Crossing problem

Sep 2018
I ran across the following problem in Barret O'Neill's Elementary Differential Geometry: "Let C be a Curve in the xy plane that is symmetric about the x axis. Assume C crosses the x axis and always does so orthogonally. Explain why there can be only one or two crossings. Thus C is either an arc or is closed."

O'Neill defines a curve as "A curve in $E^3$ is a differentiable function $\alpha : I \rightarrow E^3$ from an open interval I and stipulates that the velocity has to be everywhere nonzero. A plane curve in $E^3$ is a curve that lies in a single plane.

A "curve" made of two circles touching one another crosses orthogonally at three points. Does this mean O'Neill was wrong?


MHF Helper
Apr 2005
No, that's not a "curve" by this definition. Look at the point where the two circles meet. What is the tangent vector at that point?

(I get mildly annoyed by "physics" terminology in mathematics- instead of saying "the velocity has to be everywhere nonzero" I would say "the tangent vector has to be everywhere nonzero".)