geometry - - texas a&m university corpus christi slides modified from angel book 6e and from...

GEOMETRY Slides modified from Angel book 6e and from

Anthony Steed and Pam Perlich and cs4160

(Columbia U)

2 COSC4328/5327 Computer Graphics

Objectives

• Introduce the elements of geometry

• Scalars

• Vectors

• Points

• Develop mathematical operations among them in a

coordinate-free manner

• Define basic primitives

• Line segments

• Polygons


Basic Elements

• Geometry is the study of the relationships among

objects in an n-dimensional space • In computer graphics, we are interested in objects that exist in

three dimensions

• Want a minimum set of primitives from which we

can build more sophisticated objects

• We will need three basic elements • Scalars

• Vectors

• Points


Coordinate-Free Geometry

• When we learned simple geometry, most of us started with a Cartesian approach

• Points were at locations in space p=(x,y,z)

• We derived results by algebraic manipulations involving these coordinates

• This approach was nonphysical

• Physically, points exist regardless of the location of an arbitrary coordinate system

• Most geometric results are independent of the coordinate system

• Example Euclidean geometry: two triangles are identical if two corresponding sides and the angle between them are identical


Scalars • Need three basic elements in geometry

• Scalars, Vectors, Points

• Scalars can be defined as members of sets which can be combined by two operations (addition and multiplication) obeying some fundamental axioms (associativity, commutivity, inverses)

• Examples include the real and complex number systems under the ordinary rules with which we are familiar

• Scalars alone have no geometric properties


Vectors

• Physical definition: a vector is a quantity with two attributes • Direction

• Magnitude

• Examples include • Force

• Velocity

• Directed line segments

• Most important example for graphics

• Can map to other types

v


Vector Operations

• Every vector has an inverse • Same magnitude but points in opposite direction

• Every vector can be multiplied by a scalar

• There is a zero vector • Zero magnitude, undefined orientation

• The sum of any two vectors is a vector • Use head-to-tail axiom

v -v v v

u

w

Vectors

Length and direction. Absolute position not important

Use to store offsets, displacements, locations But strictly speaking, positions are not vectors and cannot be added: a

location implicitly involves an origin, while an offset does not.

=

Usually written as or in bold. Magnitude written as a a

9

Vectors, V (x, y, z)

v

w

v + w

Vector addition

sum v + w

v 2v

(-1)v (1/2)V

Scalar multiplication of

vectors (they remain parallel)

v

w

Vector difference

v - w = v + (-w)

v

-w

v - w

x

y

Vector OP

P

O

10

Vectors V

n Length (modulus) of a vector V (x, y,

z)

|V| =

n A unit vector: a vector can be

normalised such that it retains its

direction, but is scaled to have unit

length:

||V of modulus

Vvector ^

V

VV

11

Dot Product

u · v = xu ·xv + yu ·yv + zu ·zv

u · v = |u| |v| cos

cos = u · v/ |u| |v|

n This is purely a scalar number not a vector.

n What happens when the vectors are unit

n What does it mean if dot product == 0 or == 1?

Dot (scalar) product

a

b

?a b b a

1

cos

cos

a b a b

a b

a b

( )

( ) ( ) ( )

a b c a b a c

ka b a kb k a b

Dot product: some applications in CG

Find angle between two vectors (e.g. cosine of angle between

light source and surface for shading)

Finding projection of one vector on another (e.g. coordinates of

point in arbitrary coordinate system)

Advantage: can be computed easily in cartesian components

Projections (of b on a)

a

b

?

?

b a

b a

cosa b

b a ba

2

a a bb a b a a

a a

Dot product in Cartesian components

?a b

a b

x xa b

y y

a b

a b a b

a b

x xa b x x y y

y y

16

Cross Product

n The result is not a scalar but a vector which is

normal to the plane of the other 2

n Can be computed using the determinant of:

u x v = i(yvzu -zvyu), -j(xvzu - zvxu), k(xvyu - yvxu)

n Size is u x v = |u||v|sin

n Cross product of vector with it self is null

uuu

vvv

zyx

zyx

kji

uuu

vvv

zyx

zyx

kji

uuu

vvv

zyx

zyx

kji

uuu

vvv

zyx

zyx

kji

Cross (vector) product

Cross product orthogonal to two initial vectors

Direction determined by right-hand rule

Useful in constructing coordinate systems (later)

a

b

a b b a

sina b a b

Cross product: Properties

0

( )

