Planar Curves out of Their Curvatures in R2

Knots and Their Curvatures

I discuss an old result of John Milnor stating roughly that if a closed curve in space is not too curved then it cannot be knotted. CONTENTS 1. The total curvature of a polygonal curve 1 2. A probabilistic interpretation of the total curvature 2 3. The total curvature of a smooth closed curve 4 4. Total curvature and knotting 6 References 7 1. THE TOTAL CURVATURE OF A POLYGONAL CURVE An (orient...

Bilipschitz Homogeneous Curves in R2 Are Quasicircles

Tangents and curvatures of matrix-valued subdivision curves and their applications to curve design

Subdivision provides an efficient method to generate smooth curves and surfaces. Recently matrixvalued subdivision schemes were introduced to provide more flexibility and smaller subdivision templates for curve and surface design. For matrix-valued subdivision, the input is a set of vectors with the first components being the vertices of the control polygon (or the control net for surface subdi...

Arrangements of Planar Curves

Computing arrangements of curves is a fundamental and challenging problem in computational geometry as leading to many practical applications in a wide range of fields, especially in robot motion planning and computer vision. In this survey paper we present the state of the art for computing the arrangement of planar curves, considering various classes of curves, from lines to arbitrary curves.

Bisector curves of planar rational curves

This paper presents a simple and robust method for computing the bisector of two planar rational curves. We represent the correspondence between the foot points on two planar rational curves C1(t) and C2(r) as an implicit curve F(t; r) = 0, where F(t; r) is a bivariate polynomial B-spline function. Given two rational curves of degree m in the xy-plane, the curve F(t; r) = 0 has degree 4m 2, whi...