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Introduction to
Computing
with
Geometry
Adrian Bowyer and John Woodwark
INFORMATION GEOMETERS
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p l d I W W Q
s n o m o n q o m v d
R U o k e n
n
y P P S v f
u
o o k d n F o m
s f x I E V U R U P V E H Q E V
h p o n l l m I I H 7 n l m n
o m o n l d o n n d F x n
n m d o n o n D n d ( D n d o m
o l o n o d F
s n o m o n q o m v d I W W Q F
n d d n d o F
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F F F o m n o n l n F
h r t o n s o n
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Contents
p o o d S
I F s n o d o n T
P F q o m I T
Q F m n d Q I
R F f n n E n d S I
S F q n l m l n d T U
T F l l o n U U
U F e o m o n V V
V F o n o m I H P
W F n o m I I T
I H F s n o n I P U
I I F h n n d o ' I Q W
I P F q o m l o m c c
I Q F q o m o m m n I S W
n n d f l o I U Q
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Foreword
s n o m o n q o m n g o m n q o m
o n m o m n l o
n F o o k m l n d o n o F
g o m n o m l @ n d n o m l m d d A
( l d F r d o o l l o o m o E o E l n o m
d D m d n m o l F m n o m o
@ n l m A m o o n o l d
d o m n n l E
D o @ m A l F s n
o m o m n o n o d D n d n l E
o d l o o o o n l m n
o l d d d o o o k o n n d F o
9 k o n l l n F e n n n o d l o n
o d d o l l o o o d d n o o n o
l l F
k n o l d l l d k n d l o n
d o m n o n o o F o o k
n o o n o l n d o m n n d D d o ' o o
l z o n n D m k n d @ n l A
o o n o m o m n m n F o l d m o l
o o n o ( n d F
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1
Introduction
s o m o m n o l n m o n y d l l D n o
n o l n o m y m o n n F w o n l
' D l D l y D o n E d m n o n l
I
F r o m
@ n d o l d n l d o m o A j
l F o m D o D o m o d m n o n o l
F n o o n l " l m y o d o l l o
o n d D n o m o n o o d o n m k o l "
n d o o m o m o l o n d m n n
m l E d m n o n l n o m o n o n n l F
f n o o m l D o m o m l l o
o o m o m n d ' n m o n n m E
o n F n k d o m m n d o n n D
n d n m o n D m l n o
n m l l o n o l l d @ n A m n o o l D n d m o o o n d
o n o m o l F
m o d o m n n o m m n Y
g D f o r t r n n d p r o l o g l n o m l F
s o n o o n o o o l n D o o l d
( n d o m n o o o o m l m n D o o n D
n d l o m o m o m o n o
o m n o m F
g o m n o m l o F s n
d d d n o n m o m n d o n o m m n o E
I
y u p i n n i l i z d o t u ' o u t t i F h o n 9 t u n d t n d
t d i ' n t n g p t l n d i x p e n s e c e l D t i i l m o t t i n l o u
o l m F F F F
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s n t o d u t o n U
n D o o n l o n D o o n o " o E
n d F f l o D m o o n d o j
o l d n o n m n D
d n ( d z z E F s n o o k D d o n o o l
o n l l o n D l o o l o m
o d o m E d d d n n d m n F
Computer graphics
g o m
P
n n n m o l m n E
o n o o m n o m F m n n o d o
o o k d l n j m d l o o o l m l
X
e l o m @ f n m 9 l o m A o d n o n
n F
g l n n d n d o F
n o m @ n l d n A F
E m n d o l o n F
r d d n E l n 9 n d d d n E 9 l o m F
l l D o m o o l l m n o n d D k n o n o E
l o o o X n n l l d n
o o l F
q n n l d d l D n d
n o k d n n m n d ' n d o n D X
q l o l @ d o 9 A l n F
p l n m o n F
p l l n d m o o n o F
h n m n d o l l o n F
o o o l m m m o o m l d o m "
n l o m @ o n n o o o l o n
d o n ( n E l m n @ f e A m o d A " n d o o m o
n F r o D l o o l o j D o m o E
o o m D n o o n o m l m n F
P
g l l d i o m n o o n F
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V s n t o d u t o n
I @ A " e n m o o l o n n o m E
n o m @ d d o n A n d n E
n d l o n m z d n F e m l
d m o o n o n l o n d n "
o d " o m l d l o n
l l F x o m o o n d
l o n n d o m n o d d D
m n n n ( n n o n E o m F
e m m D d d o n l l n
l o n D m n n @ o o A
l o n o o m o m n @ o
n o m o n m " g i s A n o o n n l E
l o n F
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s n t o d u t o n W
Computer vision and image processing
g o m o n n d m o n l n d n E
n l n l l F w n m E o n n n
l d n l o n " o m l D n n o m
n d m ( l " n d o d o n o F o E
( l n l l n @ i A o o n F o m n l
n d o n o n o n o o m o n o m o
o n Y m n o o l l o l n o
F
Computer-aided design
g o m E d d d n @ d A n l m Y n l d
j l o d n n o o m F e l l
n o m n m n o m o D j m n m
d n n d o l d m o d l l n m ( n F l o m E d d
d n m o n n o l n E E l n d E
o m o n m n o l o n E o l d n o o n z l
n d l ( l d o n l l l n d o F
Computer-aided geometric design
g o m E d d o m d n @ g d A l l d o o
d o E o m n d F m o d n
@ o d o m o m n A d o o n o n
n d n i s n d n l l d m l o D z o
o o l l o n l o n n m ) Y
F g o m E d d o m d n o m n n o m m o n
X d o o o n m o n o l l
d m o k o n d n n l o o l n F
d n o o o d z o l F
Computational geometry
g o m o n l o m m o l d o o
d o o m l l o m D n d l l o o l
0 n D o o d F m n D n l o m n n I H o n d
o n I H o n D o l o n d o k o n I H H c p o n n D k
I H H o n d D l l n o d n l o m Y k I H H H H D 9
o d n2 Y k P H H D 9 O(n log n)D n d o o n F l
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I H s n t o d u t o n
n l m o m l d D o m l n
o m o n l o m o n n m l o n m l D
o o n F r o D o n n n 0 n
o l m n l o n D l o n m o o m l d l m n F
Data visualization