( ) ( )

a b b a

a a

a b c a b a c

a kb k a b

x y z

y x z

y z x

z y x

z x y

x z y

Cross product: Cartesian formula?

a b b a

a a a a b a b

b b b a b a b

x y z y z y z

a b x y z z x x z

x y z x y y x

*

0

0

0

a a b

a a b

a a b

z y x

a b A b z x y

y x z

Dual matrix of vector a


Linear Vector Spaces

• Mathematical system for manipulating vectors

• Operations

• Scalar-vector multiplication u=v

• Vector-vector addition: w=u+v

• Expressions such as

v=u+2w-3r

Make sense in a vector space


Vectors Lack Position

• These vectors are identical • Same length and magnitude

• Vectors spaces insufficient for geometry • Need points


Points

• Location in space

• Operations allowed between points and vectors

• Point-point subtraction yields a vector

• Equivalent to point-vector addition

P=v+Q

v=P-Q

23

Vectors, V (x, y, z)

n Represent a direction (and magnitude) in 3D space

n Points != Vectors Vector +Vector = Vector

Point – Point = Vector

Point + Vector = Point

Point + Point = ?


Affine Spaces

• Point + a vector space

• Operations

• Vector-vector addition

• Scalar-vector multiplication

• Point-vector addition

• Scalar-scalar operations

• For any point define • 1 • P = P

• 0 • P = 0 (zero vector)


Lines

• Consider all points of the form

• P()=P0 + d

• Set of all points that pass through P0 in the direction of the vector d


Parametric Form

• This form is known as the parametric form of the line • More robust and general than other forms

• Extends to curves and surfaces

• Two-dimensional forms • Explicit: y = mx +h

• Implicit: ax + by +c =0

• Parametric:

x() = x0 + (1-)x1

y() = y0 + (1-)y1


Rays and Line Segments

• If >= 0, then P() is the ray leaving P0 in the direction d

If we use two points to define v, then

P( ) = Q + (R-Q)=Q+v

=R + (1-)Q

For 0<=<=1 we get all the

points on the line segment

joining R and Q

28

Parametric equation of a line

(ray)

Given two points P0 = (x0, y0, z0) and

P1 = (x1, y1, z1) the line passing through

them can be expressed as:

P(t) = P0 + t(P1 - P0)

x(t) = x0 + t(x1 - x0)

y(t) = y0 + t(y1 - y0)

z(t) = z0 + t(z1 - z0)

=

With - < t <


Convexity

• An object is convex iff for any two points in the object all

points on the line segment between these points are also

in the object

P

Q Q

P

convex not convex


Affine Sums

• Consider the “sum”

P=1P1+2P2+…..+nPn

Can show by induction that this sum makes sense iff

1+2+…..n=1

in which case we have the affine sum of the points P1,P2,…..Pn

• If, in addition, i>=0, we have the convex hull of P1,P2,…..Pn


Convex Hull

• Smallest convex object containing P1,P2,…..Pn

• Formed by “shrink wrapping” points


Curves and Surfaces

• Curves are one parameter entities of the form P() where

the function is nonlinear

• Surfaces are formed from two-parameter functions P(, b)

• Linear functions give planes and polygons

P() P(, b)


Planes

• A plane can be defined by a point and two vectors or by

three points

P(,b)=R+u+bv P(,b)=R+(Q-R)+b(P-Q)

u

v

R

P

R

Q


Triangles

convex sum of P and Q

convex sum of S() and R

for 0<=,b<=1, we get all points in triangle

Barycentric Coordinates

Triangle is convex so any point inside can be represented as

an affine sum

P(1, 2, 3)=1P+2Q+3R

where

1 +2 +3 = 1

i>=0

The representation is called the barycentric coordinate

representation of P

35 COSC4328/5327 Computer Graphics 36 COSC4328/5327 Computer Graphics

Normals

• Every plane has a vector n normal

(perpendicular, orthogonal) to it

• From point-two vector form P(,b)=R+u+bv, we

know we can use the cross product to find n =

u v and the equivalent form

(P()-P) n=0

u

v

P

REPRESENTATION

COSC4328/5327 Computer Graphics 37 38 COSC4328/5327 Computer Graphics

Objectives

• Introduce concepts such as dimension and basis

• Introduce coordinate systems for representing vectors

spaces and frames for representing affine spaces

• Discuss change of frames and bases

• Introduce homogeneous coordinates

39

Vectors and Matrices

n Matrix is an array of numbers with

dimensions M (rows) by N (columns)

