Problem in generating exponential map on mesh pieces.

We are creating dcal on mesh using exponential map.

We are facing major 2 problems.

While generating exponential map, moving frame to vertices is not getting properly transform. Its gets rotated in case of sphere. Do somebody have code snippet to correctly translate frame and vertex on map?
Mesh can have islands. Exponential map should consider those scenarios. My logic finds contours along islands and stitch appropriately. Current logic fails in some cases to find contours. It ends in incorrect loop of link list of vertices. Anybody can share code to generate exponential map on mesh pieces.