γλώσσες
Σελίδες
Νομικός
Preface
ElementaryLinear
Algebra
Introduction toGeometricComputing
Data Structuresand Algorithms
ObjectOrientation
(C++)
ComputerGraphics
and
Visualization
ComputationalGeometry Robotics Computer
Vision
Computer-AidedDesign
GeographicInformation
Systems
Classical Geometry Computing
Geometric Computing
Chapter 1
P (x, y)
x
y
x
y
Point E2d P[5];
P[0]. . . P[4]
A
B
A
B
P2
x
y
P1
+?
P3
x
yP1
−→v1
P2
−→v2
−→v4
−→v3
Tuple E2d
x,y: double
Point E2d Vector E2d
flawed design
−→v1−→v2−→d1
−→d2
P
Q
R
O
P
Q
R
O
−c/a
−c/b
−→v1
−→v2
−→v2 −−→v1
(0, 0)(1, 0)
(2, 0)(0, 1)
(1, 1)(2, 1)
−→v1
−→v2
−→v1−→v2
−→v1
−→v2
−→v1−→v2 −→v1
−→v2−→v1
−→v2
−→v1
−→v2
−→v1
−→v2
|M | = 1 |M | = 1|M | = 1|M | = 1|M | = 1|M | = −1|M | = −1|M | = −1
Chapter 2
−ve
+ve +ve
−ve
0 0
left halfspace
right halfspace
+−
−→v = −→v1 ×−→v2
x
y
z
x
y
z
x
y
z
x
y
z
x
y
z
x
y
z
−→v−→v1
−→v2
−→v−→v1
−→v2
−→v
−→v1
−→v2−→v
−→v1
−→v2−→v−→v1
−→v2
−→v
−→v1−→v2
P1 P1
P2 P2
P3P3
left turn colinear right turnP1
P2
P3
x
y
z−→v3
−→v1
−→v2
p1p2
p3
q
insidecircle
oncircle
boundary
outsidecircle
p1p2
p3
qp1
p2
p3
q
L
X+
Y+
Y−
X−
P1P2 P3
P+1P−1
Chapter 3
Point E2d
x,y: double
Point E3d
z: double
Point E2d
#x,y: double
Point E3d
z: double
xy
z
px2 + y2
px2 + y2 + z2
P1
P3
P4
P2
P1
P3
P4
P2
−→N
−−−→P1P2
−−−→P2P3
−−−→P1P4
Chapter 4
θ
φ
P ′(x′, y′)
P (x, y)
x
y
−→v
T −→v′ = T (−→v ) = −→v
T
PT P ′ = T (P )
x
y
ymin
ymax
y′min
y′max
xm
in
xm
ax
x′ m
in
x′ m
ax
E
−→d
x
y
z
−→v ?
E
−→d
x
y
z
−→v−→u−→s
x
y
z
x
y
z
x
y
z
x
y
z
xz-dimetric yz-dimetric xy-dimetricisometric
x
y
z
x
y
z
x
y
z
x
y
z
x
y
z
RyRx 6= RxRy
E1
A B
A′ B′
A
B
C
A′ B′
C′
M =?E2
Chapter 5
nointersection
singlepoint
segment
O
P1
P2
splittingline
clippingregion
EN
WS
ENWS
L L LL
E
N
W
S
LLLL
LRLL
LLLR
RLLL
RRLLLRRL
LLRL
LLRR RLLR
East-HalfspaceClipping
North-HalfspaceClipping
West-HalfspaceClipping
South-HalfspaceClipping
InputPolygon
OutputPolygon
Chapter 6
Point E2
�B
ind�
doub
le
Point E2 double Point E2 float
NT
x,y: NT
�B
ind�
float
+×
3 42
P1
P2 P3
P4
A
B
C
D
E
GH
F
Chapter 7
A
B
PQ
R
(−5,−5), (0, 0) (−5,−5), (−2,−2) (−6,−6), (−5,−5) (−3,−3), (−2,−2)
0−ε ε︸︷︷︸treat as 0
> 0< 0
Chapter 8
P−→d
P
Q
R
S
T
P+
P−
antipode of P
P ′
AB
C
A ≤ B
B ≤ CC ≤ A
P
Ssource
Starget
s2s1
s2s1s2
s1
s2
s1
Chapter 9
E 2 S 2
P2 T 2
O
−→d
−−→d
−→N
O
−→Y
