# Geometric characterization of inflection points of a curve in a sphere

(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.

• 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.) Commented Jun 2, 2018 at 21:45
• @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). Commented Jun 3, 2018 at 19:13
• 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. Commented Jun 3, 2018 at 19:21