The centroid and moments of an area program provides the centroid, moments of inertia, product of inertia, radii of gyration, and area of any closed planar geometric figure. The figure must be available in graphic form and is digitized once with chart digitizer (graphic tablet). The digitizer origin may be set anywhere on the digitizer table. After digitizing, fifteen quantities are calculated and displayed: (1) area (2) moment of inertia of area with respect to digitizer x-axis, (3) moment of inertia of area with respect to digitizer y-axis, (4) product of inertia of area with respect to digitizer axes, (5) first moment of x for digitizer axes, (6) first moment of y for digitizer axes, (7) x coordinate of centroid, (8) y coordinate of centroid, (9) moment of area inertia of with respect to x axis through centroid, (10) moment of inertia of area with respect to y axis through centroid, (11) product inertia of area with respect to x and y axes through centroid, (12) polar moment of inertia of area around centroid, (13) radius of gyration about digitizer x axis, (14) radius of gyration about digitizer y-axis; and (15) variance in the x-direction

Patch, R. W.

English

NASA Technical Reports Server

General Disclaimer 
One or more of the Following Statements may affect this Document 
 
 This document has been reproduced from the best copy furnished by the 
organizational source. It is being released in the interest of making available as 
much information as possible. 
 
 This document may contain data, which exceeds the sheet parameters. It was 
furnished in this condition by the organizational source and is the best copy 
available. 
 
 This document may contain tone-on-tone or color graphs, charts and/or pictures, 
which have been reproduced in black and white. 
 
 This document is paginated as submitted by the original source. 
 
 Portions of this document are not fully legible due to the historical nature of some 
of the material. However, it is the best reproduction available from the original 
submission. 
 
 
 
 
 
 
 
