Shape representation is a central issue in computer graphics and computer-aided geometric design. Many physical phenomena involve curves and surfaces that are monotone (in some directions) or are convex. The corresponding representation problem is given some monotone or convex data, and a monotone or convex interpolant is found. Standard interpolants need not be monotone or convex even though they may match monotone or convex data. Most of the methods of investigation of this problem involve the utilization of quadratic splines or Hermite polynomials. In this investigation, a similar approach is adopted. These methods require derivative information at the given data points. The key to the problem is the selection of the derivative values to be assigned to the given data points. Schemes for choosing derivatives were examined. Along the way, fitting given data points by a conic section has also been investigated as part of the effort to study shape-preserving quadratic splines

Lam, Maria

English

NASA Technical Reports Server

1 
Y & 
FINAL REPORT 
of 
NASA RESEARCH GRANT NAG-1-760 
Title of the Grant: Investigations into the Shape- 
Preserving Interpolants Using 
Symbolic Computation 
Duration: May 1987 to October 1988 
by 
Maria Lam 
Associate Professor 
Department of Computer Science 
Hampton University 
Hampton, VA 23668 
G3/64 0183403 
n 
https://ntrs.nasa.gov/search.jsp?R=19890006300 2020-03-20T04:57:49+00:00Z
CONTENTS 
Page 
Abstract ........................................ ii 
I. Introduction .............................. 1 
11. Fitting Data by Conic Sections ............ 2 
111. Shape-Preserving Interpolation ............ 9 
IV. Contributed Papers ........................ 13 
V. References ................................ 16 
ABSTRACT 
Shape representation has been a central issue in computer 
graphics and computer-aided geometric design. Many physical 
phenomena involve curves and surfaces that are monotone (in 
some directions) or are convex. The corresponding representation 
problem is given some monotone or convex data, and a monotone 
or convex interpolant is found. Standard interpolants need not 
be monotone or convex even though they may match monotone or 
convex data. 
Several authors including Fritsch, Butland, McAllister, 
McLaughlin, Schumaker and Foley have investigated this pro- 
blem. Most of their methods involve the utilization of quad- 
ratic splines or Hermite polynomials. In our investigation, we 
have adopted a similar approach. These methods require derivative 
information at the given data points. The key to the problem is 
the selection of the derivative values to be assigned to the 
given data points. Schemes for choosing derivatives were exam- 
mined. Along the way, fitting given data points by a conic sect- 
ion has also been investigated as part of the effort to study 
shape-preserving quadratic splines. 
ii 
I. INTRODUCTION 
This project commenced in May 1987 and ended in October 1988, 
which includes a six-month, no-cost extension. Dr. Maria Lam 
was the Principal Investigator. She was assisted by a graduate 
student in the Applied Mathematics program at Hampton University. 
Hampton University has benefited by this project. Through 
this grant, the Principal Investigator was able to obtain release 
time and physical resources to conduct research into the area 
that interests her. This project has been beneficial to the 
graduate program in Applied Mathematics by way of the research 
assistantship made available to the graduate student. The work 
done under this project will constitute part of the graduate 
student's dissertation. The Computer Science Department has also 
benefited. This project helped to fulfil part of the faculty 
research criterion set forth by the ACM Accreditation Board when 
the Department applied for accreditation in the spring of 1988. 
In short, this project has been beneficial to the University in 
several ways. 
Two contributed papers were presented on research done under 
this project. The details of these are given in Sections I1 and 
111. 
t 
11. FITTING DATA BY CONIC SECTIONS 
Data Identification: 
The initial attempt has been to identify 2-D data assuming 
there is some underlying relation among the data points. Owing 
to the diversity of this problem, the study has focused on conic 
sections. A conic section can be represented either algebraically 
or geometrically /1,2/. Some of these representations require 
additional information than just the data points. In the absence 
of this information, in theory, a conic section can be represent- 
ed algebraically by a quadratic relation 
ax2+2bxy+cy2+2dx+2ey+f =O ..... (1) 
or PQPT=O where 
p=cx,Y,ll 
a b d  
Define angle s as 
2s=tan'I(2b/ (a-c) ) 
or 
s= x/4 if a=c. 
Now effect a rotation of the coordinate system given by 
X=XCOS(S) - Ysin(s) 
y=Xsin (s) + Ycos (s) . 
Thus , equation (1) becomes 
2 
AX2+CY2+2DX+2EY+F=0 ..... (2) 
By examining the coefficiets of equation (2), the type of conic 
section (including the degenerate cases) can be determined /3/. 
