Error propagation in geometric constructions
Introduction
The aim of this paper is to show how to treat some problems of error propagation in geometric constructions in a geometric way.
A geometric construction is a procedure which takes geometric objects (points, lines, circles) as input, and gives a geometric object as output. It must be invariant with respect to some geometry, which is best explained by an example.
We consider a very simple geometric construction: the intersection of two lines in the Euclidean plane. The input consists of two lines, and the output is a point. If we translate or rotate the input data, the output undergoes the same transformation. This means invariance with respect to Euclidean transformations. Another example is evaluation of a Bézier curve at a certain parameter value. If the control points of the curve undergo an affine transformation, the curve point is transformed by the same transformation. This means that evaluation of Bézier curves is invariant with respect to affine transformations.
If we speak of error propagation, we mean the following: suppose each item of the input data can vary independently in some domain (for instance, a point varies in a disk). We can think of input data given imprecisely or of tolerance zones for the input data. We ask for the set of all possible outputs. If this is not possible, we would at least want to know some tolerance zone which contains all possible outputs (cf. [1]).
We want to determine the precise locus of output points, if the tolerance zones of the input points are known. If the operation performed is a geometric construction, i.e. is a geometric invariant, then this locus of possible outputs is automatically a geometric invariant. This is very satisfactory from the mathematician's viewpoint, because the answer to this question should not contain artifacts of a coordinatization anyway.
From the computational point of view however it might be desirable to have a faster, but imprecise estimate of the tolerance zone of the output. In this paper we do not discuss fast computation, but focus on exact tolerance zones.
Interval arithmetic (see [2], [3], [4], [5], [6], [7]) is one of the basic tools, if one has bounds for input data of some calculations, and wants to compute bounds for the output. A simple example shows how interval arithmetic is not ‘geometric’ in the sense that it does not give exact error bounds.
Suppose the point (x,y) has coordinates x∈[1−ϵ,1+ϵ], y∈[−ϵ,ϵ]. If we rotate this point about 45°, we know its image (x′,y′) to have coordinates If we know only the bounds for x′, y′ independently, a further rotation about 45° gives the point (x″,y″) whose coordinates are bounded by x″∈[−2ϵ,2ϵ], y″∈[1−2ϵ,1+2ϵ], whereas rotation of (x,y) about 90° gives the bounds x″∈[−ϵ,ϵ], y″∈[1−ϵ,1+ϵ].
A computational scheme which handles error bounds and tolerances in a geometric way is expected to rotate the tolerance square of the point (x,y), but never to increase its size.
Nevertheless in Section 2.4 we show that interval arithmetic fits in a natural way into our approach.
In this paper we restrict ourselves to two different types of problems: first, we consider geometric constructions which are affine or even convex combinations of points, which is a geometric operation in affine geometry—if the input data undergo an affine transformation, the output does the same.
This includes most of the spline curves defined by control points. We assume that the control points can vary independently in closed convex domains: for a parameter value t we look for the locus of possible curve points. We will always assume that the error pertinent in the computation of the curve point is negligible in comparison to the effect of changing the control points in their various domains. So the problem reduces to the problem of affine or convex combination of planar or spatial domains.
As a second example we consider an elementary Euclidean geometric construction: the circumcircle of three points. The difference between these two is that the former is affinely invariant, involving only the linear structure of real vector space n, whereas the latter is a Euclidean construction which involves the Euclidean orthogonality relation and metric.
In general, metric constructions are not as easy to analyze as affine ones. A detailed algorithmic study of geometric constructions involving lines and circles is given by Ref. [8]. Applications to collision problems involving toleranced objects are studied in Ref. [9]. Nevertheless there is still much to do in this field.
Another topic is the inverse problem: given a geometric construction and the tolerance zones of the output, how must we choose the tolerance zones of the input points?
Section snippets
Elementary facts about convex bodies
One of the main tools for studying compact convex bodies of d is the support function. There is a large amount of literature including some monographs (see e.g. Ref. [10] for a detailed overview of the whole field of convex geometry). For the convenience of the reader we repeat some basic facts.