h l z o n n E m o n m n l o n n d
d l o l m o n o d l l o n d o m n o
l l n d o d E n n F l k m d
d m n o n l y o d Y o n n D l l o l l
l o m n d o n o m D n d o d E n n
d o n d ' n o n m n F n l o l m
l n o m o n @ o n l l d n 9 A m k n o o
m o d m n o n F e n d o o l m o n n
l o m 0 n n d l m o n o
d n o l d n m o l o n F
Numerical control and robotics
o n o l o o o m o d n 9 o n o l l n
o n n o m n o o l o n d l o o F e o m o o n E
n o m o n n n o l o o m D o o n
n l l @ l l D n l A Y n d o n n o l d m n l E
o n ( n l n D l o m n o m
@ l l D n l A F n o l m o d n m n d l n n n
' o o o m n d o d o X n l k l l n
m o m n o n D j o D n d o ' o m F
From geometry to program
o o m o m o n o o m D n d o o o
n X
o m t l l o t m o m F
i o l l o d o l m F
Geometry algebra
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p o m o m t t o o m I I
l l o ' o m d o n o o l m D
d o E n d E l n m E n d E
l n c F o n D m ( o n l n
n d l n n o n l o m n o o d n m @ E
l l g n o o d n A D n d d d o o l d o ( n d
n F l l o d o n o l l n
n l o m D l o m n o n l o m o l o n
m o n n m n d o m o F n n d d 0 E
l o o m n o m D n d o n n o
o n d n o n k n F p o n n o n
o n n d n o m n n @ o ' o l m A
d l o l l ( l k l F l l l l
o l m ( D n o o m ( n d @ l A
o o n o n n d n o m n Y 9 l o l
n l D l o F p l l n k o n o d l o o m
o l o n @ D o o n A o n o l o F
i n o o d n l o l l D o n o
n k o Y o n n D o o n l o m
o o l o m n d n o n @
l l d o A n n d o m o m l m o n d o n n m Y
o n n o n d n m n n o E D o
o E o l D n n o o o o n E
n o m ( o n F
Algebra algorithmx o 9 o n o m m o n o n d o o n ( o m E
m n F r o D o o o n o m o o o m
o l m o n o n l F p o n n D o m l E
m
Q
o m m o n n d l o o l o m n l E
o n D l l m o l d ' n o n n d n o n F q o m E
o l m n d o o d l o n o n E l n o n
k l m l k Y 9 n m n o m l F
y n l n l d 0 l n o n z d n d l o n
l z d o m l m n l F
i n o D n n l m m k o n o n o o E
l m F 9 d o d n o l l
Q
g l l d l t m o m n o o n F
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I P s n t o d u t o n
m l o m n l l o m n o n X y m o l X l m l l Y o l o m l l
n o o n D n d o k o o d o m o n
) F o d o n m k D n o m o l o
l n o n
R
F
e n l y X l l 9 l l o Y
l o m d m o d m n o n l o l o n F
x m l X l l o n n d Y o l m n
o m l d D n o l o d E o m o l o n
S
F n o n
n d d n m l m o d @ F F l o n A o m l E
n o o n F
e o m X n m n Y d o n 9 n n
o 9 l D n d o k n m l ( o n F
l l o k l l o m o l o F
n D n d l l D n d F p o n n D m o m
o m o o l m D n d n o m l n n l o l o n
o m l o m F o n d n o l m D n d
n o l o o m o l o n o n o
d F
Algorithm programy l o 9 m o d 0 l o d ' n o m n d d o o d
o m m n F m o n n d o l d E
l m n l n o l o n o o m @ n E
m l o n n X l n l l o n n m l l o n o l l d
m n o o l Y n m l l X n o o d n o n d l
n A F m n d o d o d n d o l o E l l o d o n
o o n Y o m l @ l l l A o m n o o E
o o n n l o o m n Y n d o l m
o m m o n F e n D o d n d D d l
n D o d k D n n 9 n o @ o d d n l D
F F F A n ( n l o m F
R
o u o n l j u t @ I W W Q A Y o m l D n t n i o n t o t n g l i F
S
e l o d E o m o l u t i o n i n n t t n i t t n d o n t i t Y
o m l t o o l o o k o m u l
b b2 4ac2a
i l o d E o m o l u t i o n
t o t o l m o ( n d i n t o o t o u d t i D u t t i n o l o d E o m
o l u t i o n t o t o l m o ( n d i n t o o t o u i n t i F
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p o m o m t t o o m I Q
I @ A " h m n o n l D n l n d o m n o l o m l E
n d n d n D n d m l d ( n o o
E d m n o n l 9 F
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I R s n t o d u t o n
Dimensional, analytic and combinatorial snags
r n l o o k d o m o m m n o D k
o o m o n D n l o l m n o m
n o o n Y D o m l y n o m
o l m c s n d o o n o m D n o l n X l o
o d m n o n D k o n D n d o o m n @ d ' n A o
o m o n F s l l o n I @ A D n o n
T
F
Dimensional complexity
q o m o k m k l d ' n l y n d ' n n m o d E
m n o n F p o n n D n l l l E d n n o d E
m n o n D l l d l n F @ g o m l o n d l o k F
d o n x o o l c o d o n 9 o E
l m o n l o k F A e n d o n d d m n o n n o Y o
n m o n o l d o E d m n o n l o j D d o n 9
l m D o n n o m m l Y m d m n o n
k o l k o m F
Analytic complexity
l d d o F i o n n n o
n n Y k n o d o o m n l k X l n E
D d D D o n l d D o o D
l D E d o n l D n o n D n n d n l D
o m l o l l F F F F o l m m m d l o m m
o n n o n D l n d D n d o o m n o n o E
o n m o n d d F l o o l " o D
o " o o o n d d n o n l o m E
l D n 9 n o l d F
Combinatorial complexity
o m o o o l n l o o d Y n o o d
n m o o n n o l m F @ o D n 9 o d
o n n m o n o n d m n o n D o n d n
T
s t 9 i n g o m p u t n h p e @ t n n d f i l i o o d t i l o
o o k n d m n t i o n d i n t t t A D u t t i d i o i i n l l t t i u t l
t o g l v n D l i F
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h m n o n l D n l t n d o m n t o l n I S
o m n k F A e o o d d l o o m o n l o m o
o n D n d 0 n l o m o d l n l o o m F w o
o m l d o m n o o m n o l o l m Y
m n d o m k o m o n n d o F p o m l D
o ( n d l l d o m d n m o n n D
n d o o m Y o ( n d D o F
n o d n3 l o m o F
e n d d o n l D d ' n D o m n o l l o n
l o o d ' n o o m o d l n o n o m F
