Iteration of Involutes of Constant Width Curves in the Minkowski Plane

In this paper we study properties of the area evolute (AE) and the center symmetry set (CSS) of a convex planar curve $\gamma$. The main tool is to define a Minkowski plane where $\gamma$ becomes a constant width curve. In this Minkowski plane, the CSS is the evolute of $\gamma$ and the AE is an involute of the CSS. We prove that the AE is contained in the region bounded by the CSS and has smaller signed area. The iteration of involutes generate a pair of sequences of constant width curves with respect to the Minkowski metric and its dual, respectively. We prove that these sequences are converging to symmetric curves with the same center, which can be regarded as a central point of the curve $\gamma$.Comment: 16 pages, 4 figure

Involutes of Polygons of Constant Width in Minkowski Planes

Consider a convex polygon P in the plane, and denote by U a homothetical copy of the vector sum of P and (-P). Then the polygon U, as unit ball, induces a norm such that, with respect to this norm, P has constant Minkowskian width. We define notions like Minkowskian curvature, evolutes and involutes for polygons of constant U-width, and we prove that many properties of the smooth case, which is already completely studied, are preserved. The iteration of involutes generates a pair of sequences of polygons of constant width with respect to the Minkowski norm and its dual norm, respectively. We prove that these sequences are converging to symmetric polygons with the same center, which can be regarded as a central point of the polygon P.Comment: 20 pages, 11 figure

The Projective Pedal of an Eq\"uiaffine Immersion

Given a codimension $1$ immersion $f$ together with an eq\"uiaffine tranversal vector field $\xi$, its projective pedal is a codimension $2$ immersion obtained as the centroaffine dual of the lifting of the pair $(f,\xi)$. In this paper we discuss the properties of the projective pedal, the most significant being that its asymptotic lines correspond to the curvature lines of the original immersion. As a consequence, we prove that Loewner's conjecture for asymptotic lines at inflection points of surfaces in $4$-space is equivalent to Loewner's conjecture for curvature lines at umbilical points of surfaces in $3$-space with an eq\"uiaffine transversal vector field.Comment: 11 pages. The earlier versions of this article was divided in the new version and article Arxiv: 2008.0476

Envelope of mid-planes of a surface and some classical notions of affine differential geometry

For a pair of points in a smooth locally convex surface in 3-space, its mid-plane is the plane containing its mid-point and the intersection line of the corresponding pair of tangent planes. In this paper we show that the limit of mid-planes when one point tends to the other along a direction is the Transon plane of the direction. Moreover, the limit of the envelope of mid-planes is non-empty for at most six directions, and, in this case, it coincides with the center of the Moutard's quadric. These results establish an unexpected connection between these classical notions of affine differential geometry and the apparently unrelated concept of envelope of mid-planes. We call the limit of envelope of mid-planes the affine mid-planes evolute and prove that, under some generic conditions, it is a regular surface in 3-space.Comment: 15 pages, 1 figur