Produced by the NASA Center for Aerospace Information (CASI) 
https://ntrs.nasa.gov/search.jsp?R=19770004775 2020-03-22T11:55:19+00:00Z
FRP.?IjUUUCu3t).rry
NASA TECHNICAL	 NASA TM X-73528
MEMORANDUM
c^	 (NASA—TM—X-•73528)	 CENTROID AND MOMENTS OF	 N77— 11718
can	 AN AREA USING A DIGITIZER (NASA)	 19 p HC
ti	 A02/MF A01	 CSCL 09B
X	 UncIas
G3/61 54567
I--Q
c^
QZ
6
CENTROID AND MOMENTS OF E.N AREA USING A DIGITIZER
by R. W. Patch
Lewis Research Center
C)eveiand, Ohio 14135
October 1976
Y^^,^^ac^^ ^ ^i1
2 
A
N S4 'Gib	 a,
tl
	 ^^1T^1:fl11TYc"tfn rr n...-
	 _	 `^
ii
I,
I.	 Report No. 2. Government Accession No. 3. Recipient's Catalog No
NASA TM X-73528
4.	 T!tle and Subtitle 5. Report Date
CENTROID AND MOMENTS OF AN AREA
6. Performing Orgemtebon CodeUSING A DIGITIZER
7.	 Author(s) B. Performing Organization Report No
R. W. Patch E-8947
10. Work Unit No.
9. Performing OrganizationName and Address
Lewis Research Center 11. Contract or Gran[ No, 
National Aeronautics and Space Administration
Cleveland, Ohio 44135
13. Typo of Report and Period Covered
Technical Memorandum 12, Sponsoring Agency Name and Address
National Aeronautics and Space Administration
14. Sponsoring Agency code 
Washington, D. C. 	 20546
15. Supplementary Notes
16. Abstract
The Centroid and Moments of an Area program provides the centroid, moments of inertia,
product of inertia, radii of gyration, and area of any closed planar geometric figure.. 	 The
figure must be available in graphic form and is digitized once with chart digitizer (graphic
tablet).	 The digitizer origin may be set anywhere on the digitizer table. 	 After digitizing,
fifteen quantities are calculated and displayed:	 (1) area, (2) moment of inertia of area with
respect to digitizer x-axis, (3) moment of inertia of area with respect to digitizer y-axis,
(4) product of inertia of area with respect to digitizer axes, (5) first moment of x for
digitizer axes, (6) first moment of y	 for digitizer axes, (7) x coordinate of centroid,
(8) y coordinate of centroid, (6) moment of inertia of area with respect to x axis through
centroid, (10) moment of inertia of area with respect to y axis through centroid, (11) product
of inertia of area with respect to x and y axes through centroid, (12) polar moment of in-
ertia of area around centroid, (13) radius of gyration about digitizer x axis, (14) radius of
gyration about digitizer	 y axis, and (15) variance in the x-direction.
17. Key Words (Suggested by Author(s)) 16. Distribution Statement
Digitizer; Centroid; Moment of inertia; Unclassified - unlimited
Product of inertia; Radius of gyration; Vari-
ance; Graphic tablet; Desk calculator,
Beam, Column, Mechanics of material
19. Security Classif. (of this report) 20. Security Classif. (of this page) 21, No. of Pages 22.	 Price'
Unclassified Unclassified
'For sale by the National Technical Information Service, Springfield. Virginia 22161
NASA-C-168 (Rev. 10.75)
1
i
I
g
1
!7
{
a
^l
A
1x
1
a"I
l^
r	 it
,a
r,
r.
RWRODtrGMMITY „,
i
CENTROID AND MOMENTS OF AN AREA USING A DIGITIZER
by R. W. Patch F
Lewis Research Center
INTRODUCTION
The object of this program is to provide a general-purpose digitizer t,
program for calculating area and first and second moments of any closed
planar geometric figure such as a cross section of a beam or column or a
spectral line shape that has been closed at the baseline, - For simple
figures such as circles, rectangles, or triangles, the area and moments
are easily calculated by hand from simple formulas given in handbooks.
For figures consisting of two or more simple figures superimposed, the
area and moments may be calculated by hand with somewhat more difficulty.
The principal application of this program, however, is to complicated or
irregular figures such as spectral line shapes, beams, columns, rotating
machine parts, statistical distributions, and ship hull sections, which
are difficult or impossible to calculate by hand.
_e
^	 In addition to the basic quantities of area and moments, cert$in
00
	 quantities are also calculated for convenience.
Previous published digitizer programs have merely calculated area
or circumference of closed geometric figures.
PROBLEM TASK DESCRIPTION
A closed planar, geometric figure of arbitrary shape is shown graphi-
cally in figure 1. The size of the figure is such as to fit on the
sensitive portion of the digitizer table (graphic tablet). The figure
is to be digitized by going around it once clockwise with the cursor.
Curved portions are to be digitized at closely spaced intervals (here-
after referred to as "continuous digitizing" because the intervals are	 3
determined automatically). To improve accuracy, straight portions may
be digitized by merely digitizing the two end points, 	 j	 I
The quantities to be calculated require several axes. The digitizer
axes x and y may be set anywhere on the sensitive portion of the digi-
tizer table. The closed figure has a centroid whose position is not known
initially. Through the centroid are centroidal axes x and yg
parallel to the x and y axes, respectively. Consequently, if the
angular orientation of the x and yg axes relative to the figure is
important, the figure Should ge rotated to the desired angular orienta-
tion relative to the digitizer table before doing the digitizing for
this program. This rotation may be facilitated by using the cursor and
another small program to check the figure angular orientation (ref. 1).
STAR category 09
Ar
ii
(3)cy = V
	
y dA
RErronUCIBIr.rry
i	 y
a
j!
2 1
The following quantities are to be calculated and displayed for
each case.
(1) Area of figure
(2) First moment of	 x	 with respect to 	 y axis {
(3) First moment of	 y	 with respect to 	 x axis
(4) Moment of inertia of area with respect to	 x	 axis s
(5) Moment of inertia of area with respect to	 y	 axis
(6) Product of inertia of area with respect to	 x	 and	 y	 axes
(7) x	 coordinate of centroid
(8) y	 coordinate of centroid
(9) Moment of inertia of area with respect to	
x 
	 axis
(10) Moment of inertia of area with respect to	 y 	 axis
(11) Product of inertia of area with respect to	 xg
	and
	 yg	axes
a
(12) Polar moment of inertia of area around centroid
(13) Radius of gyration about	 x	 axis
(14) Radius of gyration about y axis
(15) Variance in the x direction
Most of these quantities are defined in reference 2, but they are also
defined in, the next section or in the List of Symbols.
METHOD OF SOLUTION
Six basic integrals must be calculated.
A= j dA
	
