2}\right)\cdot X^{-1}$	$\displaystyle =$	$\displaystyle Y\cdot f\left(X,Y,\frac{1}{2}\right)^{-1}$	(10)