In this paper, we show that B -spline curves and surfaces can be viewed as digital filters. Viewing B -spline problems as digital filters allows one to predict some properties of the generated curves and surfaces. We ...
In this paper, we show that B -spline curves and surfaces can be viewed as digital filters. Viewing B -spline problems as digital filters allows one to predict some properties of the generated curves and surfaces. We find that even-order B -splines and odd-order B -splines behave differently when used in curve and surface interpolation. Even-order B -splines generate smoother curves and surfaces than do odd-order B -splines.
Rational curves and splines are one of the building blocks of computergraphics and geometricmodeling. Although a rational curve is more flexible than its polynomial counterpart,many properties of polynomial curves ar...
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ISBN:
(纸本)0819402982
Rational curves and splines are one of the building blocks of computergraphics and geometric
modeling. Although a rational curve is more flexible than its polynomial counterpart,
many properties of polynomial curves are not applicable to it. For this reason it is very useful
to know if a curve presented as a rational space curve has a polynomial parametrization.
In this paper, we present an algorithm to decide if a polynomial parametrization exists,
and to compute the parametrization.
In algebraic geometry it is known that a rational algebraic curve is polynomially parametrizable
if it has one place at infinity. This criterion has been used in earlier methods to test
polynomial parametrizability of space curves. These methods project the curve into the
plane and test parametrizability there. But this gives only a sufficient condition for the
original curve. In this paper we give a simple condition which is both necessary and sufficient
for polynomial parametrizability. The calculation of the polynomial parametrization
is simple, and involves only a rational reparametrization of the curve.
Q uadric surfaces such as cylinders and spheres play a fundamental role in CAGD.This paper describes a new method for creating triangular surface patches on a quadricsurface. The surface patches are defined using a re...
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ISBN:
(纸本)0819402982
Q uadric surfaces such as cylinders and spheres play a fundamental role in CAGD.
This paper describes a new method for creating triangular surface patches on a quadric
surface. The surface patches are defined using a restricted type of quadratic Bezier
control polyhedron. The control polyhedron and the resulting quadric surface patch
satisfy all of the standard properties of parametric Bezier surfaces, including interpolation
of the corners of the control polyhedron and the convex hull property. A new
technique for creating a C1 mesh of these quadric surface patches is also introduced.
We introduce a subdivision algorithm for shape preserving function interpolationin 111 and JR2 . The method is based on iterative knot insertion andguarantees preservation of convexity. Starting from data points, a se...
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ISBN:
(纸本)0819402982
We introduce a subdivision algorithm for shape preserving function interpolation
in 111 and JR2 . The method is based on iterative knot insertion and
guarantees preservation of convexity. Starting from data points, a sequence of
piecewise linear function is generated. The sequence can be shown to be convergent
to a C1 function. The process is specially suitted for curve and surface
generation in CAGD since it is local and the computation can be stopped whenever
the desired visual effect is attained.
A survey of some techniques that may have potential for free-form modeling with algebraic surfaces is continued. Classical results as well as several recent innovations are included. Specific attention is paid to cubi...
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A survey of some techniques that may have potential for free-form modeling with algebraic surfaces is continued. Classical results as well as several recent innovations are included. Specific attention is paid to cubic algebraic surfaces, although many of the ideas presented have application to algebraic surfaces of any degree. Topics addressed include piecewise constructions, interpolation to points and space curves, and parameterization.
Among all trajectories in the plane that have a given location and direction of endpoints, the one that minimizes the integral of the squared curvature is defined as the minimal energy trajectory. Plane trajectories m...
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Among all trajectories in the plane that have a given location and direction of endpoints, the one that minimizes the integral of the squared curvature is defined as the minimal energy trajectory. Plane trajectories minimizing a length-normalized and thus scale-invariant energy measure are discussed along with algorithms for obtaining them. It is shown that a scale-invariant measure is more natural for the design of interpolation and shape completion curves, and with this measure, circular arcs are optimal in a large number of situations. A simple numerical procedure is proposed for computing piecewise linear approximations of optimal trajectories as a solution of discrete two-point boundary value problems. Such trajectories are useful in computergraphics, geometric design, and motion planning of robots.
Discrete Beta-splines arise when a Beta-spline curve is subdivided; that is, extra knots are inserted so that the curve is expressed in terms of a larger number of control vertices and Beta-splines. Their properties a...
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Discrete Beta-splines arise when a Beta-spline curve is subdivided; that is, extra knots are inserted so that the curve is expressed in terms of a larger number of control vertices and Beta-splines. Their properties and an algorithm for their computation are given in “Discrete Beta-Splines” by Joe (computergraphics, vol. 21, pp. 137-144). We prove a stronger version of one of these properties, from which a new algorithm for computing discrete Beta-splines is obtained. This algorithm can also be used to compute discrete B-splines. We give a comparison of operation counts for this algorithm versus other algorithms, and for two methods to compute the new control vertices of Beta-spline and B-spline curves and surfaces.
A set of methods is presented for detecting complex surfaces, using collections of simple, uniformprocesses. The methods are designed to detect surfaces in range data with parameters that can beestimated from local re...
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ISBN:
(纸本)0819402982
A set of methods is presented for detecting complex surfaces, using collections of simple, uniform
processes. The methods are designed to detect surfaces in range data with parameters that can be
estimated from local regions (for example, natural quadrics such as spheres). The system uses
combinations oflocal estimates of zeroth and first derivative properties, to produce votes for specific
parameterizations. Accumulations of votes lead to hypothesized surfaces. A conflict resolution
strategy is used to separate the true surface hypotheses from the false ones. The overall approach is
based on the ideas ofthe Hough transform and parameter space methods, but is designed to explicitly
address shortcomings of these techniques, while maintaining their modularity and efficiency. Examples
of these techniques, used to detect natural quadrics in real, low resolution range data scenes,
are presented.
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