Last update: July 23, 2002
Publications in Computational Geometry on Probing :
BibTeX references.
H. Hagen, University of Kaiserslautern
IEEE Computer Graphics and Applications, pp.53-60, 1992.
Display Gaussian curvature through the use of focal surfaces:
Christof M. Hoffmann
Computer-Aided Geometric Design, vol. 7, pp. 517-532, 1990.
We propose a paradigm for analyzing problems involving complex curved surfaces in a manner suitable for practical implementation. Rather than deriving closed-form expression for certain surfaces and problems, we reformulate the problem in a higher-dimensional space with more variables but simpler equations, thus avoiding complex symbolic manipulation and numerically delicate operations. In particular, we consider the formation of offsets, spherical blends of fixed or variable radius, and Voronoi surfaces.
Advantages of a higher-dimensional space reformulation:
The envelope of a parameterized family S(a) = 0 of surfaces, with a a vector of m independent parameters, satisfies the system of equations:
where equations (2) are the m 1st-order partial derivatives of S(a) by each component u of a.
Let the surface f be given in parametric form as :
Then its r-offset is described by the system:
where S(s,t) = [x - f1(s,t)]^2 + [y - f2(s,t)]^2 +[z - f3(s,t)]^2 .
Let the surface be given by f(x,y,z) = 0. Let p = (u1, u2, u3) be a regular point of f, and let Df(p) = (a,b,c) be the surface gradient at p.
Construct the r-offset from spheres S centered on the surface:
Form directional derivatives by multiplying the vector of partials of S by the ui with 2 LI vectors perpendicular to the gradient at p. E.g., take the 2 perpendiculars (-b, a, 0) and (-c, 0, a) to get:
where DS = (dS/du1 , dS/du2 , dS/du3)
Then, the r-offset of f is described by the system of equations:
NB: The choice of various pair of perpendiculars may introduce extraneous solutions.
In principle, the implicit equation of the r-offset may be recovered by eliminating s and t (in the parametric form) or (u1, u2, u3). This can be done using resultants methods, but may lead to a final equation additional extraneous factors diificult to find and eliminate. Alternatively, Grobner basis techniques can be used to avoid this extraneous factors problem (but this leads to more complex computations).
Furthermore, in practice, one usually does NOT have exact surface equations as an input. Thus the symbolic computations work with rational coefficient approximations of uncertain accuracy, or, else, incur arithmetic errors (hard to predict).
Vor(f,g) = { p = (x,y,z) | df(p) = dg(p) }
Let the 2 surfaces be given parametrically by:
Then the Voronoi surface can be characterized via variable r-offsets :
This system gives a description of Vor(f, g) in an 8th dimensional space.
The implicit equation of Vor(f, g) can, in principle, be recovered by elimination of sf, tf , sg, tg & r . This is hardly practical but for the simplest surfaces.
Blending surface : Surface F that intersects both f anf g tangentially along some curves.
Constant radius blend : Blending surface that has a family of principal curvature lines consisting of circles of fixed radius.
Variable radius blend : Blending surface that has a family of principal curvature lines consisting of circles whose radius may vary.
Envelope surface of a family of spheres with fixed or varying radius, whose centers are constrained to lie on a space curve (the spine). A Variable radius blending surface is a canal surface (Monge).
Variable radius blend of 2 surfaces f and g (operational) :
The family of spheres whose envelope is the desired blending surface is given by:
To define the envelope of the family, we must add an equation that is the derivative of Sh in the tangent direction (to the spine); this tangent can be obtained via the cross product of the surface normals of Vor(f, g) and of h .
Joseph Hoschek, 1985.
CAGD, Technical University Darmstadt, Germany
From a travel report by Christoph M. Hoffmann (Purdue U.) on June 13, 1995 :
The problem of surface smoothing arises in CAD in a variety of contexts. For example, suppose that CAD system A generates a parametric surface. Through a sequence of operations, a derived surface is obtained that has many patches, and the problem is to reapproximate the derived surface with fewer patches. The research work I saw in this problem domain approaches the problem from two perspectives. First, given a set of points and a tolerance, find a smooth approximating surface that is within tolerance from the data points. Second, given a curve or surface, reapproximate it to within tolerance by a smoother curve or surface.
The notion of ``smoothness'' is explored minimizing an energy-based functional. Several integral formulations are considered, each derived from the intrinsic geometry of the surface, and each having a specific global effect. For instance, one formulation may favor less Gaussian curvature and so ``flatten'' the surface, while another may minimize mean curvature and so try to reshape the surface closer to a minimal surface. The exploration of these different measures of smoothness, and the techniques for combining them, are strongly guided by application problems from the automotive sector, from turbine blade design, and from precision optics.
S.S.Skiena
1989.
[Brief survey]
Springer, Heidelberg, Germany, 2002.
Graduate textbook.
Shape interrogation is the process of extraction of information from a geometric model. It is a fundamental component of Computer Aided Design and Manufacturing (CAD/CAM) systems. The authors focus on shape interrogation of geometric models bounded by free-form surfaces. Free-form surfaces, also called sculptured surfaces, are widely used in the bodies of ships, automobiles and aircraft, which have both functionality and attractive shape requirements. Many electronic devices as well as consumer products are designed with aesthetic shapes, which involve free-form surfaces. This book provides the mathematical fundamentals as well as algorithms for various shape interrogation methods including nonlinear polynomial solvers, intersection problems, differential geometry of intersection curves, distance functions, curve and surface interrogation, umbilics and lines of curvature, geodesics, and offset curves and surfaces.
Keywords: Differential Geometry, Nonlinear Polynomial Solver, Intersections, Offsets, Robustness.
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1998-2002
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