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3D Geometry for Computer Graphics

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Title: 3D Geometry for Computer Graphics


1
3D Geometry forComputer Graphics
  • Class 1

2
General
  • Office hour Sunday 1100 1200
    in Schreiber 002 (contact in advance)
  • Webpage with the slides http//www.cs.tau.ac.il/
    sorkine/courses/cg/cg2005/
  • E-mail sorkine_at_tau.ac.il

3
The plan today
  • Basic linear algebra and
  • Analytical geometry

4
Why??
5
Why??
  • We represent objects using mainly linear
    primitives
  • points
  • lines, segments
  • planes, polygons
  • Need to know how to compute distances,
    transformations, projections

6
Basic definitions
  • Points specify location in space (or in the
    plane).
  • Vectors have magnitude and direction (like
    velocity).

Points ? Vectors
7
Point vector point
8
vector vector vector
Parallelogram rule
9
point - point vector
B A
B
A
A B
B
A
10
point point not defined!!
11
Map points to vectors
  • If we have a coordinate system with
  • origin at point O
  • We can define correspondence between points and
    vectors

12
Inner (dot) product
  • Defined for vectors

w
?
v
L
Projection of w onto v
13
Dot product in coordinates (2D)
y
yw
w
yv
v
xv
xw
x
O
14
Perpendicular vectors (2D)
v?
v
15
Parametric equation of a line
v
t gt 0
p0
t 0
t lt 0
16
Parametric equation of a ray
v
t gt 0
p0
t 0
17
Distance between two points
y
A
yA
B
yB
xB
xA
x
O
18
Distance between point and line
  • Find a point q such that (q ? q)?v
  • dist(q, l) q ? q

l
19
Easy geometric interpretation
q
l
v
q
p0
L
20
Distance between point and line also works in
3D!
  • The parametric representation of the line is
    coordinates-independent
  • v and p0 and the checked point q can be in 2D or
    in 3D or in any dimension

21
Implicit equation of a line in 2D
y
AxByC gt 0
AxByC 0
AxByC lt 0
x
22
Line-segment intersection
Q1 (x1, y1)
y
AxByC gt 0
Q2 (x2, y2)
AxByC lt 0
x
23
Representation of a plane in 3D space
  • A plane ? is defined by a normal n and one point
    in the plane p0.
  • A point q belongs to the plane ? lt q p0 , n gt
    0
  • The normal n is perpendicular to all vectors in
    the plane

n
q
p0
?
24
Distance between point and plane
  • Project the point onto the plane in the direction
    of the normal
  • dist(q, ?) q q

n
q
q
p0
?
25
Distance between point and plane
n
q
q
p0
?
26
Implicit representation of planes in 3D
  • (x, y, z) are coordinates of a point on the plane
  • (A, B, C) are the coordinates of a normal vector
    to the plane

AxByCzD gt 0
AxByCzD 0
AxByCzD lt 0
27
Distance between two lines in 3D
q1
l1
p1
u
d
p2
v
l2
q2
The distance is attained between two points q1
and q2 so that (q1 q2) ? u and (q1 q2) ? v
28
Distance between two lines in 3D
q1
l1
p1
u
d
p2
v
l2
q2
29
Distance between two lines in 3D
q1
l1
p1
u
d
p2
v
l2
q2
30
Distance between two lines in 3D
q1
l1
p1
u
d
p2
v
l2
q2
31
Barycentric coordinates (2D)
  • Define a points position relatively to some
    fixed points.
  • P ?A ?B ?C, where A, B, C are not on one
    line, and ?, ?, ? ? R.
  • (?, ?, ?) are called Barycentric coordinates of P
    with respect to A, B, C (unique!)
  • If P is inside the triangle, then ???1, ?, ?,
    ? gt 0

C
P
A
B
32
Barycentric coordinates (2D)
C
P
A
B
33
Example of usage warping
34
Example of usage warping
C
Tagret
B
A
We take the barycentric coordinates ?, ?, ? of
P with respect to A, B, C. Color(P) Color(?
A ? B ? C)
35
See you next time!
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