I am reading 'Differential Geometry of curves and surfaces' by Banchoff and Lovett and I'm confused about the following statement on page 28:
"Let two curves $C_1$ and $C_2$ have a regular point $P$ in common. Given a point $A$ on $C_1$ near $P$, let $D_A$ be the orthogonal projection of $A$ onto $C_2$, i.e. the point on $C_2$ closest to $A$. (...)"
If we were talking about lines I would understand but these are more general, continuous (but not necessarily differentiable) plane curves. I understand orthogonal projection from a point $p$ on $C_1$ as projection along the line orthogonal to the tangent vector at $p$. But this doesn't have to be the shortest distance to $C_2$ it seems? Unless we are assuming something like the curve becoming a line in the limit of being very close to $P$?