Implicit Gn-blending of vertices

doi:10.1016/S0167-8396(01)00030-9

Computer Aided Geometric Design

Volume 18, Issue 3, April 2001, Pages 267-285

https://doi.org/10.1016/S0167-8396(01)00030-9 Get rights and content

Abstract

Implicit vertex blending methods are introduced which generate pencils of surfaces $G^{n}$ -continuous to three given surfaces. There are solutions for the three possible corners (suitcase-, house-, 3-beam-corner) between three transversally intersecting surfaces. The basic tools are functional splines (implicitly defined blending surfaces). Several examples show applications of functional splines and the implicit vertex blending. All implicit methods can be applied to more general surfaces, especially to parametrically defined surfaces, via the normalform of a surface.

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Implicitly defined curves and surfaces have their advantages, such as (1) the closure property under some geometric operations (intersection, union, offset etc.), (2) the implicit algebraic equation presentation captures all elements of that set, (3) implicit algebraic curve segments have more degrees of freedom than parametric curves, and (4) the algorithms for curve and surface fittings do not need the parametrization of the data. Due to the above advantages, people began to pay much attention to the study of modeling, especially blending, with implicit algebraic curves and surfaces [1,3,7,8,10–14,18,21,25,27]. The paper is organized as follows: Section 2 presents the functional splines and symmetric functional splines with different degrees of smoothness; Section 3 gives some applications of functional splines and symmetric functional splines; and Section 4 provides the conclusion of this paper.
Implicit curves and surfaces are extensively used in interpolation, approximation and blending. [Li J, Hoschek J, Hartmann E. $G^{n - 1}$ -functional splines for interpolation and approximation of curves, surfaces and solids. Computer Aided Geometric Design 1990;7:209–20] presented a functional method for constructing $G^{n - 1}$ curves and surfaces which are called functional splines. In this paper, functional splines with different degrees of smoothness are presented and applied to some typical problems.

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