s @ n m o d m n o n A n o ( n d n o n n
l n D o n o d o Y n o d l D
n d o n X l n E l n D l E l D n d l n E l F s
l l D n d o n D n d o o n F o
o m m @ y o o o n A Y d o n 9 j
' l n o m o m k F s 9 o l n n
o l l l m o n l o n D n d l o o n k
o o n o o n F F F F
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2
Geometric basics
Dimensions
v 9 m o n 9 m l n n n o n d m n o n "
l o n l n X n o d m n o n " n l n X o n
d m n o n " n F x o D o n k 9 o n o o '
n o n d m n o n d o o D o 9 m k n F s n D 9 l o o o l o n d m n o n o n d F m o m k o m m n l l o n d l o o l l o n
F
Dimensionality looks easy to extend but it isnt
y n D o D d m n o n o n d l k o n D o D l @ o
A " F F m o o m Y 9 n o o d m n o n l
o k F e d d n d m n o n o o l m l l y n
n D l o m n D n d
n d n o l F
s n o n d m n o n D n @ F F 9 l k o m E
m n m o n A Y n 9 n o l l n o m
l l F f D o n o l o m o l m n o n E
d m n o n l D n d n o n E d m n o n l
d ( n d o o l n o n d n o d n d n o d D
n 9 d o n n E d m n o n l F
s n o d m n o n D n n o n o m n D
l F w n o m l d @ F F
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h m n o n I U
o l o n d A o n E d m n o n l m d d d n
D o n o k n o n l d m n o n n d d o o n
@ F F o o d o o l o n A F
s n d m n o n D o o j n o o n o o m E
n F y n o n d D n d o j o
l n l o l d F n o o n E d m n o n l
@ A n d o E d m n o n l @ A m d d d
n o F e o l d o n " o m l " n m l o
l n @ d A n d @ A n d D n l k o l o n D
n o n o o d n m F
y n D o o d m n o n o n d l m n F n o
o D ( " o m o c w n o n n l z d l l
n o n d m n o n D d o n 9 m n n d o n n n E l m F s n l X
t m m " n d o m " n d l
m o n d m n o n D d o n 9 o l o n E
n d n d n o E d m n o n l X l o
n l n m o d d ' n l @
f n o ' 9 o o k A F
s o n d m n E d m n o n l X o n E
n m l E l v n d o l d m n E
o n n d o o ( d m n o n o n o d n
F f d o n o m n m l E d m n o n l
n l l d m n o n l n d m n E
n D n l d m n o n d o F
v 9 j m d o D n d l o o k o m l l o n F
m o m m d o o d m n o n F f n
n n m " l l o l l l " n o
l n o n n n n n d d o n l l d m n o n F
o m o D m o l o n l l
m m l n xD y D z n d m Y n d o m m E l l l F
n n m l o o n o d m n o n F
k m l o n o l @ l l
d o Y o 9 o d D o m k l A X
(x x0)2 + (y y0)2 r2 = 0.
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I V q o m t
e d d n m X n o o n o l d o m o d l l m o n
n l n F o j o m l n
x = x0 + f t
y = y0 + gt,
t m X o n d o l k F o t o n d l l l o m X
(x x0 + f t)2 + (y y0 + gt)2 r2 = 0.
p n X l t z d o D 9 n l l l l n d X n o m m l D n d n o n l k
F
v o o k n o o n d F k E
o n X
(x x0)2 + (y y0)2 + (z z0)2 r2 = 0. o n o l z t @ o m A D o c x o m o n l l l D l n n d
o d n o o m l X
r = t2 + pt + q
@ p n d q o m o o n n d d o m z0 n d r A Fi n n m l D n o o o n l n k n
m o n o E d m n o n l n d E d m n o n l
o n F s n n m o Y m o l o n 9
o E d m n o n l m o m n o n n o n o m m D o
n n o o n D m k m d 0 l o n d l F
n l m o n l z n m o n D o
n n D o o E d m n o n l o l m D m o
d m n o n Y n l o o l n
l l o D n o n n o l l @ F F q l n 9
I W V V D n d o o d k 9 l o A F
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o j t o n I W
Projection
y n o n n n d o o o n d d m n o n l
o l m o d d m n o n l o o d o n
o m o d m n o n F d o n m k
o m E d m n o n l n D n d l l d o j o n F
t o n o o d n l d o m o E
n F p o n n D n n n o o j n n
D o m l d l o n n
o n l o j o o d n n d o n n o o d n F
p D o n n o o o o d l o o d n o X
n o o d o m l o n l o m F s n o
o l d o j D o n n D o n l n o n o
F
l y o l m n n l l o o j o n
n F E n l l E k n o n n d n n Y
o j o d o j d o n o n m o l n D
o o o n d o o n o m o n o d o @ l D
o n d A o n n n o n F s n D
m l n n n o n o @ s l l o n P @ A A F
o ' o m o j d n o o k n n
n l d m n o n m n o n d o Y d o o
d ' n o j o m n o d D n d o
n n o n " l o n o n "
o o n d k l F s n D m l n n o
n n n o o o l m D o d Y
l o n E o o m d o l o o d o F
s n o D o j o n n o l F p o n n D n E
l n n o n o j D o l d m n l o o k
o n o E d m n o n l d ( n d F s n D
o n d 0 l D l o o E
d m n o n l D d o o n o l o E d m n o n l
Y o k o m ' o o l n o n n m l o E
j o o m n l d l d o d F n k o n o d
o d n o n o o n Y n m D o
n o n o n o n 9 F
o l n o n o j D n o o m
n d ( n n n d m n o n @ l n 9 l l E
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P H q o m t
P @ A " E n Y o j o j d o n o
n o n Y n n o n o d E
m n o F
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o n t n d t o P I
k n o n s i g g r p h A F e l o o k o d ( n
n o n l n d o l o n o
o j D d o n 9 n d o k n o n o @ d n d
o n d l A o o j l l l l n d F
Points and vectors
s 9 o o m d m n o n " " 9 D
o o m n o n c o n o o d X j l
o o o d n l F e o n l o d l m n n o E
o d n (x , y , . . .) o m o n (0, 0, . . .)F o m m 9 d l k o d l m n o n d D n d m o o o n o E
l o m o n 9 l d o F ) o n 9 o n
l l d o F s o d d o o o n D o d ( n n o
o n D o d d o o o n o o F F F F s n
l l l o n X
P P = VV + V = V
V V = VP + V = P
P V = PP + P = . . .
r 3 m o o o n n d ( n d
I
F
v o o k d n o D o d ( n m o m n o
n d n n ( d d o n F n o
d n n d d o n n o m l z n o F k E
d m n o n l o D (xa, ya, za)F p m n d X
|a
|= x2a + y2a + z2a.
