Two topics in topology, the comparison of plane curves and faces on geometric models, are discussed. With regard to the first problem, a curve is defined to be a locus of points without any underlying parameterization. A metric on a class of plane curves is defined, a finite computation of this metric is given for the case of piecewise linear curves, and it is shown how to approximate curves that have bounded curvature by piecewise linear curves. In this way a bound on the distance between two curves can be computed. With regard to the second problem, the questions to be discussed are under what circumstances do geometrical faces make sense; how can they be explicity defined; and when are these geometrical faces homeomorphic to the realization of the abstract (topological) face

Emery, J. D.

English

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THE DEFINITION AND COMPUTATION 
OF A METRIC ON PLANE CURVES 
THE MEANING OF A "FACE" ON A GEOMETRIC MODEL 
James D. Emery 
Allied Bendix Aerospace 
Kansas City, Mo. 
I shall talk about two separate topics as indicated in the 
title. The first topic concerns the comparison of plane 
curves. This work arose from the following problem. A 
surface on a part is generated by a plane curve, which is an 
interpolating spline. The part is manufactured by an 
outside vendor using the given interpolation points, but an 
unknown interpolation technique. The points on the machined 
curve must be checked to verify that they lie sufficiently 
close to the original definition. In this case the vendor 
supplies what is known as the APT-CL-file, which is a set of 
points defining a piecewise linear curve that defines the 
actual movement of the machine tool. APT (Automatic 
Programmed Tool) is a computer program for controlling 
numerical machine tools. The problem was to calculate a 
"distance" between these two curves. 
There are many other cases where a curve is approximated by 
some process or algorithm. For example, plotting a curve is 
usually an approximation technique, and the generation of an 
offset curve may involve approximation. Other applications 
include the comparison of surfaces and the inspection of 
parts. Surfaces may be compared by examining nets of plane 
curves. Part inspection may involve the determination of a 
curve or surface from a finite set of measurements. 
When the curves are parametric we can use function 
approximation theory to compare their coordinate functions. 
However, this would make the comparisons dependent on the 
particular parameterizations. We could use a canonical 
parameterization, such as parameterization by arc length, 
but this is hard to compute and the two curves that we wish 
to compare may not have the same total arc length. 
We shall define a curve to be a locus of points without any 
underlying parameterization. We shall define a metric on a 
class of plane curves, give a finite computation of this 
metric for the case of piecewise linear curves, and show how 
to approximate curves that have bounded curvature by 
piecewise linear curves. In this way we can compute a bound 
on the "distance" between two curves. These techniques have 
been implemented in Fortran. 
We shall prove some standard preliminary propositions. Let 
(M,d) be a metric space and A a subset of M. Define 
https://ntrs.nasa.gov/search.jsp?R=19850020259 2020-03-22T19:37:29+00:00Z
DEFINITION. Let A and B be compact sets. The distance 
between A and B is defined to be 
~ ( A , B )  = ~aximuml Sup (d(x,~)), Sup (d(x,~))} 
xeB x eA 
This is the radius of the largest circle that has center on 
one of the curves, and just touches the other. Note: we 
have given different meanings to "d." The particular 
meaning intended is determined by context. 
PROPOSITION. Either there exists an x in A so that ~ ( A , B )  =
d(x,B), or there is an x in B so that d(A,B) = d(x,~). 
PROPOSITION. Given a family F of compact sets, (F,d) is a 
metric space. 
We will prove the triangle inequality. Without loss of 
generality assume d(A,C) = d(x,C) for some x in A. Since 
f(w)=d(x,w) is continuous on the compact set B, there exists 
a y in B so that d(x,~) = d(x,y). By definition 
Also there exists a z in C so that d(y,~) = d(y,z) 
We now have 
Thus 
DISTANCE FROM A POINT TO A LINE SEGMENT 
This computation is fairly obvious. The distance is either 
the distance from the point to the line containing the 
segment, or the distance from the point to one of the end 
points of the segment. 
FINITE COMPUTATION FOR PIECEWISE LINEAR CURVES 
The piecewise linear curve is a union of line segments. We 
call the end points of the segments knots. Given two curves 
A'and B we say that they satisfy the knot condition if the 
metric is realized at either a knot of A or a knot of B. 
That is, either there is a knot a in A so that 
or there is a knot b in B so that 
PROPOSITION. Suppose the pair (A,B) does not satisfy the 
knot condition. Suppose ~ ( A , B )  = sup{d(x,B)). Then there 
exists a point a in A so that ~ ( A , B )  = d(a,~) and the circle 
of radius r = d(A,B) and center a meets B in two or more 
points. 
PROOF. Define ~ ( x )  to be a line segment that contains point 
x (there will be two such segments if x is a knot). Assume 
that for every a such that d(a,B) = ~(A,B) the circle of 
radius r = ~(A,B) and center a meets B in only one point. 
Let a be a point so that 
Then the circle of radius r = ~ ( A , B )  and center a meets B in 
only one point b and 
Let T be the line tangent to the circle at b. There are 
three cases. 
(1) L(a) and T are not parallel. 
( 2 )  L(a) is parallel to T and some L(b) is not parallel to 
T. 
( 3 )  L(a) and every L(b) are parallel to T. 
Case 1. The line T separates the plane into two half 
spaces. The circle lies in one and every L(b) lies in the 
other. This follows from the definition of d(A,B). Because 
the circle touches B only at b and T is not parallel to 
L(a), we may move the center a to a new center a' so that 
the radius of the circle increases, and does not contain a 
point of B in its interior (see the left figure). But this 
contradicts the definition of d(A,B). 
Case 2. T is parallel to L(a) and some L(b) does not lie in 
