(Background: I am trying to understand the definition of angle-preserving function..I posted a question earlier but I still have doubts)

My question is:how is the angle between two curves defined if the curves are not differentiable? If they are, then we can work with the scalar product of their derivatives at that point.

If they are not, I don't know!

Thank you

  • $\begingroup$ They would really need to be differentiable and have non zero derivative at the crossing points. $\endgroup$ – copper.hat May 5 '15 at 16:12

The concept of angle is not going to be well-defined for such curves: consider $ y = x $ and $ y = \lvert x \rvert $. What is the angle between these curves at $(x,y)=(0,0)$? It could be $0$, approaching from positive $x$, or $\pi/2$, approaching from negative $x$. And that's the least pathological case, where the left- and right-hand derivatives exist (and, indeed, one function has a standard derivative); it's obviously going to be even less sensical to try and do so if such limits don't exist.

The whole point of an angle between curves is that it is the angle between their tangents: if you can't construct tangents, you have nothing from which you can make an angle.

  • $\begingroup$ I see, thanks! :-) $\endgroup$ – Ant May 5 '15 at 16:20

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