3D Geometry forComputer Graphics

Class 1

General

Office hour: Sunday 11:00 – 12:00 in Schreiber 002 (contact in advance)

Webpage with the slides: http://www.cs.tau.ac.il/~sorkine/courses/cg/cg2005/

E-mail: sorkine@tau.ac.il

The plan today

Basic linear algebra and Analytical geometry

Why??

Why??

We represent objects using mainly linear primitives: points lines, segments planes, polygons

Need to know how to compute distances, transformations, projections…

Basic definitions

Points specify location in space (or in the plane). Vectors have magnitude and direction (like

velocity).

Points Vectors

Point + vector = point

vector + vector = vector

Parallelogram rule

point - point = vector

A

BB – A

A

BA – B

point + point: not defined!!

Map points to vectors

If we have a coordinate system with

origin at point O We can define correspondence between points

and vectors:

p p p O

v O + v

Inner (dot) product

Defined for vectors:

, || || || || cosθ v w v w

L v

wL

cosθ || ||

,

|| ||L

w

v w

v

Projection of w onto v

Dot product in coordinates (2D)

v

w

xv

yv

xw

yw

x

y

O

( , )

( , )

,

v v

w w

v w v w

x y

x y

x x y y

v

w

v w

Perpendicular vectors (2D)

v

v

, 0

( , ) ( , )v v v vx y y x

v w

v v

Parametric equation of a line

p0

vt > 0

t < 0 t = 0

( ) , ( , )t t t 0p v

Parametric equation of a ray

p0

vt > 0

t = 0

( ) , (0, )t t t 0p v

Distance between two points

B

A

xB

yB

xA

yA

x

y

O

22 )()(

,

||||),(dist

ABAB yyxx

ABAB

ABBA

2

22 2 2

2

, 0

( ), 0

, , 0

, ,

, || ||

,dist ( , ) || || || ||

|| ||

t

t

t

l

0

0

0 0

00

q q v

q p v v

q p v v v

q p v q p v

v v v

q p vq q q q p

v

Distance between point and line

Find a point q’ such that (q q’)vdist(q, l) = || q q’ ||

p0

v

q

q’ = p0 +tv

l

Easy geometric interpretation

p0

v

q

q’2 2 2

2 2 2

22

2

:

(1) dist( , ) || ||

,(2)

|| ||

dist( , ) || ||

,|| || .

|| ||

L

L

L

0

0

0

00

Pythagoras

q q q p

q p v

v

q q q p

q p vq p

v

l

L

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…

Implicit equation of a line in 2D

0, , , R, ( , ) (0,0)Ax By C A B C A B

x

yAx+By+C > 0

Ax+By+C < 0

Ax+By+C = 0

Line-segment intersection

1 2

1 1 2 2

The segment Q Q intersects the line

( )( ) 0Ax By C Ax By C

x

yAx+By+C > 0

Ax+By+C < 0

Q1 (x1, y1)

Q2 (x2, y2)

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 < q – p0 , n > = 0

The normal n is perpendicular to all vectors in the plane

n

p0

q

Distance between point and plane

Project the point onto the plane in the direction of the normal:

dist(q, ) = ||q’ – q||

n

p0q’

q

Distance between point and plane

n

p0q’

q

2

22 2 2 2

2

( ) || , R

, 0 (because is in the plane )

, 0

, , 0

,

|| ||

,dist ( , ) || || || || .

|| ||

0

0

0

0

0

q q n q q n q q n

q p n q

q n p n

q p n n n

p q n

n

q p nq q q n

n

Implicit representation of planes in 3D

0, , , , R, ( , , ) (0,0,0)Ax By Cz D A B C D A B C

(x, y, z) are coordinates of a point on the plane (A, B, C) are the coordinates of a normal vector to the

plane

Ax+By+Cz+D > 0

Ax+By+Cz+D < 0

Ax+By+Cz+D = 0

Distance between two lines in 3D

p1

p2

u

v

d

The distance is attained between two points q1 and q2

so that (q1 – q2) u and (q1 – q2) v

q1

q2

l1

l2

1

2 2

( )

( )

l s s

l t t

1p u

p v

Distance between two lines in 3D

( ) , 0

( ) , 0

s t

s t

1 2

1 2

p p u v u

p p u v v

2

2

( ), || || , 0

( ), , || || 0

s t

s t

1 2

1 2

p p u u v u

p p v u v v

p1

p2

u

v

d

q1

q2

l1

l2

1

2 2

( )

( )

l s s

l t t

1p u

p v

Distance between two lines in 3D

2

2 2 2

2

2 2 2

, , || || ,

|| || || || ,

|| || , , ,

|| || || || ,

s

t

1 2 1 2

1 2 1 2

u v v p p v u p p

u v u v

u u p p u v u p p

u v u v

p1

p2

u

v

d

q1

q2

l1

l2

1

2 2

( )

( )

l s s

l t t

1p u

p v

Distance between two lines in 3D

||)~()~(||),(dist 2121 tlslll

p1

p2

u

v

d

q1

q2

l1

l2

1

2 2

( )

( )

l s s

l t t

1p u

p v

Barycentric coordinates (2D)

Define a point’s 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, , , > 0

A B

C

P

Barycentric coordinates (2D)

A B

C

P

triangletheofareathedenotes,,

,,

,,

,,

,,

,,

,,

CCBA

BAPB

CBA

ACPA

CBA

CBPP

Example of usage: warping

Example of usage: warping

Tagret

AB

C

We take the barycentric coordinates , , of P’ with respect to A’, B’, C’.Color(P) = Color( A + B + C)

PP

1P

See you next time!