Error propagation in geometric constructions

Abstract

In this paper we consider error propagation in geometric constructions from a geometric viewpoint. First we study affine combinations of convex bodies: this has many applications concerning spline curves and surfaces defined by control points. Second, we study in detail the circumcircle of three points in the Euclidean plane. It turns out that the right geometric setting for this problem is Laguerre geometry and the cyclographic mapping, which provides a point model for sets of circles or spheres.

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 $\text{x′,y′∈[}\text{2}\text{/2−}\text{2}\text{/2·ϵ,}\text{2}\text{/2+}\text{2}\text{/2·ϵ].}$ 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 $\text{R}$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?