scholarly journals A Coons Patch Spanning a Finite Number of Curves Tested for Variationally Minimizing Its Area

2013 ◽  
Vol 2013 ◽  
pp. 1-15 ◽  
Author(s):  
Daud Ahmad ◽  
Bilal Masud
Keyword(s):  
Minimal Surface ◽  
Mean Curvature ◽  
Step Function ◽  
Variational Parameter ◽  
Curvature Function ◽  
Unit Step ◽  
Straight Lines ◽  
Boundary Curves ◽  
Percent Decrease ◽  
Natural Way

In surface modeling a surface frequently encountered is a Coons patch that is defined only for a boundary composed offouranalytical curves. In this paper we extend the range of applicability of a Coons patch by telling how to write it for a boundary composed of an arbitrary number of boundary curves. We partition the curves in a clear and natural way into four groups and then join all the curves in each group intooneanalytic curve by using representations of the unit step function including one that isfully analytic. Having a well-parameterized surface, we do some calculations on it that are motivated by differential geometry but give a better optimized and possibly more smooth surface. For this, we use an ansatz consisting of the original surface plus a variational parameter multiplying the numerator part of its mean curvature function and minimize with the respect to it the rms mean curvature and decrease the area of the surface we generate. We do a complete numerical implementation for a boundary composed of five straight lines, that can model a string breaking, and get about 0.82 percent decrease of the area. Given the demonstrated ability of our optimization algorithm to reduce area by as much as 23 percent for a spanning surface not close of being a minimal surface, this much smaller fractional decrease suggests that the Coons patch we have been able to write is already close of being a minimal surface.

scholarly journals Pansu–Wulff shapes in ℍ1

2021 ◽  
Vol 0 (0) ◽  
Author(s):  
Julián Pozuelo ◽  
Manuel Ritoré
Keyword(s):  
Mean Curvature ◽  
Isoperimetric Problem ◽  
Variational Formula ◽  
Geodesic Curvature ◽  
Vertical Axis ◽  
Singular Part ◽  
Tangent Plane ◽  
Curvature Function ◽  
Invariant Norm ◽  
The First Variation

Abstract We consider an asymmetric left-invariant norm ∥ ⋅ ∥ K {\|\cdot\|_{K}} in the first Heisenberg group ℍ 1 {\mathbb{H}^{1}} induced by a convex body K ⊂ ℝ 2 {K\subset\mathbb{R}^{2}} containing the origin in its interior. Associated to ∥ ⋅ ∥ K {\|\cdot\|_{K}} there is a perimeter functional, that coincides with the classical sub-Riemannian perimeter in case K is the closed unit disk centered at the origin of ℝ 2 {{\mathbb{R}}^{2}} . Under the assumption that K has C 2 {C^{2}} boundary with strictly positive geodesic curvature we compute the first variation formula of perimeter for sets with C 2 {C^{2}} boundary. The localization of the variational formula in the non-singular part of the boundary, composed of the points where the tangent plane is not horizontal, allows us to define a mean curvature function H K {H_{K}} out of the singular set. In the case of non-vanishing mean curvature, the condition that H K {H_{K}} be constant implies that the non-singular portion of the boundary is foliated by horizontal liftings of translations of ∂ ⁡ K {\partial K} dilated by a factor of 1 H K {\frac{1}{H_{K}}} . Based on this we can define a sphere 𝕊 K {\mathbb{S}_{K}} with constant mean curvature 1 by considering the union of all horizontal liftings of ∂ ⁡ K {\partial K} starting from ( 0 , 0 , 0 ) {(0,0,0)} until they meet again in a point of the vertical axis. We give some geometric properties of this sphere and, moreover, we prove that, up to non-homogeneous dilations and left-translations, they are the only solutions of the sub-Finsler isoperimetric problem in a restricted class of sets.

Radiation from supersonic faulting

1971 ◽  
Vol 61 (4) ◽  
pp. 1009-1012 ◽  
Author(s):  
J. C. Savage
Keyword(s):  
Initial Point ◽  
Step Function ◽  
Fault Model ◽  
Dislocation Segment ◽  
Far Field ◽  
Fault Propagation ◽  
Unit Step ◽  
Propagation Factor ◽  
Double Couple ◽  
Field Radiation

abstract The far-field radiation from a simple fault model is given by the radiation pattern associated with the appropriate strain nucleus (e.g., double couple) multiplied by a fault propagation factor. For a unilateral fault model the propagation factor is F ( c ; t ) = ζ bd [ H ( τ ) − H ( τ − ( L / ζ ) ( 1 − ( ζ / c ) cos ψ )) ] / ( 1 − ( ζ / c ) cos ψ ) where ξ is the velocity of fault propagation, b is the fault slip, d is the fault width, τ = t − r0/c, r0 is the distance of the observer from the initial point of faulting, c is the velocity of the seismic wave, H(τ) is the unit-step function, L is the length of the fault, and ψ the angle between r0 and the direction of fault propagation. This representation is valid for both subsonic and supersonic fault propagation. The latter case is important because Weertman (1969) has recently shown that spontaneous faulting may propagate at supersonic velocities. Because the propagation factor is always positive, the nodal planes for the radiation are the same as for the appropriate strain nucleus. Finally, it is shown by the application of this equation that the radiation from a screw dislocation segment is represented by the double-couple nucleus, not the compensated linear-vector dipole nucleus as recently suggested by Knopoff and Randall (1970).

scholarly journals Hypersurfaces with Null Higher Order Anisotropic Mean Curvature

Geometry ◽  
2013 ◽  
Vol 2013 ◽  
pp. 1-7
Author(s):  
Hua Wang ◽  
Yijun He
Keyword(s):  
Mean Curvature ◽  
Positive Function ◽  
Higher Order ◽  
Curvature Function ◽  
Convexity Condition ◽  
Relative Nullity ◽  
Anisotropic Mean Curvature ◽  
Set Of Points

Given a positive function on which satisfies a convexity condition, for , we define for hypersurfaces in the th anisotropic mean curvature function , a generalization of the usual th mean curvature function. We call a hypersurface anisotropic minimal if , and anisotropic -minimal if . Let be the set of points which are omitted by the hyperplanes tangent to . We will prove that if an oriented hypersurface is anisotropic minimal, and the set is open and nonempty, then is a part of a hyperplane of . We also prove that if an oriented hypersurface is anisotropic -minimal and its th anisotropic mean curvature is nonzero everywhere, and the set is open and nonempty, then has anisotropic relative nullity .

Minimal surfaces with self-intersections along straight lines. I. Derivation and properties

1999 ◽  
Vol 55 (1) ◽  
pp. 58-64 ◽  
Author(s):  
E. Koch ◽  
W. Fischer
Keyword(s):  
Minimal Surface ◽  
Symmetry Group ◽  
Minimal Surfaces ◽  
Special Kind ◽  
Branch Points ◽  
Euler Characteristics ◽  
Oriented Surface ◽  
Periodic Minimal Surface ◽  
Full Symmetry ◽  
Straight Lines

A special kind of three-periodic minimal surface has been studied, namely surfaces that are generated from disc-like-spanned skew polygons and that intersect themselves exclusively along straight lines. A new procedure for their derivation is introduced in this paper. Several properties of each such surface may be deduced from its generating polygon: the full symmetry group of the surface, its orientability, the symmetry group of the oriented surface, the pattern of self-intersections, the branch points of the surface, the symmetry and periodicity of the spatial subunits demarcated by the surface, and the Euler characteristics both of the surface and of the spatial subunits. The corresponding procedures are described and illustrated by examples.

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