(1, p.242) has the following geometric characterization of an inflection point $p$ of a curve $\gamma$ lying on a 2-dimensional sphere.

$p\,$ is an inflection point of the curve $\gamma$ if it has the following property:

Near the connected component of the intersection of $\gamma$ with the tangent great circle $C$ that contains $p$, the curve does not lie on one side of $C$.

What is meant by the curve not lying on one side of $C$?

(1): Tabachnikov, Serge. Differential and symplectic topology of knots and curves. No. 190. American Mathematical Soc., 1999.

  • 1
    $\begingroup$ Exactly the same thing it means to say $y=x^3$ lies on both sides of the tangent line $y=0$ in the plane near the point $(0,0)$. (A great circle separates the sphere into two hemispheres, just as a line separates the plane into two half-planes.) $\endgroup$ Commented Jun 2, 2018 at 21:45
  • $\begingroup$ @TedShifrin Thanks, I understood with your explanation :) This characterization is still valid if the curve $\gamma$ is itself a great circle? In this case the curve will lye exactly in the boundary of the two hemispheres, but I think in this case all the points of $\gamma$ are inflection points (by the Tennis ball theorem). $\endgroup$
    – shamisen
    Commented Jun 3, 2018 at 19:13
  • 1
    $\begingroup$ A great circle is the analogue of a straight line in the plane. Every point has intrinsic curvature $0$, and so, yes, every point is an inflection point. $\endgroup$ Commented Jun 3, 2018 at 19:21


You must log in to answer this question.

Browse other questions tagged .