o n d n o m l Y 9 l l n m o o
l F s d d l l o m o n n o o m n E
d n D n o m l z d D o X l n o I D
I
s n o t o d D t t o o i n t n d t o i n o t l o s e d u n d t o E
t i o n o d d i t i o n n d u t t i o n F
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P P q o m t
d o n n n d X
a =
xa|a| ,
ya|a| ,
za|a|
.
o m l l o n c n l m l l o
o n n D o l o n o o o n F l o o
l o o m n o X
d o o d o o o a n d b l d l @ F F
n m A l o |a||b| cos D n l n
m F o o d o ( n d o n l n o D
l l n o F n o d d l o m
o m o n n o o D j m l l n m o X
a.b = xaxb + yayb + zazb.
o o d n n o n d l o
o n m l l d D l n l o @ o d E
n A o d o l n X
a b = ((yazb ybza), (xbza xazb), (xayb xbya)) .
o o d o o l m l o n n
l o o 9 o o m o m F n d o n k l D l E
o l l n d o n o d n m o
m o o n n F n k o n l n
m n o n o n o n o Y o d o m l l n
m o l d o m o n o o d 9 o m n F
l l n o n l d o m o " o n n d d
n o n E l n 9 o d m o o m l o n " n l l n o m E
o m n l l n d 0 n n o n F
Implicit and parametric geometry
o o m m o n o m l m n c n o
o o l o o l l n d o l n y =mx + cD n d n o n d o o n o l d n 9 n l l n D d o x = k Y o D n d n l n n l
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s m l t n d m t o m t P Q
l l o mD o o 9 d o ' o m
x = m
y + c
F F F n d l o m o o n o o ' F
o explicit o n o o o m y = f(x) o x =f(y) @ n d l D z = f(x, y) F A o n l l o d n n o n D n l n m D
n l E l d " o k n o o n 9 d o l k o n m E
l F s n o @ g R A l o n l l
m o d l n m o n l o m l m n
l l n o l o o k F
s o o m l o m l m n n o d o
l n m n o l D n o o X
Implicit equations
s m l o n l y l l o n n l n D o n D
n o o Y o o o n o d ( n
o n d n o F m l o d o
o l o m l f(x, y)" o f(x,y,z) n d m n o n " o n F l n m Y 9 n D o n o n
o n d o o Y 9 o D o n o n
o F n l o n k o m n o m o n o
o n E d m n o n l l n D o n n s l l o n P @ A F
o l o o n m o n o
o n o o n E d m n o n l l n X o o
f(x, y) = 0 o f(x,y,z) = 0F l l d m l n d D m l d o n l ( o n F
l o d o l E D d d o o d n
n o o l X o n l ( d o n d o n
l ( d n F o l n o n n n l D o
o D n d o n m o n d d n d o n o @ n
o l A o o n o n d d @ n o l n A F
r o D l E d d D m
d m n o n l o n l o n n
m d d d F o D n l n D l l m l o n d Y
n E d m n o n l D l l d Y n d n o
d m n o n d o l m @ l z o n A F
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P R q o m t
P @ A " g m n X n m l l n m
o m o o o n d m n o n Y m o n m
o D o m o n o o d m n o n F
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s m l t n d m t o m t P S
Parametric equations
m o n o n d n o d n o n o m o E
l D o m D n d l l n xD y " n d z F " n o n o m X
x = 1(t ,u ,v , . . .)
y = 2(t ,u ,v , . . .)
z = 3(. . .)F
F
F =F
F
F
o n n k o m n o o o o d n F
@ s m m xD y D z o o d n D n o l d n o m o n " g W F A e m
o n l o n m n n o o n o
n m l o n X n o " l o o k s l l o n P @ A
n " o n o m o n E d m n o n l l n o o E o E
d m n o n l F
r o D n d m n n m o o o d n
n d n m o m n d n d n l D n o ( d E
l o n n o o o m l l m n n d
m o n n d n o k n F
n o m m n m d o E d m n o n l
n o E d m n o n l D o l n F f
m o l m o n n o n o n l E
m n o n d d m n o n @ l n n d A n d
n o n o o m n @ A F
o m m z X
s m l o n g l o n p d d m n o n l
n l o
m o n q n o n o n e n d m n o n l
l m n
s n n l D n o o o n n m l n d E
m o n o o m l m n @ g I I A F r o D
m l d o o m l n d m o n D n d
l l n d o o n l o o k n m l o n X
l n D l n D l n d F s o n m o d l D
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P T q o m t
( n n l o n d o n o m o o m m n d o ' o
o o X e o m m 9 q o m y F
The straight line and plane
m l l n ax + by + c = 0Y m l n X
x = x0 + f t
y = y0 + gt.
f n o n l n ax + by + cz + d = 0 n d
x = x0 + f1t + f2uy = y0 + g1t + g2u
z = z0 + h1t + h2u.
The circle and sphere
l d n m l o n o l
(x x0)2 + (y y0)2 r2 = 0, l n d d o X
(x x0)2 + (y y0)2 + (z z0)2 r2 = 0. n (x0, y0, z0) n d d r F l m o n o l X
x = x0 + r cos
y = y0 + r sin
n d l D n o E o o E l D X
x = x0 + r cos cos y = y0 + r sin cos
z = z0 + r sin .
n l n d l d n d l o n d l F o o l m o m n o n n F p D o o m o n l
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f o u n d n o m t P U
o n d o n 9 l k n o n @ k o o l o n A D o o m E
m o n l l
cos n d
sin l E n l o m l X
sin =2t
1 + t2
cos =1 t1 + t2
where t = tan
2.
