Prove Strict Tangents: Exercises & Real Curves

[mathbalarka]

Call any straight line that is tangential with a curve which cuts the curve nowhere a strict tangent respect to that curve. Complete theses exercises :

1. Prove that any real, smooth curve has no more than 2 strict tangents.
2. Find a smooth curve with 3 strict tangents. Can you find one with exactly 4?
3. Find a real, smooth curve with no strict tangents at all.

Have fun!

 

[mathbalarka]

Here is my solution to this problem given below :

Spoiler

1. As two types smooth and real of curves are possible, one with the limit at \(\displaystyle \pm \infty\) either of \(\displaystyle \pm \infty\) or a closed curve, i.e., an ellipse. For the first one, the maximum number of strict tangents possible to draw is 2 as the possible areas of tangentiality are closed by the strict tangents drawn from a point, and for a closed curve, the curvature of the curve decays faster than the tangents, thus proving this case.

2. Consider an elliptic curve, which has exactly 3 strict tangents drawable from a point. A closed form for one with 4 of these is the Cramer's curve. I conjecture : this is the smallest such degree.

3. An Archimedean spiral is one such curve.

Balarka
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