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Abstract
We present a Bzier representation of visually continuous quartics and quintics. Explicit formulas are given for the conversion of the Bzier representation to and vice versa a Hermite-like representation, defined by the continuity conditions. Positivity conditions which insure properties like convex hull and variation diminishing properties are given.Gegeben wird eine Bzier-Darstellung visuell stetiger quartischer und quintischer Kurven. Es werden explizite Ausdrcke fr die Konversion der Bzier in und vice versa eine Hermite Darstellung, definiert durch die Stetigkeitsbedingungen, angegeben. Es werden Positivittsbedingungen, die Eigenschaften wie z.B. die Eigenschaften der konvexen Hlle und der Variationsreduktion sichern, gegeben.
... This gives a wide ranging class of already well known methods but also certain new combinations offairness criteria, constraints and representations, characterizing new methods not developed yet. It can be seen that among others the conventional e(2) cubic splines (with minimal strain energy), the change of curvature minimal splines [13], the segment weighted cubic splines [6], the point weighted cubic splines (v-splines, [14]), the geometric splines [9] and the quintic ,-splines [8] are comprised as special cases proving the great flexibility of this approach. ...
A knowledge base for geometric design has been developed including mainly two types of knowledge for problem formulation resp. problem solution. The problem formulation knowledge serves to classify the problem type according to statements given by the user about geometric data, mathematical representation, criterion function and discrete as well as integral constraints. Problem solution knowledge pertains to the choice of adequate problem solvers for each problem type, all of them regarded as constrained optimization problems. A prototype system was implemented for the design of curves using the declarative programming language PROLOG embedded in a hybrid environment with MATHEMATICA, X11 and C, enabling the user to choose multitudinous combinations of criteria functions and constraints at session time. The presented approach lends itself to achieving great flexibility within a large class of shape generation problems.
The interpolation of a discrete set of data, on the interval [a, b], representing the functionf is obtained using explicit splines. Estimations of interpolation accuracy are obtained.
This paper presents a bibliography of over 1600 references related to computer vision and image analysis, arranged by subject matter. The topics covered include architectures; computational techniques; feature detection, segmentation, and image analysis; matching, stereo, and time-varying imagery; shape and pattern; color and texture; and three-dimensional scene analysis. A few references are also given on related topics, such as computational geometry, computer graphics, image input/output and coding, image processing, optical processing, visual perception, neural nets, pattern recognition, and artificial intelligence.
We say that a curve has geometric continuity if the curve, its unit tangent vector and its curvatures relative to a Frenet frame are continuous. Here we study locally supported basis functions in spaces of piecewise polynomials which can be used for the Bézier representation of geometrically continuous curves. For spaces with identical connection conditions at each knot, we study the dependence of the basis functions on the parameters defining the connection conditions, and give a method for the computation of these functions. We also investigate the effect of the various parameters on the shape of the Bézier curve. For connection conditions prescribed by a totally positive matrix, we obtain conditions for the solvability of the cardinal interpolation problem and give a new proof of the uniqueness and non-negativity of the basis functions.
Im Anschluß an Scheffers [18] wird für ebene Kurven, die in Parameterdarstellung (insbesondere in der Form y = y(x)) gegeben sind, eine allgemeine Theorie der Berührung höherer Ordnung dargelegt (die auch für nicht ebene Kurven im euklidischen Raum Rk (k > 2) gilt). Von berührenden Kegelschnitten einer Kurve werden Bestimmungsstücke angegeben sowie Kriterien dafür, von welcher Art sie sind, wenn sie von höchstmöglicher Ordnung berühren.
Following Scheffers [18] a general theory of higher order tangents is presented for plane curves given in parametric form (in particular in the form y = y(x)); the theory holds also for non-plane curves in a euclidean space Rk (k > 2). The parameters of conic sections tangent to a curve are given as well as criteria for the kind of section in case the conic section is a tangent of the highest possible order.
Visual continuity means tangent and curvature continuity of composed curves and surfaces. Several investigations have been made in the last six years. A historical overview is given and some main results are presented.
Parametrized polynomial spline curves are defined by an S-polygon, but locally they are Bézier curves defined by a B-polygon. Two algorithms are given which construct one polygon from the other and vice versa. The generalization to surfaces is straightforward. This may be of some interest in CAD because of the good local properties of the B-polygon
Parametric polynomial curves in Bézier-Bernstein representation are considered as prohections of rational norm curves of degree n in n-space; from this point of view the singularities of a planar Bézier cubic are determined and expressed by its affine invariants. Secondly, for an arbitrary pair of adjacent parametric curves in homogeneous coordinates, the general conditions for geometric continuity of any order k, Gk, are established.This result generalizes the corresponding conditions in the non-homogeneous (affine) case, recently obtained by [Goodman '84]. Some applications are given for Bézier curves. In particular, for γ-splines [Boehm '85], the existence of a global rational parameter that makes it to a C2 parametric curve is shown. Furthermore, for two adjacent rational Bézier curves the complete conditions for G3 are stated using the projective properties of the control points only.
The paper introduces an effective method for approximate spline conversion. The method uses mainly parameter transformations and nonlinear optimization techniques. Geometric continuity conditions are used as parameter invariant spline conditions. For geometric continuity of order 1, 2, 3, 4, algorithms are introduced for approximate reducing of the polynomial degree of a given spline segment (and splitting into as few spline segments as possible) or elevating the polynomial degree (and merging as many spline segments as possible). The method is extended to spline approximation of offset curves (and splitting into as few new spline segements as possible).
Smooth curves and surfaces Geometric Modelling: Algorithms and New Trends
- W Böhm
Geometric continuity of parametric curves) I-2] B6hm, W: Smooth curves and surfaces Geometric Modelling: Algorithms and New Trends
- B A Barsky
Barsky, B A, De Rose, T D: Geometric continuity of parametric curves. University of California, Berkeley, Computer Science Division (EECS), Technical Report # UCB/CSD 84/205 (1984) I-2] B6hm, W: Smooth curves and surfaces. In: Farin, G. (ed.) Geometric Modelling: Algorithms and New Trends. SIAM (1987) 175-184
W: Visual continuity
- B6hm
B6hm, W: Visual continuity. Computer-aided design, Vol. 20, No. 6 (1988) 307-311
Properties ofbeta Splines
- T Goodman
Bézier Representation of Geometric Spline Curves. A general concept and the quintic case
- D Lasser
- M Eck
Curves and Tensor Product Surfaces with Third Order Geometric Continuity
- H Pottmann
- H. Pottmann