(1)
cx
 =	 x dA
	 (2)
n3
Ix
 = J y2 d_A
Iy =	 x2 dA
i
(4)
(5)
RRPROD^ ^^IEIS POOR
F.
li
i!
Ixy - 	 xy 
dA	
(6)	 1	 pJ
	(Symbols are defined in the List of Symbols.) From these integrals, all
	
p
other required quantities may be found. To minimize storage requirements
the six integrals are found by accumulating 12 sums as the closed figure
is traversed with the cursor, assuming for curved portions that the fig-
ure consists of a succession of straight lines between digitized points.
The six integrals are calculated from the 12 sums at the end of the pro-
gram.
The area A may be expressed
	
A = / /
	 dy dx = J y dx - J y dx	 (7)
	
JJ	 upper	 lower
where "upper" and "lower" refer to the parts of the figure for which a
perpendicular outward vector is in the +y and -y directions, respectively.
Using the trapezoidal rule, equation (7) becomes
A = 2 ^ (yo + Yn)(xn - xo)	 (8)
9
i
where subscripts o and n refer to the first (old) and second (new)
points, respectively, of an adjacent pair of digitized points, and the
summation is over all pairs of adjacent points all the way around the
closed figure, For xn = xo there is no contribution to equation (8).
The first moment o
(
f
^ 
x with respect to the y axis is
cx =	 / x dy dx = J xy dx -	 xy dx
	 (9)
Jupper	 lower
or
cx = ((Yoxn - xoYn)(xn + xo) + 3
	
(Yn - Yo}(xn + xnxo + xa) (10)
I+ 4 ^ (Yn - Yo ) (xn + xo)(xn + 
xn)
4
Again, there is no contribution if x n = xo.
The first moment of y with respect to the x axis is
cy = ff  y dx dy-_ ^yx dy - ^yx dy
	 (11)
right
	 left
where "right" and "left" refer to the parts of the figure for which a
perpendicular outward vector is in the +x and -x directions, respec-
tively. Equation (11) becomes
cy = 2
	
(Yoxn - x0Yn) (Yn + Yo ) - 3	 (xn - xo )( Yn + YnYo + yo)
(12)
There is no contribution to the last term when n = xo,
The moment of inertia of area wi(t'1h respect to the x axis is
Ix =	 o y 2 dxdy= J y 2 xdy- J y2xdy
J	 right	 left
or
Ix 3 ^ (Yo n - xoyn)(yn + 
Ynyo + Yo)
4	 (xn - xo )(Yn + Yo)(Yn + yo)
There is no contribution to the last term when xn = xo,
The moment of inertia of area with respect to the y axis is
^i
Iy = r r 2 dy dx = J x2
 y dx - J x2
	
x	 y dx
^f J	 upper	 lower
or	
/
Iy = 3	 (Yoxn - xoyn)lx2 + xnxo + x2)
(13)
(14)
(15)
(16)
i
Ir
iI
^f
i^
,
i.
n
I
f`
7
5
There is no contribution if xn = xo.
The product of inertia of area with respect to the x and y axes
is
	
Ixy = r / xy dy dx = 2 [JO
 
xy 	 dx - J xy 2 dx]	 (.l7)J J	 upper
	 lower
or
- 1Y 
(yoxn - xo
Ixy 4	
Y")2(xn + xo)
xn
 - xo
12	 2\
1	 (Yoxn - xoyn)(Yn - Yo) X  + xnxo + xo/
+ 3 —	 xn - xo
1E (Yn - Yo)2(xn + xo)(xn + xo)
There is no contribution if x = xo and, if this occurs, each rerm must
be set equal to zero to avoid gividing by zero.
The other required quantities may be found from the six basic inte-
grals. The x coordinate of the centroid is
_	 d x dA. cx
X = ^l A	 = A	 (19)
The y coordinate of the centroid is
_ or y dA c
Y =	 A	 = A	 (20)
The moment of inertia of area with respect to the x  axis is (ref. 2)
1 g = Ix - Ay-2 	(21)
The moment of inertia of area with respect to the y  axis is (ref. 2)
Igy = Iy - A3 2	(22)
''	
a
{	 ,
i!
x:
l;
j:
1'^. _.
6 d
The product of inertia of area with respect to the	 xg
	and	 yg	 axes is
(ref,- 2) '
Igxy = Ixy - AxY	 (23)
The polar moment of inertia of area around the centroid is (ref. 2)
s
g%j  = Igx -1- Igy	 (24)
The radius of gyration about the	 x	 axis is (ref. 2)
I
k	 =	 A	 (25)x
S1
The radius of gyration about the	 y	 axis is (ref. 2)
k  = 6—Ay	(26)
The variance in the x direction is (ref. 3)
I
sX =	
AY	
(27)
PROGRAM DESCRIPTION
The program is written in BASIC for a Hewlett-Packard 9830A, calcu-
lator and 9864A digitizer.
The flow chart for MOMENT is given in figure 2. 	 First an initial
i
point
	
