Definite integrals are evaluated for the cardinal functions of an interpolation method which provides C sup 1 continuity over a triangular grid

Lawson, C. L.

English

NASA Technical Reports Server

I 1 i
NATIONAL AERONAUTICS AND SPACE ADMINISTRATION f
•_' Technical Memorandum 33-808
Integrals of a C'-Compatible Triangular
,Surface Element
(NASA-CR-149277) INTEGRALS OF A N77-I_797
C-I-COMPATIBLE TRIANGULAB SURFACE _LE_ENT
(Jet Propulsion Lab.) 20 p HC A02/MF A01
CSCL 12A Unclas
G3/6q 58971 _
/2 JAN1977 _
!_ RECEIVED'_I
JET PROPULSION LABORATORY 4,?/, 6_;.>
CALIFORNIA IN,TITUTE OF TEOHNOLOQY _
PASADENA, CALIFORNIA
Deceml_r15, 1976 "'_
,?-
1977007854
https://ntrs.nasa.gov/search.jsp?R=19770007854 2020-03-22T12:29:29+00:00Z
y, o
33-808
PREFACE
The work described in this report was performed by the
" Information Systems Division of the Jet Propulsion Laboratory. --
:i
iii
1977007854-002
33-808
CONTENTS
1. Introduction .................... 1
2. Preliminary Integration Formulas .......... i
3. Determining fPi for the Rational Function 0i .... Z
4. Determining fPi for the Piecewise Cubic
_: Function 0i .................... S -
5. Integrals of the Cardinal Functions ai' 8i'
! and Yi ....................... 7
6. Example: An Isosceles Right Triangle ....... 8
7. Example: An Equilateral Triangle ......... i0
8. Corroboration of Results by Numerical
Integration ................... 13
TABLES
I. Integration of Pl over T I ........... 6
2. Integration o£ Pl over T 2 ........... 6
3. Parameters of the Triangle of Fig. 3 ..... 9
4. Mean Values of, Auxiliary and Cardinal
Functions ................... 9
5. Parameters of the Triangle o£ Fig. 4 ..... I0
I 6. Mean Values of Auxiliary and Cardinal
• Functions ................... ii
7. Nunencal Integration of the Zienkiewicz Rational!
? Function ................... 14
8. Numerical Integration of the Clough-Tocher
Piecewise Cubic Function ............ 14
REFERENCES ....................... iS
v
L
1977007854-003
1 33-808ii /
_l
! ,
I
?
i !
: Abstract _
2 :
Definite integrals are evaluated for the cardinal
functions of an "nterpolation method which provides
C1 continuity over a triangular grid. !
vi
1977007854-004
' i
!
i. Introduction
In Ref. I, algorithms are given foz cl-compatible in-
terpolation over a plane triangle given values of f' fx' and fy
at the vertices of the triangle. In particular on p. 30 of
Ref. l, functions ai,B i, and Yi are defined such that with
Bi=Si/2 and yi=Yi/2 the interpolated value w can be ex-
pressed as
(1) w := Z (fill + fy,iYi )i=l fx'iBi+
In this note we derive expressions for the definite
integrals of the cardinal functions ei,Si, and Yi' i=i,2,3
over the triangle. These integrals are needed, for example,
if a surface representation problem includes conditions on
volumes to be encompassed under the surface.
2. Preliminary integration formulas
From Ref. 2, we obtain the following formulas for in-
tegration of polynomials over a triangle:
i
(2) IPl = A pI[I'I'_]
(3) /P2 = 1A{P2[['2-'0]+P2 +P2[0'_'_]}
c4) -;p3 __A(zsP3[31 1 .1 3 1, .1 _ 3. 1 1 1= ,_,_]+2SP 3 I_,_,_I+2SP 3[_,_,_I" 27p 3 [_,_,_1 }
' 1
1977007854-005
i: i 33-808
i In the above formulas the integration is over a triangle in the /
' II xy plane. The integrand Pi denotes a polynomial of degree at
_ _ most i in x and y. The symbol A denotes the area of the triangle.
The notation Pi[rl,r2,r3] denotes the value of the polynomial
Pi at the point in the triangle whose barycentric coordinates
are rl, r2, and r3. For discussion o£ the affine invariance of
quadrature formulas stated in terms of barycentric coordinates
see Ref. 2.
i
The above formulas can be used to obtain the following
integrals for the fundamental linear, quadratic, and cubic func- "_
tions used in Ref. i.
: (5) /r i = A/3
(6) /@i _ fri+iri-I = A/12
(7) fgi _ fri+iri-l(ri+iri-i ) = 0
In the following two sections we determine fPi for
the two different definitions of Pi given in Re£. i. Formulas
for the integrals of the cardinal functions are then derived
in Section 5.
