Skip to content
Merged
Changes from 3 commits
Commits
File filter

Filter by extension

Filter by extension

Conversations
Failed to load comments.
Loading
Jump to
Jump to file
Failed to load files.
Loading
Diff view
Diff view
12 changes: 9 additions & 3 deletions docs/src/concepts/using_manifolds.md
Original file line number Diff line number Diff line change
Expand Up @@ -130,8 +130,14 @@ Retractions are numerically efficient approximations to convert a tangent vector

### Q7) Projection vs. Log map

Projections are somewhat ambiguous, since on the one hand it might mean that a point from the manifold is projected up onto a nearby tangent space.
The term projection can be somewhat ambiguous between references. In Manifolds.jl, projections either project a point in the [embedding](https://en.wikipedia.org/wiki/Embedding) to a point on the manifold, or a vector from the embedding onto a tangent space at a certain point.

Confusion, can easily happen between cases where there is no ambient space around a particular manifold. Then the term projection may be moot.

In Manifolds.jl, an _inverse retraction_ is an approximate logmap of a point up from the manifold onto a tangent space -- i.e. not a projection. It is important not to confuse a point on the manifold as a point in the ambient space, when thinking about the term projection.

It is best to make sure you know which one is being used in any particular situation.



Confusion, however, is never too far away since there is a second type of projection that people sometimes use: that is when a vector in the ambient space around a manifold (if one exists) is projected down onto a nearby tangent space.

There is of course also the confusion on whether a vector is projected up onto the tangent space from the manifold, or whether a tangent vector is projected down onto the manifold... It's an overloaded term, so best is to make sure you know which one is being used in any particular situation.