Feature Matching and Heat Flow in Centro-Affine Geometry

Olver, Peter J.; Qu, Changzheng; Yang, Yun

doi:10.3842/sigma.2020.093

Cited by 8 publications

(7 citation statements)

References 45 publications

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“…Andrews [3] studied an affine-geometric, fourth-order parabolic evolution equation for closed convex curves in the plane and proved the evolving curve remains strictly convex while expanding to infinite size and approaching a homothetically expanding ellipse. More recently, similar results were obtained for the heat flow in centro-equi-affine geometry [52] and in centro-affine geometry [38]. Interestingly, the heat flow for the centroaffine curvature is equivalent to the well-known inviscid Burgers' equation.…”

Section: Introductionsupporting

confidence: 68%

The maximal curves and heat flow in fully affine geometry

Yang¹

2022

Preprint

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In Euclidean geometry, the shortest distance between two points is a straight line. Chern made a conjecture (cf. [10]) in 1977 that an affine maximal graph of a smooth, locally uniformly convex function on two dimensional Euclidean space R 2 must be a paraboloid. In 2000, Trudinger and Wang completed the proof of this conjecture in affine geometry (cf. [47]). (Caution: in these literatures, the term "affine geometry" refers to "equi-affine geometry".) A natural problem arises: Whether the hyperbola is a fully affine maximal curve in R 2 ? In this paper, by utilizing the evolution equations for curves, we obtain the second variational formula for fully affine extremal curves in R 2 , and show the fully affine maximal curves in R 2 are much more abundant and include the explicit curves y = x α α is a constant and α / ∈ {0, 1, 1 2 , 2} and y = x log x. At the same time, we generalize the fundamental theory of curves in higher dimensions, equipped with GA(n) = GL(n) ⋉ R n . Moreover, in fully affine plane geometry, an isoperimetric inequality is investigated, and a complete classification of the solitons for fully affine heat flow is provided. We also study the local existence, uniqueness, and long-term behavior of this fully affine heat flow. A closed embedded curve will converge to an ellipse when evolving according to the fully affine heat flow is proved.

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Section: Introductionsupporting

confidence: 68%

The maximal curves and heat flow in fully affine geometry

Yang¹

2022

Preprint

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“…The latter consists of the subgroup of the affine transformation group that fixes the origin, identified by the general linear group [28]. In this case, the flow (1.1) leads to the well-known inviscid Burgers equation related to the centro-affine curvature k and centro-affine arc-length s k t = kk s .…”

Section: F Xia and Yh Yumentioning

confidence: 99%

Numerical Simulations of a Curve Flow Related to Centro-affine Invariants

Xia,

2024

Preprint

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In this paper, we propose a novel numerical implementation for visualizing the evolutionary process of a curve flow. This flow is an invariant second-order flow in centro-affine geometry. The key of the scheme proposed in this essay is to indirectly obtain the position vector by solving the evolution equation of the support function. Additionally, to improve efficiency, we choose a specific tangential velocity which can eliminate the necessity to update normal angles at each iteration. The reason is that the normal angles do not change over time under this tangential velocity. Finally, we conduct multiple experiments to demonstrate the feasibility and efficiency of our proposed method.

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“…The latter consists of the subgroup of the affine transformation group that fixes the origin, identified by the general linear group [23]. In this case, the flow (1.1) leads to the well-known inviscid Burgers equation related to the centro-affine curvature k and centro-affine arc-length s k t = kk s .…”

Section: Introductionmentioning

confidence: 99%

Numerical Simulations of a Curve Flow Related to Centro-Affine Invariants

YU,

XIA

2023

Preprint

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Feature Matching and Heat Flow in Centro-Affine Geometry

Cited by 8 publications

References 45 publications

The maximal curves and heat flow in fully affine geometry

The maximal curves and heat flow in fully affine geometry

Numerical Simulations of a Curve Flow Related to Centro-affine Invariants

Numerical Simulations of a Curve Flow Related to Centro-Affine Invariants

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