X1,Y1	 is digitized.	 Then a loop is entered which is traversed
once for each new digitized point.
	 During each traverse the sums
	 T1
and	 T2	 are always added to, but the sums 	 SO	 through	 S9	 are only
added to if	 X2	 is not equal to 	 X3.
Closure occurs when the operator brings the cursor back sufficiently
close to the initial point.
	 To check if this has occurred, two tests
are made each time the loop is traversed: 	 (1) has any point been outside
-
the closure distance 	 E	 from the initial point? (this test prevents
closure before we have moved appreciably away from the initial point at
the start), (2) is the last point inside the closure distance
	 E	 from
the initial point?	 E	 is arbitrarily set at 0.03 inch, and the dis-
tances	 D	 from the initial point are approximated by
D 
= Ix 	 - x I + IYn - Yi )	 (28)i
,i
When both conditions are met, closure is forced by making the new point
the same as the initial point, and a last pass is made by the calculator
through the upper part of the program loop.
1
220 Q = Y3 * Y3 + Y2 * Y2 a^
230 N = Q + Y2 * Y3
24d T1 = Tl + B * G
25d T2= T2+G*N
26d IF X2	 X3 THEN 420 !:
27d F = X3 - X2
280 H = X3 + X2
i
f
296 I = Y3 - Y2
300 M = X3 * X3 + X2 * X2 ''	
n
316 L =M +X3 * X2
320 S0 = S0 + B * F "{
330 Sl= Sl+G *H
34d S2 =S2+I *L
350 S3= S3+G*L
.	 y
360 S4 =S4+I *H*M rt
37d S5= S5+F*N y
38d S6 = S6 + F * B * Q
390 S7 = S7 + G * G * H/F "'1
Y
400 S8= S8+G*I*L/F
410 S9= S9+I*I*H*M/F
420 IF F2 = 1 THEN 550
e
.v
Y	
`i
REPRODUCIBILITY OF	 f
011IGR4AL PAGE IS PA"''
ry-
^r
1`	 k
7 r
When the last pass is complete, the loop is exited, and the digi-
tizer gives two beeps to signal closure. The six integrals are then cal- 	 ^)
culated from the 12 sums, and the nine derived quantities are calculated.'i
The six integrals and nine derived quantities are then displayed three at
a time.
The listing of MOMENT is
10 E _ .03
20 F1 = 0
30 F2 = 0
40 ENTER (9,*) Xl,Y1
50 X2 = X1
60 Y2 = Y1
70 SO = 0
86 S1 = d
96 S2 =6
100 S3 = 0
110 S4 = 0
120 S5 = 0
130 S6 = 0
144 S7 = 0
151 S8=0
160 S9 = 0
170 T1 = 0
180 T2 = 0
19P ENTER (9,*) X3,Y3
266 B = Y2 + Y3
210 G=Y2 *X3 -X2 * Y3
i8
4 43q X2 = X3
11 449 Y2 = Y3
f 45d D = ABS(X3 - Xl) + ABS(Y3 - Y1)
460 IF D > E THEN 48d
-j 470 GOTO 45¢
, 48d F1 = 1
t
499 IF D < E AND Fl = 1 THEN 51d
f
1i
59d GOTO 19¢
j 510 F2 = 1
S,
j 521 X3 = X1
j^ 539 Y.3 = Yl
549 GOTO 200
ii
55d WRITE (9,*)
,i 56d WAIT 3dd
^1.
f!
if
570 WRITE (9,*)
580r A = Sd/2
59d Cl = S1/2 + S2/3
60 12 = S3/3 + S4/4
'I 61d C2 = T112 - S5/3
! 620 Il = T2/3 - S6/4
639 13 = S7/4 + S8/3 + S9/8
649 G1 = Cl/A
659 G2 = C2/A
669 14=I1-A *G2 *G2
670 I5=I2-A *GI *01
I(
d
td
t^
1
I €
1}
!	 3
i
^s
^-	
S
9
680 I6=I3 — A*G1*G2
690 J = 14 + 15
700 K1 = SQR(I1/A)
710 K2 = SQR(I2/A)
720 V = I5/A
734 FIXED 2
740 DISP "XAVE"; Gl; "VAR"; V; "A"; A
