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.