The conjugate locus on convex surfaces

TL;DR

This work analyzes the conjugate locus $C_p$ of a point $p$ on a smooth strictly convex surface by linking its rotation index $i$ to the number of cusps $n$ via $i=(n-2)/2$, and derives the vierspitzensatz $n\ge 4$. It develops a plane-evolute framework through the distance function $R$ and distance curve, and extends to plane convex curves where the analogous relation holds, with a generalized form $i=(n+2I)/2$ when the base curve has rotation index $I$. A central contribution is the projection-based analysis that yields $I=-1$ and hence $i=(n-2)/2$ for the convex-surface case, together with a counting-branching method (the McIntyre–Cairns approach) that bounds the cusp count. The paper also provides a concrete formula for the geodesic curvature of $C_p$, $k_g(\psi)=\frac{\xi_{,s}(R(\psi),\psi)}{R'(\psi)}$, and discusses higher-order conjugate loci and the role of caustics, along with corrected statements about the existence of smooth loops in evolutes. These results yield topological restrictions on conjugate loci and offer tools for understanding their geometric structure on convex surfaces.

Abstract

The conjugate locus of a point on a surface is the envelope of geodesics emanating radially from that point. In this paper we show that the conjugate loci of generic points on convex surfaces satisfy a simple relationship between the rotation index and the number of cusps. As a consequence we prove the `vierspitzensatz': the conjugate locus of a generic point on a convex surface must have at least four cusps. Along the way we prove certain results about evolutes in the plane and geodesic curvature. (Note: this is a corrected version of the original paper, see comment on page 5 and Appendix B).

Figures (7)

• Figure 1: The orientations of $\hbox{\boldmath $\gamma$}$ (blue) and $\hbox{\boldmath $\beta$}$ (red) as defined in the text.
• Figure 2: The distance curve in $T_p\mathcal{S}$ and its image in $\mathcal{S}$ under the exponential map: the conjugate locus of $p$, $C_p$.
• Figure 3: How the 'count' changes as we pass from one component of $\mathcal{S}/\hbox{\boldmath $\beta$}$ to another.
• Figure 4: On the far left a typical conjugate locus (taken from TWbif) with the count marked in some regions; then shaded are $S_2, S_3$ and $S_4$, each of which are discs and have Euler charateristic 1. Thus the rotation index is 3, and the curve has 8 cusps as expected.
• Figure 5: Examples of $C_p^1$ (left) and $C_p^3$ (right) on the triaxial ellipsoid. Note the four cusps on the left as predicted by the Itoh Kiyohara theorem, and the smooth loop on the right. Blue lines are geodesics, black lines are the conjugate locus and the red dots are conjugate points.

Theorems & Definitions (14)

• Theorem 3
• Theorem 4
• Lemma 1
• proof
• Theorem 1
• proof
• Theorem 2
• Theorem 3
• proof
• Theorem : McIntyre and Cairns