750 STOP
760 DISP "IX"; 11; "IY"; I2; "IXY"; I3
770 STOP
780 DISP "IGX"; 14; "IGY"; I5; "J"; J
790 STOP
800 DISP "IGXY"; I6; "CX"; Cl; ' ICY"; C2
810 STOP
820 DISP "KX"; K1; "KY"; K2; "YAVE"; G2
830 END
The storage capacity required is 818 words.
OPERATING INSTRUCTIONS
Assuming the program is recorded on file 0 of a cassette, the fol-
lowing is a check list for operating MOMENT:
1. Turn on calculator and digitizer.
2. Insert cassette.
3. Line up axes of figure (if any) with axes of digitizer.
4. Press SCATCH A EXECUTE.
5. Press LOAD 0 EXECUTE.
6. Set origin.
J
f j
a
r
p
A
6
L
a
s
x
i E
10
7. Press RUN EXECUTE.
8. Place cursor at initial point of figure (anywhere on figure
will do).
9. Press C on cursor to start continuous digitizing.
10. Move cursor around curve clockwise back to initial point.
11. Get two beeps to indicate closure.
12., Press C on cursor to turn off continuous digitizing.
13. Read X, S2 , A.
14. Press CONT EXECUTE.
15. Read IX, Iy , IXy.
16. Press CONT EXECUTE.
17. Read Igx , Igy , Jg.
18. Press CONT EXECUTE.
19. Read Igxy , cx, cy.
20. Press CONT EXECUTE.
21. Read kX , ky , y.
22. Go back to step 6 or 7 for another case.
If the figure contains a straight segment, approach the segment in the
continuous C mode, press C to turn off the digitizer at the beginning of
the segment, go to the other end of the segment by any path desired,
press C to turn on the digitizer at the end of the segment, and proceed
around the rest of the figure. The corners of a polygon may be digitized
one at a time by pressing S once at each corner.
A test case is given in figure 3, but should be laid out with draw-
ing instruments. The correct results in appropriate powers of inches are
RAVE	 1.00	 VAR	 0.67	 A	 8.00
IX	 34.00	 IY	 13.33	 IXY	 14.67
IGX	 7.11	 IGY	 5.33	 J	 12.44
IGXY	 0.00	 CX	 8.00	 CY	 14.67
KX	 2.06	 KY	 1.29
	 YAVE	 1.83
REPRODUCIBILITY Oh' 'i
ORIGINAL PAGE IS P& 1; .
r
r
However, do not expect to obtain accuracy to two decimal places on all
quantities because the digitizer is only accurate to 0.01 inch and there
is additional error in positioning the cursor.
REPRODUCIBILITY OP' 'l'
ORIGINAL PAGE IS AY
a
z
8
^P
_v
('	 r
12
APPENDIX - SYMBOLS
BASIC Display Mathe- Description
name name matical
symbol
A A A Area
B yo + y 
C1 CX
c 
First moment of
	 x	 with respect to	 y axis
C2 CY
c 
First moment of	 y	 with respect to	 x axis
D D Distance from initial point
E E Closure distance from initial point
F
xn - xo
F1 Flag to indicate if any point outside of closure
distance
F2 Flag for last pass
G yoxn - xoyn
G1 XAVE x x	 coordinate of centroid
G2 YAVE y y	 coordinate of centroid
H x + x
n	 o
I yn - yo
Il IX Ix Moment of inertia of area with respect to	 x
axis
I2 IY Iy Moment of inertia of area with respect to	 y
axis