−→X
−→Z
O
−→c
−→B1
−→B2
O
P
Q
−→N
−→N
O
P
Q
−→M
O
P
Q
(P, Q,−→M ) (P, Q,
−→N )
P1
P2
P3
−→c1
−→c2
−→c1 ∩ −→c2
−→c2 ∩ −→c1
Chapter 10
Chapter 11
l
m
E
A
A′
B
B′
C
C′
AB C
B′ = f(B)
A′ = f(A) C ′ = f(C)m∞
E
l
f(l∞)f−1(m∞)m∞
l∞m
l∞
A
B
C
A′
B′
C′B′′
C′′
AB
C
A′ = B′ = C′
B︸︷︷︸ ︸︷︷︸CA
lm
A
B
C
A′
B′
C ′
E
l
m
A B C
A′B′
C′
E
A
B
A′
B′
image plane Eye
object plane
l∞
P (l∞)
l
l′ = P (l)
l∞
l1l2
horizon line
ideal pointsP (S∞1 )
S∞1
S1
S2 S3
horizon lineide
alpo
int
l
m lm l
m
A
B
C
C
B
A
AB
C
A′
B′
C′
1-point
y
x
z
2-point
3-point
(b)(a)
AA′
BB′
C
C ′
O
A
A′
C ′
BC
B′
Chapter 12
x
w
idea
lpoi
nt
P1
x
w
canonical representatives
x
w
x
w
A
BR
x
yww
=1
x
yw
24 x1
y1
w1
3524 x2
y2
w2
35
w
l
x
y
w=
1 −→N
w
x
yw
=1
x
yw
A
Bl
−→A
−→B
l−→B
l−→A
x
y
(0, y, 0)
(x, 0, 0)(0, 0, w)
w = 0
x y z w
0 xy xz xw
−xy 00
0−xz
−xw
−yz
x
y
z
w
yz yw
zw
−yw −zw
x
y
z (x, 0, 0, 0)
(0, y, 0, 0)
(0, 0, z, 0)
(0, 0, 0, w)
w = 0
(wx, w/k)
(wx, w)(x, 1)(kwx, w)
(kx, 1)
(wx, w)(x, 1)(wx + wt, w)
(x + t, 1)
(x, w)
(x, px + w)
slope = p
slope = −p
E3 P 3 P 3 P 3 P 3 E3 E2M V P D+1 \z
x
y
z
(x, y, z)
(x′, y′,−N)−
N
x
y
z
2664−1−111
37752664
11−11
3775
LB−N1
R
T−N1
x
y
z −N−F
x
z
x
z
x
z
x
z
x
z
x
z
y
x
z
A
B
A′
B′
B′′
E3 P 3 P 3 P 3 P 3 E3 E2M V P D+1 \z
E3 P 3 P 3 P 3 P 3 E3 E2M V P D clip \zE3+1
E3 P 3 P 3 E3 E2M P D \z+1E3 E3Vclip
E3
E3 P 3 P 3 E3 E2M V P Dclip \zE3 +1
E3 E3
AB C
A′
B′ C′
O
Chapter 13
AB
a
b
ABab
AB
ab
AB P
ba
a + b
︸︷︷︸︸︷︷︸a b:
A
B
P (a, b)
b
a
A′
B′
AB ab
Q
A
B
C
P
c
a
b
l
A
B
C
P
c
a
b
A
B
C
P
Q
A
B C
A
B
C
Chapter 14
D
A
C
C′
D′
E
E
C′
E′
A′
B
view
input
A
C
A′
C′
D
E
C′
E′
D′
A
C
A′
C′
D
E
E
C′
E′
D′
x
w
AB
C
D
EF
+∞
+∞
x
w
x
w
lim
w→
0+(−
1,w
)
lim
w→
0−
(−1,w
)lim
w→
0+(1
,w)
lim
w→
0−
(1,w
)
x
y
w
x
y
w
+ve+ve
N
Chapter 15
x
y
z
A
B
B′
C
x
z
A′ C′
w
x
(a)
P2P1
(b) (d)(c)
w
x
w
x
w
x
w
x
(a) (b) (d)
w
x
w
x
w
x
(c)
(a) (b) (d)(c)
m
n
l
m
l
n
l
m
n
l
w
x
P0
P1
w
x
P0
P1
w
x
P0
P1
P0P1
P0P1
P0P1
E3 T 3 E3 E2V P D \z+1T 3 T 3M
T 3
E3 T 3 E3 E2V P Dclip \z+1T 3T 3 T 3M
T 3
E3 T 3 E3 E2V P Dclip \z+1T 3T 3 T 3M
T 3
E3 T 3 E3 E2V P Dclip \z+1T 3 T 3 T 3M
T 3
x
z
x
z
x
z
x
z
x
z
x
z
S 2 P2 T 2S 2 E 2 S 2
Chapter 16
Chapter 17
Geometry Module 1 Geometry Module 2
Test 1 Test 2
Client code (actual system)Input Module Output Module
Chapter 18
Chapter 19
Chapter 20
1
32
2?