Hence a conic section is completely determined by five data 
points since only five of the six coefficients in equation (1) 
are independent. 
One can indirectly solve for these coefficients by the fall- 
owing method /4/. Assume the five data points are (XirYi), 
(xi+lrYi+l) t (xi+2r~i+2) (xi+3r~i+3) and (xi+4r~i+4) Form the 
6x6 matrix M given by 
M =  
The equation 
det (M) =O ..... (3) 
yields a quadratic equation in x, y. When x=xk and y'k 
(i<=k<=i+4), two of the rows of the matrix are identical, and 
equation (3) is satisfied. This implies that equation ( 3 )  
represents the unique conic passing through these five points. 
However, this algebraic method is found to be numerically 
unstable. The determination of the type of conic section depends 
largely on whether certain coefficients are positive, negative or 
3 
zero. It is extremely sensitive to small perturbations in coeff- 
icients, which may alter the type of the conic section. When it 
is implemented in a computer program, the situation is even 
worse. Such method routinely involves considerable amount of 
algebraic operations. Their floating-point implementation on a 
computer almost invariably produces perturbations in the coeff- 
icients. The selection of reasonable tolerance is difficult. 
Computer algebras such as MACSYMA and SMP are ideal tools for 
this type of investigations because actual evaluation of 
expressions is delayed until the final step. This leads to 
improved accuracy, but, however it does not eliminate the 
problem. 
Several sets of conic data are generated. For each data set, 
five consecutive data points are used for fitting. A rotation is 
performed to eliminate the mixed term. Then the ceofficients of 
the result are observed. This process is repeated until all 
points of a data set are exhausted. 
Results: 
The results of this study are mixed. As one would expect, it 
is difficult to identify these figures definitely. 
To be identified as data that come from a circle, the gener- 
ated coefficients A ,  C must be equal. This rarely happens. The 
best one can expect is to identify the data as elliptic. Our 
study shows that one set of circular data is identified as ellip- 
tic. A second set is identified as elliptic and hyperbolic with 
the elliptic outnumbering the hyherbolic. A third data set prov- 
4 
ides similar result. Elliptical data exhibit similar behavior. 
Parabolas are identified by A=O or C=O but not both. For one 
set of parabolic data with the values of x several units apart, 
we are able to identify this trend every time. A second set of 
data, with closely spaced x values, does not fare well at all. 
It is correctly identified as parabola only on two occasions. For 
most part, it has been identified incorrectly as hyperbolic 37 
times and elliptic 5 times. The third set is the most curious 
one. The first 18 fits indicate the data is hyperbolic, this is 
followed by 22 fits that indicate poarabolic data. 
If the data is hyperbolic, the result of algebraic method 
leads one to conclude that the data is either hyperbolic or 
elliptic; however the former conclusion (i.e. hyperbolic) occurs 
more frequently. 
In spite of this lack of positive identification, one pattern 
emerges. Parabolic data are identified as either parabolic or 
hyperbolic. It is the only type of data that will draw parabolic 
conclusion. Circles and ellipses are classified as ellipses or 
hyperbolas with the elliptical conclusion appearing more often 
than the other. This allows one to draw rough estimates on the 
type of conic section that the data could have originated from. 
In summary, when given a set of conic data and algebraic 
method is used in attempt to identify it, one would assume the 
data is hyperbolic initially. If on some occasions the data is 
identified as parabolic and as hyperbolic on other occasions, 
one should accept the data as parabolic. If the data is ident- 
5 
ified as ellipse and hyperbola with ellipse more frequent than 
hyperbola, one should conclude it to be elliptic. 
Data Fittinq: 
Our next attempt is to fit arbitrarily given data sets by 
conic section. Several data sets including some standard data 
sets to test shape-preserving interpolants are used. The proced- 
ure described in the previous subsection minus the rotation is 
repeated on these data sets. At the end of each fitting, the 
derivative at the point P(i+2) is calculated. These derivative 
values will be examined later. 
When conic data is used, the result is rather consistent if 
consecutive data points are no more than a few units apart. The 
equations of the fits are approximately the same. The changes 
in coefficients are consistent and gradual. However, we 
are unable to recover the equation used to generate the data. 
These fits do not resemble the original equation whatsoever. For 
other data sets, if the five points used to do the fitting do not 
conform to conic shape, the fit will be poor if not downright 
impossible. This is usually indicated by some coefficients having 
values close to zero. The derivatives obtained from these conic 
fits generally preserve the monotone property of the data set. 