o n o l m @ e o m m 9 q o m y A Y
o n l d o 90 o o l F o n d D ( m l o n m z o n
P
F q o o n d
w o l d l l E k n o n o l m Y n
o m l " o o " n o o l o n X n o
n n l l l ' o F
Bounding geometry
s n D l d o m n n n ( n Y 9 o o l o
n o D o n o n l F n E l n E
m n D l D n d o o n d o o o F
o n d n n o o l n
o o m n D n d o n F s m l o m
l ( n D n d o o o o o l o j o F
e n l m n o m l o m l ( n o o F
o o n d n d m n d
o n o o m o j F n o n
o n d o m l l o o D l "
n n l n d o m l d " m o o D
F o o n " n l n
l n " n d o n o d o o n o m n
o n l ( d l o d n 9 l o n F s
o n d n m l o o m o o n D n
P
x o t t t t o d p m e t e z t o n t o m n i n X o i o m E
t @ A n d o n i o n o m i m l i i t t o m t i o m @ t o o i t
o i m l i i t i z t i o n A F m n o o u n d t i t m i n o l o i l t F
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P V q o m t
o m n n o o l d
n o Y l l n d o n l n o n o o D Fp o m l D o d o n l l n d
o o d n X
x = x0,
x = x1 (x1 > x0),
y = y0,
and y = y1 (y1 > y0),
n n o o o n o m n l X
x x0x x1y y0y y1
n d o m n m X
(x x0) (x x1) (y y0) (y y1).e n o n ( l n d @ o k D " o
o n d o o n o A n l F
t o m k n m o o m l d D n o o l d
d m o n d l n
m n m m n o n X
min ((x x0), (x1 x), (y y0), (y1 y)) 0.e l o n o o n n o l d ' n l n o n D n
m l o n n o n F n o m o n o l k o
n l n o n @ g S A l l o m n o
o o m l n o o n m l F g o m n n m l
n o n E o o o n d D n d n
o n d o n o E o @ o o n o l d o m "
s g A o l d m o d l l n D m o n o n d n o
F F F F
o m o n c n l o n d d
m l o n n m o m @ tD uD n d o o n n o o o d n " o m l o n n
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f o u n d n o m t P W
m A F e d D o n E d m n o n l
m D o o n d n n 9 l l o o o m E
l l F j n n n n d n n d n m
o " t0 n d t1 F f o o l l n n o n o o o n E d m n o n l m l n o n X
t t0t t1.
s n o d m n o n D o m l 9 o n o n F l l D
m n d D n d
o d l n d n l @ n d 9 l l A
n m F e n D n j l m n
l o o m @ l l 0 n d 1A D n d d o n 9 d o o n n d d n o o m y o o n d F
o l d o n d n o n n m D n d n l
o m m o n
Q
F o m
n m d " n o l o n o l l l o o k " o
n d o n E d m n o n l l 9 F k
n d n n m n l o n D n l l l l D
D n o n D n d l l n o
n l o o o n o n o n d n
d F e o n D o n n o o E
o o n d n o n D n m o l F
n o o o o k n D m o o n n n
@ o n n n n o m n n o n D o l l n
g I H A F
m o n n l o o n d n D m
k n Y n o D n m l o n D o E o o m E
n o n o m l o n D n o o F s n
m k o n o o o o o
@ F F m A l m n D n d o l E F e
m l m l o l o n m d o m l n
m n F e l o n o l y o n o m l D k n o d E
( n n n d n d n o d F o l m l n o n D
Q
n d d o t o n o o j t i t o u n d d d Y t
n o t D i n t D u u l l d o n t t i n u t i n u l o u n d i n t o
t t o m t D u t t t i n u m t i t m n t h e t n l e t s e l f F
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Q H q o m t
P @ A " n E 9 o ( n d o o n
n d o l o n X n m o n o n o d d D
F
n n o m l o n F 9 o o d o n m o
o l D o o n o l n o m o n o n
o n ( n D n d o n n o m n m o o l o n @
s l l o n P @ A A Y t o d n o m l l
( n l o n o d d o n d m n 9 n d
o l o n o n o F l o @ l l n
d m n o n A D n d n n d n o m 9 d
l k n o n o o o l d n o n D o n d
m o d l F F F F
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3
Parametric
curves and surfaces
n n o d m n o n l n n d l
n d o n o x n d y D n m o n l m tF e l o t n D n l o x n d y n d F s n d m n o n D n d o n l d d n o o n
o z F l n n X
x = x0 + f t
y = y0 + gt
z = z0 + ht.
o m @ o m n o A ( n d l l o n k o m
o n m n o m o n E d m n o n l o o d n m
n l o o d n t n o o l o E o E d m n o n l F D o n o l n
I
o d o
o m n l o t o l o xD y o z @ n o o o m l o x,y,z o l o tD o n l o n o n o o n d n l o tA F
s n o d o n d m n o o n o D n
m n o m o E d m n o n l @ 9 l l t n d u o o d m n o n A n o l F f d o n o n n d
m o m t, u n o n o o E d m n o n l D n o E d m n o n l n o m X o D E
o n n d m D E d m n o n l n o m F o n
n k o g W n o n m o m E
o n m k o F s m k m o o l l o l
D o o n 9 n F
I
y u D i t2, t3 n d o o n i n t u t i o n F
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Q P m t u n d u
v 9 k Y m o m t, u n o E d m n o n l o o d n m @ F F
x,y,z l A D
F m o n o l n
x = x0 + f1t + f2u
y = y0 + g1t + g2u
z = z0 + h1t + h2u
l d d F
s 9 m n o m o d n o o n n m
o t o t n d u n o o n F s l l o l n o m o m m k
o n o n o n o D n d o o n d m
o l n o m l F s n l
o n d D n n o n Y n d n
m n l n @ n l A o l o tD o n n l n t, u l n F y o d o n 9 m n o o o n o @ m o o l A F
e l m l o n d m o n o
m k n o n o m @ A n o l n o m l X
x = a0 + a1t + a2t2 +
y = F
F
F =F
F
F
o D o m D
x = a0 + a1t + a2u + a3tu + a4t2 + a5u
2 + a6t2u + a7tu
2 + a8t3 +
y = F
F
F =F
F
F
e l n l m o n d o n l o l n o m l Y o n k o
o n m " o k n l l m l " X
x =a0 + a1t + a2t
2 + b0 + b1t + b2t2 +
y =
F
F
F =F
F
F
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s n t o l t o n Q Q
@ o n d l n D l l F F F A o n l o o n
o n o n o n l D n
t = tan2
m E
z o n n l F l n o o m
m n l E d o l n o m l F s n o n @
o m A n d o o o n D
