计算Grassmannian geodesic

Solving for geodesic is a classical problem in the calculus of variation.

In Grassmann manifold, there exists relatively simple method of computing geodesics using the relatively simple method bsed on the SVD.

计算Grassmannian Geodesic:

Given 2 points on grassmann manifold, $X,Y\in \mathcal{G}\left( D,p \right)$  may be parametrized by a function $\varPhi \left( t \right) :\left[ 0,1 \right] \rightarrow \mathcal{G}\left( D,p \right)$, where $\varPhi \left( 0 \right)=X$ and  $\varPhi \left( 1 \right)=Y$. The parameter $t\in\left[0,1\right]$ controls the location on the geodesic on Grassmannian.

First,