They also preserve the convexity of nice, well-behaved data sets, 
but fail under severe testing when this method is applied to some 
standard data sets used to test shape-preserving property of 
interpolants. 
6 
Data Used: 
The following are some of the data sets used in this invest- 
igation. 
Set 1 
Set 2 
Set 3 
7 
Set 8 
Akima data 
Set 9 
RNP 14 data (Fritsch and Carlson) 
8 
111. SHAPE-PRESERVING INTERPOLATION 
The problem is: Given points x1<x2<. . .<Xn and values (Yi}y=l, 
find f such that 
f(xj.1 = Yi i=1,2,...,n 
and f preserves the monotonicity and/or convexity of the data. f 
is usually found by piecing fils together where fi is defined 
over [Xi,Xi+l] and 
fi(xi) = Yir fi(xi+l) = Yi+l* 
Let hi=Xi+l-Xil di=(yi+l-yi)/hi, i=1,2,...,n-i. 
There are two general approaches to this problem. One employs ' 
geometric constructions which could be quite involved /5,6/. The 
other approach uses quadratic or cubic splines, together with 
carefully chosen derivative values or tension parameters /7,8,9/ 
to control the shape of the curve. Quadratic spline is chosen for 
this investigation because quadratic polynomials have one less 
extremum than cubic polynomials, hence better preserve the 
geometric properties of the given data. The derivatives of 
quadratics are linear, this makes them easier to study and 
manipulate. Hence we focus on a simple two-point Hermite inter- 
polation problem involving quadratic polynomials: to find a 
quadratic polynomial h, if possible, such that 
h (Xi) =Yi, h' (Xi)=Si, i=1,2 ..... ( 4 )  
for some preassigned values (Si}T=1. 
9 
Conditions on the Derivatives: 
Schumaker /6/ proved that there exists a quadratic poly- 
nomial solving this problem if and only if 
..... (5) 
and explicitly defined this polynomial. Moreover, this poly- 
nomial is monotone on I=[xl,x2] if sls2>=0, it is convex on 
I if sl<s2, and concave if ~1x32. 
When condition ( 5 )  fails, Schumaker showed that corres- 
ponding to every u in ( ~ 1 ~ x 2 ) ~  it is possible to choose a 
functional value y* and derivative s* at u, and define a 
quadratic spline over I that satisfies condition ( 4 ) .  If 
sls2>=0, then it is monotone on I if and only if sls*>=O. 
If s1<s2, it is convex on I if and only if s~<=s*<=s~. 
Similarly, if SI>S~, then it is concave if and only if 
q>=s * >=s2. 
Selection of Derivatives: 
The conditions stated in the previous subsection are 
deceptively simple. However, when both monotonicity and 
convexity are taken into consideration, the problam takes 
on a new dimension because one cannot satisfy one condition 
at the expense of the other. One generally approaches this 
problem by first assigning some carefully chosen derivative 
value to each data point, then modify some of these values 
if necessary to satisfy the monotone and/or convexity 
10 
conditions, then the knots are inserted and their functional 
values and derivatives are determined, and finally the coeff- 
icients of the quadratic polynomials constituting the spline 
are calculated. 
Schumaker /6/ used a weighted average of di-1 and di 
as the value of Si. McAllister and Roulier /5/ used a very 
involved geometric construction to obtain their shape- 
preserving quadratic splines. But their method can be 
identified as Schumaker's formulation with start-up deri- 
vative Si defined as the harmonic mean of di-1, di and a 
modification scheme for Si. DeVore and Yan / 8 /  improved 
upon their modification. 
This investigation has focused on the selection of 
the values of Si. The following method looks promising. 
The derivative Si is defined as: 
di-ldi 
if di,ldi>O ------------- 
Si = tdi-l+(l-t)di I. otherwise 
where 
This formula is symmetric in di-1, di because one can simplify 
(1-t) to obtain 
11 
This makes the formula independent of the order of the data 
points, i.e. the formula gives the same value when the data 
is processed from the left or from the right. This Si also 
thus the slope of the curve lies between the slopes of 
the two adjacent data segments, 
2. Isil<=min(3ldi,ill31dil) and 
3. si reduces to the harmonic mean of di-1, di when Xi-1, 
Xi, Xi+l are equally spaced (i.e. when hi-l'hi). 
All these properties make it very desirable. 