1 + 1 = 2 =⇒ outside?
A
BC
D
E
F
≤<
ignored
Chapter 21
−→E
−→N −→
Lφ
−→E
−→N
ψ
−→H −→
L
−→N
−→L
−→N
−→N
ambient+
diffuse (per light)
−→L
−→L
specular (per light) a + d + s
scanline
A
BC
D
pixels
A
BC
D
Chapter 22
kernel
d 1
d 2
35
2
4
1
4
41
L
L
3 2 5
3 2
L 2
3
35
2
4
1
4
41
32 5
2
1
5
E3 T 3 E3 E2V P Dclip \z+1T 3 T 3 T 3M
T 3
E3 T 3 E3 E2(N)V P Dclip Vis+1T 3 T 3 T 3M
T 3
E3 T 3 E3 E2(N)V P Dclip Vis+1T 3 T 3 T 3M
T 3 ShadeT 3
Chapter 23
d1
d2
d3
−→u
−→v −−→n
−→E
−→N
−→R
θ θ
−→E
−→N
−→R
θ
ξ
Chapter 24
Nk −Rk
< 0
R
> 0
LL
LR RR
RL
N
Chapter 25
Chapter 26
z
y
x
000
100
001 011
010
111
110
101
V V EV FV
V E EE FE
EF FFV F
x
y
z
x
y
z
x
y
z
1
1
t
V
E
F
1
0 1
211
000
21
3 3
23
Tetrahedron Cube Octahedron Icosahedron Dodecahedron
Tetrahedron CubeOctahedron IcosahedronDodecahedron
{4, 4} {3, 6} {6, 3}
. . .
Chapter 27
outgoing edge
sourcetarget
succpred
twin
faceedge F
F
EV
V E EE FE
EF
TF
TF
TF
TFTF
T
F
T
FTF
lexicographicorder
std::pairorder
Ffir
st
second second
first
T
kvfs()
mvfs() v1 S
f
kef(e1)
mef(e2,e3)mev(e1)
kev(e4)v1
v2
e1
e3
e2
v3
e4
v1
v2
e1
e3
e2
v1
v2
e3
e2
mev mev mev mef
mvfs
mev
mevmevmevmef
mef
mef mef
mev
3 × mev mef kemr
4 × mefkfmrh
4 × mev
Chapter 28
−3 0 2 5
−3
0
25
a
c
b
a
b
c
−3 0 2 5
−3
0
25
A− +
− + − +
− +
− +
− +
A B A B
B
A
B
C
D
DC
A
B
B
A
A B
− + − +
− +− +
− +
− +
− +
− +
B
B
C
D
DC
C
C D
C D
L R S
L RS
L RS
L
R
L R
1:
2:
splitno splitting
L R
S
L R
S
1s:
2s:
L R
boundary/splitter coincidence
AB B
A
− +
− +
B+
B−A+
B−A−
B−A+B−A−
B+
P
P+ = φP− = S
−
+P −
+P
split
−
+P −
+P
P+ = SP− = φ
−
+P −
+P −
+P −
+
split
st s
ts
t s t s t s
tstst
case b case c1 case c2 case c3 case d case a1case a3case a2
ABB
A
B
A C
− +
− +
− +
− + − +
l
m
n l
AB
C
l
m
n
mn
p
p
p
lp nm
AB ∪BC
B
C
C
B
− +
− +
l
m
n m
AB
C
l
m
n
np
p
l
pC
B
− +
− +
m
np
l
BC ∪AB
A
B B− +
E2
+ +
+
E3
+ +
+
+
+−
+−
+
−
+
−
+
−+
−
M
N
M
N
L+−
+−
+−
L
+
−
(a)
(b)
(c)
(d)
(e)
1
+
−+
−
+
−+
−
+
−+
−
+
−+
−
+
−+
−
(f)
(g)
(h)
(i)
(j)
I
I
+
−+
−
+
−+
−