n n l E d o n n o F r o D l
o n l l l o o d o m o l o E d o n D E
n o d n o m n o n o o o l m X n
l l o d o o m l o o D o o o D z o c n d o
o l n o m l n d o m n o D 1 + t2
m n l m z o n D o m l n n
o m n o k n d o n o o l F
s m k n l o d D o D n d o o
l l n o o n l o n F
e m o m m d o l m D l o n o o
l l o n n a D b F c l l o n o n l o D o m m n n D n d m o n o n o d F
Interpolation
n l o o k d l n n d l n D d n o o n
o o X x = x0+f tY n o n d o x = a1+a2t+a3t2+ F o m n k 9 j n o o l D n d 9 D m n o n D n o n o l n
n d l n D o o 0 n d m n n F k
l n n D o n n Y (x0, y0, z0) o n t = 0 n d(f ,g ,h) o n d o n o l n F
n d d n m n o o tD o 0 n n o n m n n F m o o n o l n d
n l d F m o o m m o n m o d o m k
o o n o n n D n d m o o m E
m o n o n n o o n n d n n d o n D l o o
o n n D d n d D l o d F
g o n n o m l m n o o o n n o o
l l d n o l o n F
s n n l D m o o 0 n n o n o
o D m o o n n l o m F
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Q R m t u n d u
m n o n X
n d n o o 0 n o j o n n d @ n l 9
o n o m o n o n o " l A F
d o n 9 n o o m n o 0 n D d o
d o m o d o @ o n m A
o m o d o m o F
o D l 9 l o o k o o n o l o n n d o o n D
l n n d l F e l n l o l ( l o o n E
n X n o o o o n D o o o o n n d
n n o l F e l d o d o m D o
n o o o n D o o o n d n n o
l n D n n o o l n d l n D n d l o d m o Y
o n ( d D n d o d o d o m l k
l n F e l o o m n o n D E l n n d l o n E
o n m l D o n o n n n
n d n d n o o n n d n n o n n n m n d ' n
o m n o n F
m o n l m o l n o m l D n o m l l E
d o n o o n n d n n l l l E
o m F
Lagrange interpolationv n n n o l o n m k o o
n m o o n F s n o o n o o 0 n n
o n F p o n n D d l l o o F
d d l l o o 0 n l l o l n
o m l n o o n n o 0 n F o n d
n o o d n o o n " (x , y , . . .)" n d l n m @ A o o n " t , . . ." n o o o n F
Hermite interpolations n r m n o l o n D d ' n o n o
o D n d o l m l n o o n o o o o n n d
n n l o o n n n o l d F
m n o 0 n d n v n F e E
l l m o n o n n n o n d
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s n t o l t o n Q S
Q @ A " v n n d r m n o l o n d o o n E
m m n F
o n D k n o n n n l F
o o 0 n n o n D Y
l o n l n n n o o n o n D
n d I T o 0 n F g n n n o m n
o m X s l l o n Q @ A F
The problem of parameterizations n o l n m D d d n m l E
o n l l " o m z o n " l F
@ o l m d o n o o l D n l E l d D
n d X n d o n d o
d n n d o o n F e n d n o m n 9 l E
o n l l l o l m o n D o l d
o n o n l n m o m o n l o m o l m F A
o D o l m n o l n m D l
o o n n d n n d o n m o d d D o
m m l o l d 9 n
o n D n d m n d l l d o n o d E
F s n o v n n o l o n D m l o
o m l l l n o n d F o k
o n m l n l d Y o o m E
n n d d F n m z o n l d o
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Q T m t u n d u
l n D o l d l k o k n o l n o l l
n d o n Y n o
o D n 9 o F y n o l d m l m n
n o ( n m n " o n D
l n o m D n d o n n m l d
o n D n d " m o l n o l l E
m d Y o l d o l d 0 l n o o o l d
o n F
l o l o n o d E l n m z o n D
m l o n d o n l n o
E l n m n o n n n m F o o d o k o D
n o l o n l n o n 9 m l d
d D n d n o n F p ( n m n n o l k n
n l n n n o o n @ p n 9 o o k
g n d o g o m e d d q o m h n o m o
d l A F
r m n o l o n D m l o l m o Y n d m
m m d m n d o d o o m
dx
dt F D
l d o l z o n n o l n n
d F D l l n l o r m
o 0 n D m l d l l F 9
n o o m l l n D o n o m c
e l l m k n d d d o o l m o n E
o l o n D l o l o ( n F e n D o
o k l l l o m D n d l l l m l E
@ n o m l z o n o l m A F m
o m D n o d d n d n m l
o n l l o o n d o F f o n l l d n D
d 0 l X o d E l n m z o n n l l F
o n l d o n g o d o v n n d r m
n o l o n F ( o d o k o v n n n E
o l n m o l n o m l o o o n n
d m n o n F o n l l l d n D D n d z F
m o n ( o n l l 0D n d m E l o n 1Y o 0 n o o l n o m l l l n d n o l D o l D n d o l z Y o l Q o 0 n o
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s n t o l t o n Q U
t3 n x n d o o n F m l m d d l o o n o n l l n d n t I n d t P F
5 n l u ` m t F b
5 n l u ` t o F b
G B e o l u t l u m o B G
5 n @ A @ @ @ A ` H F H A c @ E @ A A X @ A A
G B e l m o t H E j u t o o u l t o n B G
5 n e g g @ I F H E T A
n t l n @ D D z D t I D t P D o l D o l D o l z A
l o t R D o l R Y
l o t R D o l R Y
l o t z R D o l z R Y
l o t B t I D B t P Y
{
l o t D D z D l Y
n t Y
G B
u m t t n t n t o n t t o u t o
l t I n t P F x o t t t m l t o m
t n t t n t o o n u l l o t
t n g m t l n o u l F
B G
l a H F H Y
o @ a I Y ` R Y C C A
{
a E E I Y
a E E I Y
z a z E z E I Y
l a l C @ l o t A t @ @ o u l A
@ B C B C z B z A A Y
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Q V m t u n d u
@ a a I A B t I a l Y
@ a a P A B t P a l Y
}
@ l ` e g g A
{
n t @ t D
4 v n X u t o o o t X 7 n 4 D l A Y
t u n @ I A Y
}
B t I a B t I G l Y
B t P a B t P G l Y
G B
g l l t o u t o o m u t t o n t n
o o n t F
B G
@ l n o @ D o l D t I D t P A A t u n @ P A Y
@ l n o @ D o l D t I D t P A A t u n @ Q A Y
@ l n o @ z D o l z D t I D t P A A t u n @ R A Y
t u n @ H A Y
} G B l n B G
G B
o u t o o m u t t o n t n o n