T. Again we may slightly alter a to a' to increase the 
radius of the circle (see the right figure). This is a 
contradiction. 
Case 3. L(a) is parallel to T and every ~ ( b )  lies in T. 
Then we may translate the circle along L(a) until one of the 
following occurs. 
(i) The center becomes a knot of A. 
(ii) The circle contacts a second point of B. 
(iii)  Some L(b) does not lie in T. 
E v e n t ( i )  cannot  happen because o u r  cu rves  do n o t  s a t i s f y  t h e  
kno t  c o n d i t i o n ,  and (ii) cannot  happen because  o u r  assumption 
was that a could not be chosen so that the circle contacts B 
in two or more points. Therefore (iii) must occur. 
However, we have shown that cases (1) and (2) above are 
contradictions. Thus (iii) is a contradiction. Therefore 
our original assumption is false and the proposition is 
proved. 
With the aid of this proposition we can construct an 
algorithm to compute ~(A,B). Obviously the circle can not 
contact a single segment in two points. Thus we need only 
compute all circles that have centers on A and just touch B 
at two points. The points where the circle contacts B must 
be either tangent points, or knots. That is if the circle 
C(a,r), which has center a and radius r, contacts B at b 
where b is an element of segment I, and b is not an end 
point, then C(a,r) is tangent to I. There are. only three 
cases. 
(1) The center a lies on the locus of points equidistant 
from two knots of B. 
(2) The center lies on the locus of points equidistant from 
two lines. The lines contain segments of B. 
( 3 )  The center lies on a locus of points equidistant from a 
knot of B and a line, which contains a segment of B. 
Thus we need only compute perpendicular bisectors of 
segments, angle bisectors of angles, and parabolas. The 
parabolas are defined by the knot-line pairs of B. The knot 
is the focus and the line is the directrix. 
To find all the candidates for centers, we intersect the 
various loci with the curve A. We must also of course 
reverse the roles of A and B in case d(A,B) = d(b,~) for 
some b in B. There are then four cases in total, including 
the case of a center occurring at a knot. These cases are 
illustrated below. 
The various loci are computed using projective 
representations and being careful to avoid excessive 
round-off error and making sure that special cases are 
handled correctly. 
We can avoid doing the computations for all point-point, 
point-line, and line-line pairs by computing certain 
distances between segments and establishing certain 
inequalities. 
THE PIECEWISE LINEAR APPROXIMATION 
We will show that if we know a bound on the curvature then 
we can construct a piecewise .linear curve that approximates 
a given curve arbitrarily closely. 
Let g be a function whose graph is a circular arc of radius 
r. ~ k e  center of the arc is below the x-axis and it passes 
through the points (a,O) and (b,O) where a < b. Let ~ ( f )  be 
the magnitude of the curvature of f. 
PROPOSITION. Let f be a twice differentiable function such 
that f(a) = f(b) = 0. If f is greater than g, anywhere on 
the interval (arb), then there is a point in (arb) where 
PROOF. Let h = f - g,. Suppose that f is greater than g, 
at some point of (arb). Let x be a point of (a,b) where the 
maximum value of h is attained. Therefore 
h1(x) = 0 and h"(x) I 0 
Theref ore 
fl(x) = grl(x), because h'(x) = 0. It follows from the 
expression for the curvature that 
We can repeat the argument with a slightly smaller r. Thus 
COROLLARY. If ~ ( f )  I l/r on (arb) then f Ig, on (a,b). 
PROPOSITION. Let C be a curve connecting two points. Suppose 
K(C) < 1 .  Let L be the line segment joining the two 
points. Suppose any line passing through the segment and 
perpendicular to it meets C in exactly one point. Let the 
length of the segment be 4. If 
then 
This proposition provides an obvious construction of a 
piecewise linear curve that approximates a given curve to 
arbitrary accuracy provided we know a bound on the 
curvature. 
Perhaps these techniques can be generalized to surfaces 
using approximating polygons. However, for curves the 
piecewise linear approximation in some sense approximates 
both curvature and arc length. For surfaces the analog is 
not true. It is well known that triangulations can be found 
for surfaces in such a, manner that the triangles 
approximate the surface arbitrarily closely, yet have 
arbitrarily large surface area (see [I]). 
FACES 
The second topic I wish to talk about concerns faces on 
geometric models. Systems for representing the surface of a 
solid model have been constructed (see [4] for example). 
These systems give both a topological, that is an abstract, 
and a geometric representation. The concepts involved are 
such things as vertices, edges, loops, faces, and surfaces. 
The topological representation leads to an obvious 
triangulation and thus to a simplicia1 complex. Therefore 
the model may be treated conventionally. This simplicia1 
complex has a geometric realization. There is a dual 
representation involving faces on specific surfaces. These 
faces are bounded by "loops." The questions to be discussed 
are, "Under what circumstances do these geometrical faces 
make sense?" and "How can they be explicitly defined?" 
Also, "When are these 'geometrical' faces homeomorphic to 
the realization of the abstract (topological) face?" 
Faces here are essentially discs with holes. Algebraic 
topology in effect treats faces as discs with no holes. 
Storage and computational constraints force us to deal with 
this new type of face. 
REFERENCES 
1. Smith, Kennen, Primer of Modern Analysis, Bogden and 
Quigley, New York, 1971. 
2. Agoston, Max, Algebraic Topology, Marcel Dekker, New 
York, 1976. 
3. Giblin, P. J., Graphs, Surfaces and Homology, Chapman 
and Hall, 1977. 
4. Geometric Modeling Project Boundary File Design (XBF-21, 
R-81-GM-02.1, Computer Aided Manufacturing International, 
Arlington, Texas, 1981. 


The Definition and Computation of a Metric on Plane Curves. The Meaning of a Face on a Geometric Model

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