+
−+
−
+
−+
−
+
−+
−
BC
AB
CA
EF
F D
BC
AB
CA
A
B C
D
F
E
A
BC
Chapter 29
G: GeometryCA: Combinatorial Algorithm
E1 CA S1 CA E2 CA S2 CA
�B
ind�
E1
�B
ind�
S1
�B
ind�
E2
�B
ind�
S2
CA G
B C
D
EF
A
CA
BC
+
+
+
−−
−
EF
FD
DE
+
+
+
−−
−
AB
D
F
B C
A CA
BC
+
+
+
−−
−
EF
FD
DE
+
+
+
−−
−
AB
B C
D
EF
ACA
BC
+
+
+
−−
−
EF
DE
+
+
−−
AB
BE
DA
C
F
CA
BC
+
+
+
−−
−
EF
DE
+
+
−−
AB
source
target
A
B
A
B
Algorithm
Geo
met
ry
1 2 3 4 5ABCD
?
?
?
?
?
Chapter 30
Node
Primitive node Transformation node Boolean operation node
Cylinder ConeCube
Leaf node Interior node
2
1
PrimitiveNode Transformation node Boolean operation node
Cylinder ConeCube
Node2
1
Component
Leaf1 Leaf2 Composite
⋃
Cube
T
Cube
T
empty(outside)
partial(boundary)
full(inside)
edge vertex
Chapter 31
h
hc
ggc
d(gc) < d(hc)
E
E′
Chapter 32
P
Q−→N
O
A
B a
b P
Appendix A
letter
210 mm8.5”
A4
297m
m11”
Line Join 0 Line Join 1 Line Join 2
Lin
eC
ap0
Lin
eC
ap1
Lin
eC
ap2
Cogito ergo sum
Cogito ergo sum
Appendix B
colorbuffer
rasterdisplaydevice red
bluegreen
redblue
greenred
green
blue
alpha
GL LINES GL LINE STRIP GL LINE LOOP
GL TRIANGLES GL TRIANGLE STRIP GL TRIANGLE FAN
GL QUAD STRIPGL QUADS GL POLYGON
GL POINTS
frontbuffer
backbuffer
displaydevice
glutSwapBuffers()
colorbuffer
VertexTransformation Rasterization
glBegin(..);...glEnd();
glP
ush
Matr
ix()
R1∼ =
glR
ota
ted(a
ngle
1,0,1
,0)
glP
ush
Matr
ix()
glP
ush
Matr
ix()
S1∼ =
glS
cale
d(5
,0.5
,0.5
)
T1∼ =
glT
ranslate
d(0
.5,0
,0)
glu
tSolid
Cube(
1.0
)
glP
opM
atr
ix()
T2∼ =
glT
ranslate
d(5
,0,0
)
M
MR1
MR1
MR1
M M
M
M
MR1
M
MR1
MR1
M
MR1
MR1
MR1S1
M
MR1
MR1
MR1S1T1
M
MR1
MR1
M
MR1
MR1T2
Appendix C
GlowComponent
GlowSubwindow
GlowWindow
HelloSegmentWindow
GlowWindow
HelloSegmentWindow
GlowCheckBoxReceiver
GlowPushButtonReceiver
GlowIdleReceiver
Scene
GlowComponentGlowWindow
GlowViewManipulator
Class diagram Object diagram
myArcballWindow
myManipulator
mySceneArcballWindow
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