The investigation was still in progress when this project 
came to an end. It will be carried on under the new project 
as grant NAG-1-948. 
12 
IV. CONTRIBUTED PAPERS 
The following papers were presented on the research 
conducted under this project. 
1. Shape Preserving Interpolants 
Maria H. Lam 
Annual Meeting of Virginia Academy of Science, Norfolk, 
Virginia, May 1987 
Virg. J. Sc., Vol. 38, No. 2, 71 (1987) 
2. Fitting Data by a Conic Section 
Maria H. Lam 
Annual Meeting of Virginia Academy of Science, 
Charlottesville, Virginia, May 1988 
Virg. J. Sc., Vol. 39, No. 2, 103 (1988) 
Abstracts of these papers are shown on following pages. 
13 
SHAPE PRESERVING INTERPOLANTS 
Maria H. Lam 
Department of Computer Science 
Hampton University 
Hampton, VA 23668 
Recently there has been immense interest in shape-preserving 
interpolations. Most of these algorithms are quite involved. H. 
McLaughlin has proposed a simple, local method to generate inter- 
polants which preserve the monotonicity and convexity of the 
given data. F. Fritsch and J. Butland have proposed a method to 
select values of derivatives at data points such that local mono- 
tone piecewise cubic interpolants will be obtained. These two 
methods are implmented by using computer algebra SMP. They are 
applied to several sets of data. Curves produced by these two 
methods are compared. Some of their properties are discussed. 
(Supported by NASA under grant NAG-1-415 and NAG-1-760) 
14 
L 
FITTING DATA BY A CONIC SECTION 
Maria H. Lam 
Department of Computer Science 
Hampton University 
Hampton, VA 23668 
ABSTRACT 
A conic section can be represented either algebraically or 
geometrically. In theory, it can be represented algebraically 
by a quadratic relation of the form 
ax2+2bxy+cy2+2dx+2ey+f=0. 
It is completely determined by five data points as only five of 
the six coefficients are independent. These coefficients can be 
determined by solving a system of five linear equations in six 
unknowns indirectly. This method is found to be numerically 
unstable. Its floating point implementation on a computer almost 
invariably produces perturbations in the coefficients which may 
alter the character of the conic section. Several sets of conic 
data are generated. For each data set, five consecutive data 
points are used for fitting. A rotation is performed to eliminate 
the mixed term. The coefficients of the result are observed and 
compared. This process is repeated until all points of a data 
set is exhausted. Result of this research will be presented. 
(Supported by NASA under grant NAG-1-760) 
15 
V. REFERENCES 
1. 
2. 
3. 
4. 
5. 
6. 
7. 
8. 
9. 
Goldman, R.N., Two Approaches to a Computer Model for 
Quadric Surfaces, IEEE Comp. Graphics and Appl., 
Vol. 3, NO. 9, 1983, pp.21-24. 
Miller, J.R., Geometric Approaches to Nonplanar Quadric 
Surface Intersection Curves, ACM Transaction on 
Graphics, Vol. 6, No. 4, 1987, pp. 274-307. 
Gellert, W., Kustner, H., Hellwich, M. and Kastner, H., 
The VNR Concise Encyclopedia of Mathematics, Van 
Nostrand Reinhold, New York, New York, 1977, pp. 318- 
319. 
Pratt, V., Direct Least-Squares Fitting of Algebraic 
Surfaces, Computer Graphics, Vol. 21, No. 4, 1987, 
pp. 145-152. 
McAllister, D.F. and Roulier, J.A., Interpolation by Convex 
Quadratic Splines, Math. Comp., Vol. 32, 1978, pp. 1154- 
1162. 
McLaughlin, H.W., Shape-Preserving Planar Interpolation: An 
Algorithm, IEEE Comp. graphics and Appl., Vol. 3, 1983, 
pp. 58-67. 
Schumaker, L.L., On Shape Preserving Quadratic Spline Inter- 
polation, SIAM J. Numer. Anal., Vol. 17, 1983, pp. 238- 
246. 
DeVore, R.A. and Yan, Z . ,  Error Analysis for Piecewise 
Quadratic Curve Fitting Algorithms, Computer Aided 
Geometric Design, Vol. 3, No. 3, 1986, pp. 205-215. 
Pruess, S., Properties of Splines in Tension, J. Approx. 
Theory, Vol. 17, 1976, pp.86-96. 
16 


Investigations into the shape-preserving interpolants using symbolic computation

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