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I ran across the following problem in Barret O'Neill's Elementary Differenetial 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."

I see that $(cos(t),sin(t)cos(t))$ makes a figure eight with three crossings; the middle crossing is not orthogonal, but it seems that you could deform the curve in a continuous fashion so that the crossing would be orthogonal.

Does this mean O'Neill was wrong?

O'Neill defines a curve as a differentiable function from an open set of R into $E^3$ and mentions in a remark that the velocity should be nonzero. My "figure 8" has zero velocity periodically.

Gene Naden
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    Just to be certain, how does O'Neill define a "curve"? It may be a small technicality that makes his statement correct (otherwise, you can take your idea as just making circles adjacent at only one point and crossing the $x$-axis...you could construct a "curve" in this fashion that cross any prescribed number of times). – Clayton Aug 29 '18 at 00:58
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    This is true if the curve has no self intersections. – Marco Aug 29 '18 at 01:13
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    O'Neill defines a curve as "A curve in $E^3$ is (an infinitely differentiable) function $\alpha:I\rightarrow E^3$ from an open interval I into $E^3$" – Gene Naden Aug 29 '18 at 01:17
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    Does O'Neill give a separate definition of a curve in $E^2$ ? – coffeemath Aug 29 '18 at 08:33
  • "A plane curve is a curve that lies in a single plane of $E_3$" Otherwise, I don't find a separate definition. – Gene Naden Aug 30 '18 at 18:16
  • Your statement is clearly true for the class of curves:Let $f(t)=(f_1(t),f_2(t))$. Let the range of $f_2(t)$ be symmetric about $0$.Let $f_1(t)=\sum_{n \geq i \geq 1} a_i (f_2(t))^{2i}$. – Balaji sb Sep 14 '18 at 07:38

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