m n o n o t v n n u t o u o u
o n t F o l t t l F
B G
n t l n o @ D o l D t I D t P A
l o t R D o l R Y
l o t B t I D B t P Y
{
l o t I D P D Q D t I D t P D t I D t P D n o m D t t Y
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s n t o l t o n Q W
I a I E H Y
P a P E H Y
Q a Q E H Y
t I a @ B t I A B @ B t I A Y
t P a @ B t P A B @ B t P A Y
t I a t I B @ B t I A Y
t P a t P B @ B t P A Y
n o m a @ t P E @ B t P A A B t I Y
@ @ n o m A ` e g g A
{
n t @ t D
4 v n o X n m n t t o o o t X 7 n 4 D
n o m A Y
t u n @ I A Y
}
t t a @ E t P C @ B t P A A B t I C @ t P E t P A B @ B t I A Y
o l Q a @ Q B @ B t P A E P A B t I C
@ E Q B t P C P A B @ B t I A C
I B t P E I B @ B t P A G n o m C t t Y
o l P a @ E Q B @ B t P A C P A B t I C
@ Q B t P E P A B @ B t I A C
I B t P C I B @ B t P A G n o m C t t Y
o l I a @ Q B t P E P A B t I C @ E Q B t P C P A B t I C
I B t P E I B t P G n o m C t t Y
o l H a H Y
t u n @ H A Y
} G B l n o B G
o n d o d o m r m n o l n n
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R H m t u n d u
d m n o n o o o n o d n o
n d F o n H n d I D n d d n H n d
I F o 0 n o n o l n o l n o m l n d n
o l D o l D n d o l z F l n m m l
n o v n n n o l o n F
o m t @ H D I D H D I D o l D o l D o l z A
l o t H Q D I Q D H Q D I Q Y
l o t o l R D o l R D o l z R Y
{
o m t o @ A Y
m t o @ H H D I H D H H D I H D o l A Y
m t o @ H I D I I D H I D I I D o l A Y
m t o @ H P D I P D H P D I P D o l z A Y
} G B m t B G
o m t o @ H D I D H D I D o l A
l o t H D I D H D I Y
l o t o l R Y
{
l o t D Y
a I E H E H Y
a I E H Y
o l H a H Y
o l I a H Y
o l P a Q F H B E Y
o l Q a E P F H B C Y
} G B m t o B G
Surface patches
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u t R I
Q @ A " e m Q = F(t, u) d ( n d o n l 0 t 1, 0 u 1F
m o o m
x = f1(t, u)
y = f2(t, u)
z = f3(t, u)
@ n l o o o 0 n D Q = F(t, u) E n m l l n n l d n A F
e n d ( n d o n m o o n o (t, u) m D o d l o o E d m n o n l
n l X
0 t 10 u 1.
m o n n o n o
o n d o l d m o n d o d j n F
n m n o X
h ( n n m o o n d Y o
k n n g o o n @ g o o n 9 I W T U A D
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R P m t u n d u
n o o l o l n d n o o n E
o m d o n o o n d F
i l l o o o n k n o n o
o n d F
g n o d o n n m
o d o o o t o m 0 o 3 @ F F 1, t , t2, t3 A n d m o o u @ F F 1, u , u2, u3 A F I T m n o n o o n d n o o o d n X
x = a0t3u3 + a1t
3u2 + a2t3u + a3t
3
+ a4t2u3 + a5t
2u2 + a6t2u + a7t
2
+ a8tu3 + a9tu2 + a10tu + a11t+ a12u
3 + a13u2 + a14u + a15.
y o o o n d n o n n y n d z D m k n R V m n l l F s 9 o l l d o
F e d u = 0 D o n n D o n d n o X
x = a3t3 + a7t
2 + a11t + a15.
s n D o n ( d l o t o uD o o E m n o m F
e l l k n o n o o n o o n d D n d
o m n n " n d m d D d o E
F " o o n d F v 9 ( n d n o n o
n n o u = 0 d o g n o d F p d ' n o uX
x
u= 3a0t
3u2 + 2a1t3u + a2t
3
+ 3a4t2u2 + 2a5t
2u + a6t2
+ 3a8tu2
+ 2a9tu + a10t+ 3a12u
2 + 2a13u + a14.
n u = 0 n X
x
u u=0= a2t
3 + a6t2 + a10t + a14.
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u t R Q
e n o m o n d n d d
o l n o m l
P
d l l m
u = 0 d Y
m l o n n l o d
Q
F
s n o m m o n D n o k o d E
d n d F e n n o n l D o k o j
o n o o m n d o d o m @ n d m n o n A o
n o k o l o d d o n l d F @ r E d
l o o d o d o m D d d o n 9
n o F A s n d m n d r m E
n D o o m l o n n l l o l o o n
o n n l o n n d j n @ o " l o o k n d "
o o n d n o f z o n o l m A D n
d o d o m m o E l l d o
o n X
2Q(t, u)
tu.
s l l o n Q @ A o k o o o F d
o l o n o o l m d d d o m n F @ 9
9 l l d E d F F F F A
p o n o D l 9 l o o k o m n m l F r o d o d c
m l o m k l n d n o n o E m d F
v 9 l o o k o m o d o d o F o l d m l o n
o l xD y n d z n o o D n d k l l n Y 9 n o 0 n D l o n o E m D to u ( d D n d o o l d d o n l o o l l o n F o d o l l o d n o E m l n ( d
l o uD n t n n F m o 0 n l l o l I T D I T D z I T D
o o n d o n o n o n d
o o d n F
r 9 l Y n o n d o r o n o m
R
o E
P
p o o m l i t i o n D o n t i n u i t o n i d i t i i u i d F
Q
g o n t i n u i t o m t i d i t i i n o t n n t i l o n d i t i o n o m o o t E
n Y t m i t l o j o i n m o o t l i t t n n t d i t i t t m u t u l
d i n t m d i t i o n u t d d i ' n t m n i t u d Y o t d i E
t i m i t n o t m t t l l D u t t o u l d t i l l o m m o n n o m l @ i F F t
u o u l d l o l l o l n A t t j o i n t F
R
r o n e f o m o o l n o m i l a0+ a1t+ a2t2 + a3t
3+ i a0+ t(a1+
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R R m t u n d u
Q @ A " d o d o m o n
n o l n o n o k o X o n
r m D n d o n n o n n n o n o o n n
v n n o l o n F i o n n d
o d j n D l d n n o o n
D d o d F
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u t R S
o n n F s n E l l o n o u2 n d
u3
n n n o l o o n d o
t2
n d
t3
n F
l o n o m d o o m n o o n l o n l l d
m o @ D D z A n d l o t @ D D z A D m o n 9 n n 9
n d n d o n 9 m o d l F m k n d l
o n o d l o j o n D l n n d o o n o F
l o t I T D I T D z I T Y
G B
B n t o n t t o u n t n n
B l o o D o o n 9 t n t l o t o E u t