3. Determining J0i for the rational function 9i
In Section 7, pp. 22-23, of Ref. 1 Pi is defined
as t_e Zienkiewicz rational function
• 0 i = rir2 r 2i+l i-I/[(l'ri+l)(l-ri-l)] i=1,2,3
Consider the particular triangle T O having vertices V1 = (0,0),
: V2 = (1,0), V3 = (0,1). In this triangle we can relate car-
_, tesian xy coordinates to barycentric coordinates as follows:
rI = l-x-y
r2--x
r3=y
1977007854-006
35-808Then
I = x2y2(1-x-y)/[(1-x) (l-y)]
Pl
Expand the term (1-x-y) as (1-x)+(1-y)-l. Then f p 1 is the
sum of the following thrcc i**tegrals: TO
_1 12_-_y ll-Yx2dx dyII = 0l
i2 fl x 2 1-x y2dy= 1-x f dx -.
0 0 _i
2 2
%I3 = - 1---£0
Evaluating Ii:
Ii- 3 l-y
_ 13 IoI(y2-2y3+y4) dy
i 11 2 .___1= _[_ - _ + ] 90
The integral 12 has the same value as 11. The inner integral
o£ I is3
slx _ dy --{ c_Y)2- + 2 (l-y) £n(l-y) }l-x0 1-y 0
_x 2 1= -- + 2x - £n x + _ 2
= -{(x2-4x+3) - Z. x
= -{(x-l)(x-3) - £. x
5
1977007854-007
33-808
/
Thus
1 Ix2 X2_M.XI3 = -_ 0f (x-3) dx + fl0 -xi:W-- dx
= 3_ + .r1 x2£nx dx
8 0 1-x
dx - f £. x dx - x£nx dx "
0 a_-'
• _ 3 _2 x2 x2 1 , ..
8 6 [x£nx-x + _ £.x - 4"]0
2
' _ 3 _ + 5_ = _..,-:39"4_r2_/24
Combining results we obtain:
(8) f Pl = I1 + I2 + I3- 10 + _-'O"+ 3924_2 -(593-60Tr2)/360
To
Since the area of triangle TO is 1/2, the mean value of Pl
over TO, and thus over any arbitrary triangle, is (539-60_2)/180.
• Furthermore, since the integrals of pl,P2, and P3 are the same
we may drop the subscripts and simply write the mean value of
any of these three functions as
( 9 ) P" = (593-60'e2)/180 'l
@
= 0.00457 63107 47992
The author thanks Dr. E.W. Ng for doing the analytic in-
tegrations of this section. The result given in Eq.(9) has
been corroborated by a numerical integration of Pl over the
triangle TO which gave agreement in the first seven significant
digits, See Section 8 for more details on th_ numerical integration.
1977007854-008
........................i ..1................I 1 ! .. .I . ..........
33-808 l
4. Determining /Pi for the piecewise cubic function Pi
In Section 8, pp. 24-26, of Ref. 1 , pi is defined
for i=l as the Clough-Tocher piecewise cubic function
rl[6r2r3+rl(Srl-3)]/6 if r 1 = min{rl,r2,r 3}
Pl =_r_(-r2+3r3)/6 if r 2 = min(rl,r2,r 3}
!
_r§(-r3+3r2)/6 if r 3 = min{rl,r2,r 3}
Thus Pl is defined as a different cubic polynomial in each of
the three subtriangles indicated in Fig. 1.
i
Figure 1
V2 V 3
The common intersection point of the three subtriangles has bary-
fl,l,l.
centric coordinates _ _ _).
Within each subtriangle the integration formula, Eq. ( 4 ),
can be applied. The evaluation points for this formula in sub-
triangles T 1 and T 2 are indicated in Fig. 2.
V 1
i
Figure 2
\
v2 v3 i_'
5
1977007854-009
i.........................................I .....I I I I 1 !,
: 33-808
_ /
Table 1. Integration of p I over T 1. p I ; lrl[6r2r3 )]
P " I ....
