Abstract
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.
Highlights
We observe that the nature behaves in a way that certain quantity is either a maximum or a minimum of some quantity, in various phenomena occurring in our universe
The variational minimization of the Dirichlet functional to obtain a quasi-minimal Bézier surface with shifted knots is quite useful for further geometric analysis of the surface obtained, in particular, for geometry-related quantities like Gaussian and mean curvature
A surface is said to be minimal if its mean curvature vanishes everywhere on the surface, which is the outcome of variational minimization of area functional
Summary
We observe that the nature behaves in a way that certain quantity is either a maximum or a minimum of some quantity, in various phenomena occurring in our universe. The minimal surfaces can be built through a variational technique by finding solution of vanishing condition of gradient of a constraint usually in the form of an integral having the property of minimizing something, for example, minimizing an energy integral These variational techniques are advantageous for a class of surfaces, better known as Bézier surfaces admitting useful properties appropriate for applications in CAGD. For given prescribed border as the control points for a Bézier surface, we can find the gradient of the constraint integral (area integral or some energy integral) and the vanishing condition of the gradient of the functional gives system of linear algebraic equations expressing the unknown interior control points in terms of known boundary control points. The related Plateau-Bézier surface with shifted knot problem comprises of identifying the shifted knots Bézier surfaces of minimal area amongst all the shifted knots Bézier surfaces with accepted boundary
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