We call a plane ϵ a support plane of K if K has a point in common with ϵ and K is entirely contained in one of the two closed half-spaces defined by ϵ. For all unit vectors n there is a unique plane ϵ(n
Circles in plane and space
In this section we study a geometric problem of Euclidean geometry: the circumcircle of three points. It occurs e.g. in the construction of a Delaunay triangulation. It turns out that the problem is more complicated than the affinely invariant problems of the previous section, and we will not study it in its maximum possible generality.
We describe the set of circles in the Euclidean plane which meet three circular disks K1, K2, K3. The geometric setting where this type of problem is most easily
Conclusions
In the first part of the paper we studied affine and convex combinations of points from the viewpoint of error propagation. An affine or convex combination of points, which are known to be contained in certain convex domains, is itself contained in an appropriate combination of these domains. This leads to precise bounds for Bézier and spline curves with toleranced control points.
The second part of the paper deals with a geometric construction of Euclidean geometry: the circumcircle of three
Acknowledgements
This work was supported in part by the Lithuanian Foundation of Studies and Science, and by Grant No. 12252-MAT of the Austrian Science Foundation.
Johannes Wallner is a member of the mathematics department at the University of Technology in Vienna, Austria. He received his PhD in 1997. His research interests include geometry and its applications to Computer Aided Geometric Design. Copies of recent publications can be downloaded from http://www.geometrie.tuwien.ac.at/wallner.
References (19)
- et al.
Efficient and reliable methods four rounded-interval arithmetic
Computer-Aided Design
(1998) - et al.
Polynomial/rational approximation of Minkowski sum boundary curves
Graphical Models and Image Processing
(1998) - et al.
Offset of polynomial Bézier curves: Hermite approximation with error bounds
- et al.
Disk Bézier curves
Computer Aided Geometric Design
(1998) - et al.
Applications of Laguerre geometry in CAGD
Computer Aided Geometric Design
(1998) Towards a theory of geometry tolerancing
International Journal of Robotics Research
(1983)- et al.
Approximation of measured data with interval B-splines
Computer-Aided Design
(1997) - et al.
Robust interval algorithm for surface intersections
Computer-Aided Design
(1997) - et al.
Numerical and geometric properties of interval B-splines
International Journal of Shape Modeling
(1998)
Cited by (21)
Collision free region determination by modified polygonal Boolean operations
2013, CAD Computer Aided DesignCitation Excerpt :used interval arithmetics to ensure robustness. Wallner et al. [22] showed that interval arithmetic is not geometric in the sense that it does not give exact error bounds.
Two approaches to geometrography
2010, Journal for Geometry and GraphicsSemantic tolerance modeling with generalized intervals
2008, Journal of Mechanical DesignSolving interval constraints by linearization in computer-aided design
2007, Reliable ComputingMinimal enclosing hyperbolas of line sets
2007, Beitrage zur Algebra und GeometrieA geometrical approach to the numerical stability analysis of some projective collinear mapping methods
2007, Journal for Geometry and Graphics
Johannes Wallner is a member of the mathematics department at the University of Technology in Vienna, Austria. He received his PhD in 1997. His research interests include geometry and its applications to Computer Aided Geometric Design. Copies of recent publications can be downloaded from http://www.geometrie.tuwien.ac.at/wallner.
Rimvydas Krasauskas is an associate professor in the Department of Mathematics and Computer science, Vilnius University, Lithuania. His current research interests are in geometry of real rational surfaces and their applications in computer aided geometric design. He graduated from Moscow State University in 1982, where he studied computer science and numerical mathematics. In 1988 he received a PhD in pure mathematics (algebraic topology). Since then he has been the head of the chair Geometry and Topology at Vilnius University. The change from algebraic topology to CAGD in the period of 1994–1996 was partially supported by an American Mathematical Society grant for fSU.
Helmut Pottmann is professor of geometry at the University of Technology in Vienna, Austria. His research interests include classical geometry and its applications, especially to Computer aided Geometric Design, kinematics, and the relations between geometry, numerical analysis and approximation theory. A list of recent publications can be found at http://www.geometrie.tuwien.ac.at/pottmann.
- 1
Tel.: +43-1-58801-11301.