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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 $P$?

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    For instance, I am not sure this would hold for plane curves like $c(t)=(t,pt)$ and $C(t)=(t,t^2 sin(1/t))$ (extended by $(0,0)$ at $t=0$) for $P=(0,0)$.2012-11-03

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