Spherical curves and quadratic relationships for special functions

New special curves and their spherical indicatrices

From the view of differential geometry, a straight line is a geometric curve with the curvature κ(s) = 0. A plane curve is a family of geometric curves with torsion τ(s) = 0. Helix is a geometric curve with non-vanishing constant curvature κ and non-vanishing constant torsion τ [4]. The helix may be called a circular helix or W -curve [9]. It is known that straight line (κ(s) = 0) and circle (κ...

Integral Properties of Zonal Spherical Functions, Hypergeometric Functions and Invariant

Some integral properties of zonal spherical functions, hypergeometric functions and invariant polynomials are studied for real normed division algebras.

Quadratic Forms and Elliptic Curves

ARC-LENGTH ESTIMATIONS FOR QUADRATIC RATIONAL Bézier CURVES COINCIDING WITH ARC-LENGTH OF SPECIAL SHAPES

In this paper, we present arc-length estimations for quadratic rational Bézier curves using the length of polygon and distance between both end points. Our arc-length estimations coincide with the arc-length of the quadratic rational Bézier curve exactly when the weight w is 0, 1 and∞. We show that for all w > 0 our estimations are strictly increasing with respect to w. Moreover, we find the pa...

Isoperimetric Inequalities for Immersed Closed Spherical Curves

Let a: Sl ->S2 be a C2 immersion with length L and total curvature K . If a is regularly homotopic to a circle traversed once then L2 + K2 > 4n2 with equality if and only if a is a circle traversed once. If a has nonnegative geodesic curvature and multiple points then L + K > An with equality if and only if a is a great circle traversed twice.