, Barycentric Multiplier
IPointl Coordinates Value of 01 [See Eq.( 4)] Product
1 2 2 7
a
15, 5, 5 5.125 25 189/(81.5) ,
1 10 4 4b
115, 15, 15 8l,S 25 100/(81"5)
1 4 10 4
c I15 ' 35, l--ff 81.5 25 I00/C81"5)
1 4 4 37d
19, 9, 9 81.27
_ 2 7
|
Sum: 204/(81-S)=68/(27.5)
1 2
: Table. 2. Integration o£ Pl over T2. 01 = _-r2(-r2+lr3 )
Barycentric Multiplier
Point Coordinates Value of 01 [See Eq. (4)] Product
2 1 2 1
e _, _, _ 2.3.25 25 135/(2"81"5)
f 4 1 I0 29
1--5, _-5, 1-ff 2 .81.125 25 29/(2 .81.5) 't
10 1 4 11 25 11/(2.81.5)g 15, _-_, I-ff 2"8'1'125
h 4 1 4 11 -27 -55/(2"81"5)
9, 9, _ 2.57
Sum: 120/(2-81.5) = 4--27
6
1977007854-010
I I i
i 3380
;!
The arithmetic involved in applying Eq. (4) to Pl in sub-
triangles T 1 and T2 is summarized in Tables 1 and 2. The
integral over T3 will be the same as that over T 2.
Let A denote the area of the whole triangle. Then
_ach subtriangle T. has area A/3. It follows thatl
;Pl = _ _ + = 18--_
The integrals IO 2 and /P3 have the same value as IO 1. Thus
the mean value of any one of these functions Pi is given by
, (lO) _ = 1/18o = o.oossssssss...
The result given in Eq. (10) has been corroborated by a num-
erical integration of 01 which gave agreement in the first
seven significant digits. See Section 8 for more details on the
numerical integration.
5. Integral° of the cardinal functions ai, 8i, and Yi
Referring to p. 30 of Ref. i, introduce 8 = 8
I
and 7 = _7 so that the final equation for w can be written
simply as
3
w := E [fiai + fx iSi + f iYi ]i=l , Y,
Use a bar over the symbols Pi' qi' ai' _i' and Yi
to denote the average "value of each of these functions over the
Pi = -_ IPi' In the case of the functionstriangle. For example,
the integral over the triangle is independent of i. ThusPi I
we simply write V = _/Pi for i=i,2,3. From Eq.(9)and(10) respectively
we have _ = (593-60_2)/180 if Pi is the Zienkiewicz rational
function and _ = 1/180 if Pi is the Clough and Tother piecewise
cubic function.
1977007854-011
I i I l I I ......IL
33-808
//
1
Let
= - z'2l) /z2" iXi (/_+i i- i
Then, using the notation and equations o£ Re£. i, p. 30, and ; ..
Eqs. (5-7) of the present paper, we obtain
Igi = 3lilPi = 3Ai_A
qi-- 3xi_ - 1i2-
c_i = g + 3(Xi_ I - Xi+l) P
7
- 1 - + - ]
Yi = 2- [Vi-lPi-i Vi+lqi+l
Finally we have
3
(ii) lw --A i--17 [£i_i + fx,igi + £y,iYi-] i_ :
3 ;
Note, as a matter o£ interest, that _. _. -- 1, independent I
i--i 1
o£ the value o£ _ .
J
6. Example: An isosceles right triangle
Consider the isosceles right triangle o£ Fig. 3.
whose essential geometric parameters are given in Table 3. This
triangle has area A-- a2/2.
a) Figure 3 !
V2 (a,O)
VI(O,O) _.
8 :
?
1977007854-012
3
t
_- 33-808 ,
_,_ /
it,
s Table 3. Parameters of the triangle of Fig. 3.
• 1 1 li
i_, i u. v: _.
_ 21/2_ 0. 1 -a a
2 I 0 -a a -1
3 a 0 a 1
Choosing the functions p i to be the Clough-Tocher
piecewise cubic functions we have P = 1/180. Using the form-
ulas of Section 5, we obtain the values in Table 4.
Table 4. Mean values o£ auxiliary and cardinal functions
i Pi qi _i _i _i
1 5/60 -5/60 22/60 6a/120 6a/120
2 4/60 -6/60 19/60 -9a/120 Sa/120
3 6/60 -4/60 19/60 5a/120 -9a/120
These coefficients should provide exact quadrature for
any quadratic function. As a specific example let a=120 and
consider the function f(x,y) _ x 2.
By the formula of Eq.(3) the integral of £ over
the triangle in question is given by
_ (12) I - _ 1202 2: 3 _ {60 • 602 } - 10.1203
i 9
i
1977007854-013
....I................................I.........If 1.....A_
33-808 i/
i "
!
Alternatively, using Table 4, we note that only two
of the nodal values are non-zero namely f(120,0) = 1202 and !
fx(120,0) = 240. Thus using Eq.(ll) and a2 and _2 from i: i
Table 4, we compute the integral i :
_ _02r_z0_ ]3 L 60 9.240
= 10.1203
,
: i
which is in agreement with ]q.(12). i
7. Example: An equilateral triangle
Consider the equilateral triangle of Fig. 4, whose
essential geometric parameters are given in Table S. This
triangle has area A = 31/2a2/4."
