Bézier surfaces of minimal area

The Plateau-Bézier problem with shifted knots is to find the surface of minimal area amongst all the Bézier surfaces with shifted knots spanned by the admitted boundary. Instead of variational minimization of usual area functional, the quasi-minimal Bézier surface with shifted knots is obtained as the solution of variational minimization of Dirichlet functional that turns up as the sum of two integrals and the vanishing condition gives us the system of linear algebraic constraints on the control points. The coefficients of these control points bear symmetry for the pair of summation indices as well as for the pair of free indices. These linear constraints are then solved for unknown interior control points in terms of given boundary control points to get quasi-minimal Bézier surface with shifted knots. The functional gradient of the surface gives possible candidate functions as the minimizers of the aforementioned Dirichlet functional; when solved for unknown interior control points, it results in a surface of minimal area called quasi-minimal Bézier surface. In particular, it is implemented on a biquadratic Bézier surface by expressing the unknown control point P 11 as the linear combination of the known control points in this case. This can be implemented to Bézier surfaces with shifted knots of higher degree, as well if desired.

Download Full-text

Interpolation of data in ℝ3 using quartic triangular Bézier surfaces

10.1063/5.0040457 ◽

2021 ◽

Author(s):

Krassimira Vlachkova ◽

Krum Radev

Keyword(s):

Bézier Surfaces ◽

Bezier Surfaces

Download Full-text

Geometric Design and Fabrication of Developable Bezier Surfaces

18th Design Automation Conference: Volume 2 — Geometric Modeling, Mechanisms, and Mechanical Systems Analysis ◽

10.1115/detc1992-0172 ◽

1992 ◽

Author(s):

R. M. C. Bodduluri ◽

B. Ravani

Keyword(s):

Geometric Design ◽

Developable Surface ◽

Developable Surfaces ◽

Surface Patches ◽

Bézier Surfaces ◽

Point Representation ◽

Cad Cam ◽

C2 Continuity ◽

Type Construction ◽

Bezier Surfaces

Abstract In this paper we study Computer Aided Geometric Design (CAGD) and Manufacturing (CAM) of developable surfaces. We develop direct representations of developable surfaces in terms of point as well as plane geometries. The point representation uses a Bezier curve, the tangents of which span the surface. The plane representation uses control planes instead of control points and determines a surface which is a Bezier interpolation of the control planes. In this case, a de Casteljau type construction method is presented for geometric design of developable Bezier surfaces. In design of piecewise surface patches, a computational geometric algorithm similar to Farin-Boehm construction used in design of piecewise parametric curves is developed for designing developable surfaces with C2 continuity. In the area of manufacturing or fabrication of developable surfaces, we present simple methods for both development of a surface into a plane and bending of a flat plane into a desired developable surface. The approach presented uses plane and line geometries and eliminates the need for solving differential equations of Riccatti type used in previous methods. The results are illustrated using an example generated by a CAD/CAM system implemented based on the theory presented.

Download Full-text