13 IXY Ixy Product of inertia of area with respect to	 x
and	 y	 axes
I4 IGX I Moment of inertia of area with respect to	 x
g 
x
gaxis
I5 ICY Igy Moment' of inertia of area with respect to	 y
gaxis
I 13
BASIC Display Mathe-
name name matical 4
symbol
I6 IGXY I
gxY
Product of ine rtia of area with respect to
	 xg4 and	 yg
	axes
IS S 1 Polar umment of inertia of area around the
centroid
K1 KX k Radius of gyration about	 x	 axis
K2 KY ky Radius of gyration about
	 y	 axis
f
L xn+ 	 xo + xnxo
A
M rn + xo
3
N 2	 2Yn + yo + yoYn
a
1
Q yn + yo
s0 ^(yo + yn)(xn _ xo)
3
S1
^(yoxn - xoyn ) (xn + xo)
S2 Dyn - Y.)(xn +xnxo + xo)
2	 2\\
R
S3 L(Yoxu - xoyn)(xn + x x + xo l 3n o
S4 L^(Yn - Yo)(xn + xo )(xn + xo)
SS
^(xn - xo)(yn + ynyo + yo)
S6
Dxn - xo ) (Yn + Yo) (
Yn + YO/
2
(Yoxn - xoyn) (xn + xo)
S7
xn. - X s
a
a
o-^
I
4
t
t
1W
t
h'.
j:
i
14
BASIC Display Mathe-
name	 name	 matical
symbol
(Yoxn - x0Yn)^Yn - Y0 )(xn + xnxo + xo)58
xn - xo
(Yn - YO) 2(xn + xo )(xn + xo)S9
xn - xo
T1
Dyoxn - 
xoYn)+ Yo)
/
(Y.
T2
DYoxn - x
0yn)lyn	 ynyo + yn,+
s2 \xV	 VAR Variance in the 	 direction
X
X1,Y1 xi,yi Coordinates of initial point
X2,Y2 xo,yo Coordinates of old point
X3,Y3 xn,yn Coordinates of new point
x,y Digitizer coordinates or axes
xg ,yg Coordinates with respect to centroid or axes
through centroid
g
d
3
Is
^.rI
3
1
3
4i
T
i
r
r
4
15
REFERENCES
1. Fairman, Seibert; and Cutshall, Chester S.: Engineering Mechanics.
Second ed. John Wiley & Sons, Inc., 1946.
2. Dixon, Wilfrid J.; and Massey, Frank J. Jr.: Introduction to Statistical
Analysis. Second ed. McGraw-Hill Book Co., Inc., 1957.
3. Hewlett-Packard 9830A Calculator 11272B Extended I/O ROM, Operating
Manual. Chapter 6, Hewlett-Packard, Calculator Products Division, 1972,
pp. 6-3 to 6-4.
r
1i
^I
E^i,
^I
l^
j
l
Y
I:
k
'i
Y9
i
i^
q
X
J
3
i
FIGURE I. -CLOSED PLANAR GEOMETRIC FIGURE TO BE DIGITIZED,
SHOWING DIGITIZER AXES AND CENTROIDAL AXES. THE FIGURE
MAY BE ANY SHAPE.
t
1
;i
i
t
9
I ^.
I	
/tl
START
E=0.03
FI=0
F2.0
DIGITIZE XI, YI
X2=XI
Y2=YI
SET SO-S9 AND TI-T2 T:070:]
DIGITIZE X3, Y3
CALCULATE 4 FACTORS NEEDED
FOR TI AND T2
TI=TI + TERM
t
GCALCULATE
 TERM
NO 
	 5 FACTORS NEEDEDX2-X3?	 #X3
YES	 TERM li=0,91
F2^ 
	
COMPLEX	 TE
0 GIVE 2 BEEPS
X2=X3
TE 6 INTEGRALS
=ANDCUDLERIVED QUANTITIESY2 Y3
DISPLAY	 GI, V, A
D-ABS(X3-XI)+ABS(Y3-YI)
DISPLAY
	 II, 12, I3YES
D>E? — -
DISPLAY =14,15, J
NO	 FI=I
NO DISPLAY I6, Cl, C2
D<E AND FI=1?
YES (CLOSURE) DISPLAY	 KI, K2, G2
F2=1
EN D
ii
^Y
I
T
X3=XI
Y3=Y1
FORCE CLOSURE BY MAKING	
LAST PASS
NEW POINT SAME AS
INITIAL POINT
FIGURE 2.-FLOWCHART FOR 'MOMENT'.
rt^'FIIUllUC133ILI'i'Y U
owc; NAL PAGE IS PUt-:
Xv
FIGURE 3.-TEST CASE (ALL DIMENSIONS IN INCHES).
i	 l	
^Y


Centroid and moments of an area using a digitizer

Abstract

Similar works

Full text

Available Versions

NASA Technical Reports Server