• V3(a/2 , 31/2a/2)
"A
/_ Figure 4
Z_____A
_I(0,0) V2(a,O}
r
Table S. Parameters of the triangle of Fig. 4.
: i u i v i £i _i
1 -a/2 31/2a/2 1 0
2 -a/2 -31/2a/2 1 0
3 a 0 1 0
I
10
1977007854-014
33-808
i Since all ti are zero the functions Pi will not
enter into the problem. Using the formulas of Section 5, we
obtain the values in Table 6.
I
Table 6. Mean values of auxiliary and cardinal functions [
i
i Pi qi ai 8i 7i i--
i 1/12 -1/12 1/3 a/16 31/2a/48
2 1/12 -1112 1/3 -a116 31128148
I
3 1112 -1112 113 0 -2 "3112a148
These coefficients should provide exact quadrature for
any quadratic function. As a specific example let a = 16 and
2
consider the function f(x,y) = y .
By the formula of Eq.(3 ) the integral of f over
the triangle in question is given by
(13) I = A 1/2)2 1/2)23 { (4.3 , (4.3 }
= 32A = 211 ' 3112
Alternatively, using Table 6, we note that only two of
the nodal values are non-zero, namely f(8,8"3 I/2) = 3"64 and
fy(8,8.31/2) = 16.31/2"
Thus using Eq.(ll) and _3 and _3 from Table 6, we
compute the integral
ii
1977007854-015
33-808
/
• !
J = A {3.64 16.31/2"2"31/2"16}3 48
= 32A = 211 31/2
?
which is in agreement with Eq.(13).
°. _-
• i
i
!
L
• I p
i
12 *,
1977007854-016
...... ] i...............J ill...............iii .....
33-80s 1
/
i
8. Corroboration of results by numerical integration
As a check against possible blunders in the determination
of values of p in Sections 3 and 4, direct numerical integration i
Was used to obtain estimates of p for the two different definitions i
i
I The quadrature subroutine used was the JPL Univac llO8 ]--i
library subroutine RMBI/RMB2 (Ref. 3). This subroutine effects 1
quadrature over a two-dimensional region (a triangle in the present !
d
case) by making an appropriate sequence of calls to the library's !
one-dimensional quadrature st_broutine ROMBS/ROM2. This system of !
subroutines was developed by W. R. Bunton and M. R. Diethelm of i
IJPL using an adaptive algorithm based on Romberg quadrature.
|
Results of these numerical integrations are summarized
in Tables 7 and 8. For each of the two functions being studied,
the analytically determined value of p is known. For each of the
two function_ the numerical integration was done five times with
five different values of the accuracy tolerance parameter.
Letting p denote the value obtained from the it__hhnumerical inte-
gration, values of the difference Pi - _ are given in Tables 7 6 8.
The results of these numerical integrations are consistent
with the analytically determined values of p.
13
1977007854-017
I r I 1
33-808
:/i
l)
!
Table 7. Numerical integration of the Zienkiewicz rational )
function.
I
= (593-60_2)/180 • 0.00457 63107 47992
Absolute Number o£ _.
accuracy evaluations Pi - _
i requested o£ integrand
1 10-3 113 -0.00000 14834
2 10-4 113 -0.00000 14834
-53 10 215 *0.00000 01164
4 10 -6 263 *0.00000 00059
-7
5 i0 1413 +0.00000 00007
,-
Table 8. Numerical integration of the Clough-Tocher piecewise
cubic function.
= 1/180 = 0.00555 55555 ...
Absolute Number of
evaluations Pi " paccuracy
i requested of inte_rand
-31 I0 133 -0.00000 31391
_ 2 10 .4 141 -0.00000 59111
-5
.: 3 10 231 -0.00000 00555
4 10 "_ 681 +0.00000 01064
5 10-7 895 -0.00000 00055 '_
14
1977007854-018
I. Lawson, C. L., CI-Compatible Interpolation Over a Triangl e ,
Technical Memorandum, 33-770, Jet Propulsion Laboratory,
Pasadena, Calif., May 1976.
2. Hammer, P. C., Marlowe, O. J., and Stroud, A. H., "Numerical
Integration Over Simplexes and Cones," Math. Tables and Aids
to Computation, Vcl. 10, pp. 130-137, 1956.
3. JPL FORTraN V Subprogram Directory, Edition 5, Document
! No. 1846-23; Rev. A, February, 1975, (JPL internal document).
1- 5 NAIIA-JPL-ComI,,LA, Gab!
1977007854-019


Integrals of a C-1-compatible triangular surface element

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