7 Interpolation

Consider a Lie group $G$, with two elements $a,b\in G$. We would like to interpolate between these elements, according to a parameter $t\in\left[0,1\right]$. Define a function that will perform the interpolation:

$$f:G\times G\times\mathbb{R} \rightarrow G$$ (206)

$$f\left(a,b,0\right) = a$$ (207)

$$f\left(a,b,1\right) = b$$ (208)

The function is defined by transforming the interpolation operation into the tangent space, performing the linear combination there, and then transforming the resulting tangent vector back onto the manifold. First consider the group element that takes $a$ to $b$:

$$d \equiv b\cdot a^{-1}\in G$$ (209)

$$d\cdot a = b$$ (210)

Now compute the corresponding Lie algebra vector and scale it in the tangent space:

$$\mathbf{d}(t)=t\cdot\ln\left(d\right)$$ (211)

Then transform back into the manifold using the exponential map, yielding a ``partial'' transformation:

$$d_{t}=\exp\left(\mathbf{d}(t)\right)$$ (212)

Combining these three steps gives a definition for $f$:

$$f(a,b,t) = d_{t}\cdot a$$ (213)

$$= \exp\left(t\cdot\ln\left(b\cdot a^{-1}\right)\right)\cdot a$$ (214)

Note that the result is always on the manifold, due to the properties of the exponential map. No projection or coercion is required. Furthermore, the linear transformation in the tangent space corresponds to moving along a geodesic of the manifold. So the interpolation always moves along the ``shortest'' transformation in the Lie group.