Conformal arc-length as $\frac12$ dimensional length of the set of osculating circles

Surfaces of osculating circles in Euclidean space

We introduce a new class of surfaces in Euclidean $3$-space, called surfaces of osculating circles, using the concept of osculating circle of a regular curve. These surfaces contain a uniparametric family of planar lines of curvature. In this paper, we classify those ones that are canal surfaces and Weingarten surfaces.

通过圆模式间的密切莫比乌斯变换得到的 CMC-1 曲面

CMC-1 surfaces via osculating Möbius transformations between circle patterns

本文研究平面内相同组合圆模式间的莫比乌斯变换，导出双曲空间中对偶图的实现，刻画其在共享剪切坐标或交角时的一一对应关系，类比常平均曲率曲面的 Weierstrass 表示，并在三角格上建立收敛性。

Osculating curves: around the Tait-Kneser Theorem

The Tait-Kneser theorem states that the osculating circles of a plane curve with monotonic curvature are pairwise disjoint and nested. We discuss this theorem and a number of its variations.

Variations on the Tait-Kneser theorem

The classical Tait-Kneser theorem states that the osculating circles of a smooth plane curve, free from curvature extrema, are pairwise disjoint. We prove a number of analogs of this theorem, e.g., for ovals of osculating cubics, osculating polynomials and trigonometric polynomials; in each case, we will obtain a non-differentiable foliation with smooth leaves.

Variations on the Tait-Kneser theorem

The Tait-Kneser theorem, first demonstrated by Peter G. Tait in 1896, states that the osculating circles along a plane curve with monotone non-vanishing curvature are pairwise disjoint and nested. This note contains a proof of this theorem using the Lorentzian geometry of the space of circles. We show how a similar proof applies to two variations on the theorem, concerning the osculating Hooke and Kepler conics along a plane curve. We also prove a version of the 4-vertex theorem for Kepler conics.

Space of subspheres and conformal invariants of curves

A space curve is determined by conformal arc-length, conformal curvature, and conformal torsion, up to Möbius transformations. We use the spaces of osculating circles and spheres to give a conformally defined moving frame of a curve in the Minkowski space, which can naturally produce the conformal invariants and the normal form of the curve. We also give characterization of canal surfaces in terms of curves in the set of circles.