Fairness Criteria for Algebraic Curves (original) (raw)
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This paper details the use of automatic differentiation in form parameter driven curve design by constrained optimization. Computer aided design, computer aided engineering (CAD/CAE), and particularly computer aided ship hull design (CASHD) are typically implemented as interactive processes in which the user obtains desired shapes by manipulation of control vertices. A fair amount of trial and error is needed to achieve the desired properties. In the variational form parameter approach taken here, the system computes vertices so that the resulting curve meets the specifications and is optimized with respect to a fairness criteria. Implementation of curve design as an optimization problem requires extensive derivative calculations. The paper illustrates how the programming burden can be eased through the use of automatic differentiation techniques. A variational curve design framework has been implemented in Python, and applications to CASHD curve design are shown. The new method is robust and allows great flexibility in the selection of constraints. Offsets, tangents, and curvature may be imposed anywhere along the curve. Form parameters may also be used to define straight segments within a curve, require the curve to enclose specified forms, or specify relationships between curve properties.
A General Frame for the Construction of Constrained Curves
Proceedings of the Conference on Applied Mathematics and Scientific Computing, 2005
The aim of the present paper is to review the basic ideas of the so called abstract schemes (AS) and to show that they can be used to solve any problem concerning the construction of spline curves subject to local (i.e. piecewise defined) constraints. In particular, we will use AS to solve a planar parametric interpolation problem with free knots.
Scale-Invariant Minimum-Cost Curves: Fair and Robust Design Implements
Computer Graphics Forum, 1993
Four functionals for the computation of minimum cost curves are compared. Minimization of these functionals result in the widely studied Minimum Energy Curve (MEC), the recently introduced Minimum Variation Curve (MVC), and their scale-invariant counterparts, (SI-MEC, SI-MVC). We compare the stability and fairness of these curves using a variety of simple interpolation problems. Previously, we have shown MVC to possess superior fairness. In this paper we show that while MVC have fairness and stability superior to MEC they are still not stable in all configurations. We introduce the SI-MVC as a stable alternative to the MVC. Like the MVC, circular and helical arcs are optimal shapes for the SI-MVC.
Minimum curvature variation curves, networks, and surfaces for fair free-form shape design
1983
Traditionally methods for the design of free-form curves and surfaces focus on achieving a specific level of inter-element continuity. These methods use a combination of heuristics and constructions to achieve an ultimate shape. Though shapes constructed using these methods are technically continuous, they have been shown to lack fairness, possessing undesirable blemishes such as bulges and wrinkles. Fairness is closely related to the smooth and minimal variation of curvature.
Cubic algebraic curves based on geometric constraints
Computer Aided Geometric Design, 2001
Methods for curve modeling with cubic algebraic curves based on geometric constraints are introduced in this paper. A 1-parameter family of cubic curves with four given points as well as two tangent lines at the endpoints is constructed in the first part of the paper. A 1-parameter family and 2-parameter family of cubic curve constructions are presented for interpolating two given endpoints and two given tangent lines as well as two given curvatures at the endpoints in the last part of the paper.
Constrained design of polynomial surfaces from geodesic curves
Computer-aided Design, 2008
In a recent work, Wang et al. [Wang G, Tang K, Tai CH. Parametric representation of a surface pencil with common spatial geodesic. Computer-Aided Design 2004;36(5): 447-59] discuss a constrained design problem appearing in the textile and shoe industry for garment design. Given a model and size, the characteristic curve called girth is usually fixed, and preferably should be a geodesic for manufacturing reasons. The designer must preserve this girth, being allowed to modify other areas according to aesthetic criteria. We present a practical method to construct polynomial surfaces from a polynomial geodesic or a family of geodesics, by prescribing tangent ribbons. Differently from previous procedures, we identify the existing degrees of freedom in terms of control points, and our method yields parametric polynomial surfaces that can be incorporated into commercial CAD programs. The extension to rational geodesics is also outlined.
Computer-Aided Design, 1994
Variational geometry is a powerful method for the definition and modification of geometric models, as it constrains object geometry as sets of functional constraints rather than nominal Cartesian elements. This allows one to capture design intent by specifying geometric and engineering constraints that, when resolved using a nonlinear equation solver, define the geometry of the object. The paper explores some of the issues raised when rational B6zier curves are used to generate the edges of a model, that is, specifically, the representation of basic model edges: straight lines and conics. Rational B6zier curves are investigated as they can be used to represent conics precisely in addition to being well equipped to represent and manipulate free-form curves and surfaces.
Regular algebraic curve segments (III)—applications in interactive design and data fitting
Computer Aided Geometric Design, 2001
In this paper (part three of the trilogy) we use low degree G 1 and G 2 continuous regular algebraic spline curves defined within parallelograms, to interpolate an ordered set of data points in the plane. We explicitly characterize curve families whose members have the required interpolating properties and possess a minimal number of inflection points. The regular algebraic spline curves considered here have many attractive features: They are easy to construct. There exist convenient geometric control handles to locally modify the shape of the curve. The error of the approximation is controllable. Since the spline curve is always inside the parallelogram, the error of the fit is bounded by the size of the parallelogram. The spline curve can be rapidly displayed, even though the algebraic curve segments are implicitly defined.