B t m t Y n n o R t F
B G
l o t H D I D P D Q D H D I D P D Q D
z H D z I D z P D z Q Y
l o t t D u D t D D D z Y
n t D n Y
H a Q C u B @ P C u B @ I C u B H A A Y
I a U C u B @ T C u B @ S C u B R A A Y
P a I I C u B @ I H C u B @ W C u B V A A Y
Q a I S C u B @ I R C u B @ I Q C u B I P A A Y
H a Q C u B @ P C u B @ I C u B H A A Y
I a U C u B @ T C u B @ S C u B R A A Y
P a I I C u B @ I H C u B @ W C u B V A A Y
Q a I S C u B @ I R C u B @ I Q C u B I P A A Y
z H a z Q C u B @ z P C u B @ z I C u B z H A A Y
z I a z U C u B @ z T C u B @ z S C u B z R A A Y
z P a z I I C u B @ z I H C u B @ z W C u B z V A A Y
z Q a z I S C u B @ z I R C u B @ z I Q C u B z I P A A Y
t(a2+ t(a3+ )))F s t i t m t i n t o l n o m i l i i n l u t d F
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R T m t u n d u
t a H F H Y
t a I F H G @ l o t A @ n E I A Y
m o @ Q D Q D z Q A Y
o @ a I Y ` n Y C C A G B n t t o l o o n E I t m B G
{
t a t C t Y
a Q C t B @ P C t B @ I C t B H A A Y
a Q C t B @ P C t B @ I C t B H A A Y
z a z Q C t B @ z P C t B @ z I C t B z H A A Y
l o t @ D D z A Y
}
Splines
s n o l o m k l o n o o m l d
n l E d o l n o m l F y n o l m o m E
z o n Y o o l o n m l d o n m k
m o n d m o l d o n F
e l o D E d o l n o m l @ o l l o m E
n d T n d I H A n o o m "
o n m o m " n d m l n o n n
o 0 n F @ m o o n n F A
o o o l o n @ l l D l o o o l o n A o k n o l o
o m l o o o D n d l o o n
n d o m F c s D o m
n k D o m k n o n D o n d o n
o n o m l n d n d d l l d ( n d Y E
l l n o F w o n l o m d ( d o
n D l l o o n Y n d o m n D l l
n n o o n n D ( d o o l
o F o n d o n n l o l o l d
l o l o n n D n d o l l o o m l n o m d ( n d
F n o m n m z o m m z n l o o m
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l n R U
n o o D o 9 l l ( d
o m l n d n 9 o n o n F
o d l n o l l n d E
n D n d 9 l l l l n n o X
Sequential splining
m 9 j m d X o d
m l o o n o n o n d F
o n o o " l l l
n d m o m o n " n d n j o n o F
l l n d l n o d o d o m o m
o n n m o d 9 l d n l D n d o 9 d
o m m o n n d D o n o m o l l
d m n d D n d l l o l o o l l l d l
o l m n o o m o n l o o m F
n E d o o o d n n l o n
D 9 m l Y n m n m n o D o n D
o o d ( n d n d E o n d o n D n d n d r m n E
o l o n o m l o F p o D m o d o
o n o n o m m o n o n Y d d n n m o o m l E
d D n m o o n d o n o m o n m
o d d l d n d
@ s l l o n Q @ A A F D k n o n o d n o
d n ( n o d n D n d o
m o d d 0 l o o m F
Local splining
o o d o d E d n d n D o l d l k o d m n l l n
o o m m F y n o d o n o
k n o o n n m o d o n n " n o
o n o d j o n n " l n o o F f E l n
l l m n n d o l D l o n
o f E l n D j n D o o l l m
d n l o l n d n l l n n o l o n F f
n l f E l n D n
m l d o m o m l D n l X l l D 9
l l k F
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R V m t u n d u
Q @ A " n o n n l d D n o
o l o o d j o n o o n @ A n d
o @ A d F m o @ A n d
o @ A d n o n d n d n F
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o o n t n u t R W
s o n n @ o n l l A m l l o l l n X o
y @ o o ' o e o m m 9 q o m E
y A F d o ( o m o d l l
m l m n F s n ' n n n n o E
d d o m o o n d l d F o d n d n
o n d D n n n d o o n n o n E n " o l d
j l o o d ' n m n d D o m l F y n
n n n d d d ( n d D n n
n o l d l F
Global splining
l w g o F x o o d m n o l
S
n
n l o F n n D o o m m d
k n o n n D d o n o
D n d l o o n l o l d D l d n
n d o n d o n o m n n o n d F
n ( o o l o o n o 9
Y o m z n l l o m z d o o l
D n o j o m m n F d d n n
o d d ( n n l o l l n o l " l o o m
n " l o n o l D o o l n m o E
m n o n o n l l l k l F l o
l o m l n o o n o o l D o m
m o d d Y m o n l d o n l z d n d
l l o d l @ o o k x m l A F
Types of continuity
e k n o o l n @ o " o n "
j o n d A d o n o l l m o m l l n o n E z o d E
n o F g n D o n n D
n o n E z o d d Y o m m o n l o n l o o n n d (
d " n d o l o n d o o " n o d o k n o F
d o o n n o m m o n l n C0 @ o o n A D
S
u n o t l o l l l i n d D u t o m t n i u d o i t o l i n i n
m o u D t k n o t i n t t o t o i n i d n t n d t n n t
l n F
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S H m t u n d u
C1 D C2 F
o l d n m n d l n o n l l o n d
l @ m n d X y = f(x) A n d m z o n " n d o o m ) l " l l F D
o n n n m d n o m o n n n
o m n Y n o m z o n o E
l m n l o n F p E n d o n d E d o m
o n n o n n o n n n o D n d o n n o
F o l d m m o n n
d E l n m z o n Y n n d o D 9
n o n n @ p o k n d k k l 9 o n m E
l n m d y o n o d o D l o n l
o l n o m l o n o l n A F
n n n l n d o n X
dQdtdQdt .
o m n o l l o n o E
m l D n o l o o d n ( n d o n d
d X
=
dQdt d2
Qdt2 .dQdt
3 o n d o m m o o m m o n l d
o m n n m n D l o l m o
o n n d " n d n E d " n o l d o
n F s n d m n o n D m n o o o n n d
( d n o m d ' n F o n d d m n o
m d n d o n l l m n d F s n D ( D
o n d n d d d o n d m n o n o m
l o l o o d n l l d p n m D n
o n o n l o n F e l l D n o
o n o o o n X p n m o
o n d 9 F l n n m d o m n o l l
o ' n o o n o o m n o n F
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