• 3 by 6 matrix

• element 2,3

is (3)

n Vector can be considered a 1 x M matrix

•

zyxv

100025114311212003

40

Types of Matrix

n Identity matrices - I

n Diagonal

1001

1000010000100001

n Symmetric

• Diagonal matrices

are (of course)

symmetric

• Identity matrices are

(of course) diagonal

4000010000200001

fecedbcba

41

Operation on Matrices

n Addition

• Done elementwise

n Transpose • “Flip” (M by N becomes N by M)

sdrc

qbpa

sr

qp

dc

ba

389

724

651

376

825

941T

42

Operations on Matrices

n Multiplication

• Only possible to multiply of dimensions

– x1 by y1 and x2 by y2 iff y1 = x2

• resulting matrix is x1 by y2

– e.g. Matrix A is 2 by 3 and Matrix by 3 by 4

• resulting matrix is 2 by 4

– Just because A x B is possible doesn’t mean

B x A is possible!

43

Matrix Multiplication Order

n A is n by k , B is k by m

n C = A x B defined by

n BxA not necessarily

equal to AxB

k

l

ljilij bac1

*

*

*

*

*

*

*****

*.

*.

*.

*.

*.

*.

*****

*.

..

*.

*.

*.

*.

*.

*****.....

44

Example Multiplications

____

____

1011

12

101132

______

______

______

010001100

111013322

Computation: A x B = C

A a11 a12

a21 a22

B b11 b12 b13

b21 b22 b23

232213212222122121221121

2312131122121211 21121111

babababababa

babababababaC

[2 x 2]

[2 x 3]

[2 x 3]


A

2 3

1 1

1 0

and B 1 1 1

1 0 2

[3 x 2] [2 x 3] A and B can be multiplied

1 1 1

3 1 2

8 2 5

12*01*1 10*01*1 11*01*1

32*11*1 10*11*1 21*11*1

82*31*2 20*31*2 51*31*2

C

[3 x 3]


1 1 1

3 1 2

8 2 5

12*01*1 10*01*1 11*01*1

32*11*1 10*11*1 21*11*1

82*31*2 20*31*2 51*31*2

C

A

2 3

1 1

1 0

and B 1 1 1

1 0 2

[3 x 2] [2 x 3]

[3 x 3]

Result is 3 x 3

What is a matrix

Array of numbers (m×n = m rows, n columns)

Addition, multiplication by a scalar simple: element

by element

1 3

5 2

0 4

Matrix-matrix multiplication

Number of columns in first must = rows in second

Element (i,j) in product is dot product of row i of first

matrix and column j of second matrix

1 33 6 9 4

5 22 7 8 3

0 4





9 21 3 7 3

5 2

0 4

1339

618

3

319

28 2

426

3

644

17

28 2





9 21 3 7 3

5 2

0 4

1339

618

3

319

28 2

426

3

644

17

28 2





9 21 3 7 3

5 2

0 4

1339

618

3

319

28 2

426

3

644

17

28 2



Non-commutative (AB and BA are different in general)

Associative and distributive

A(B+C) = AB + AC

(A+B)C = AC + BC

1 33 6 9 4

5 2 NOT EVEN LEGAL!!2 7 8 3

0 4

Matrix-Vector Multiplication

Key for transforming points (next lecture)

Treat vector as a column matrix (m×1)

E.g. 2D reflection about y-axis (from textbook)

1 0

0 1

x x

y y

55

Inverse

n If A x B = I and B x A = I then

A = B-1 and B = A-1

Matrix Inversion

B1

B BB1

I

Like a reciprocal

in scalar math Like the number one

in scalar math

Transpose of a Matrix (or vector?)

1 21 3 5

3 42 4 6

5 6

T

( )T T TAB B A

Identity Matrix and Inverses

1 1

1 1 1( )

AA A A I

AB B A

3 3

1 0 0

0 1 0

0 0 1

I

Vector multiplication in Matrix form

Dot product?

Cross product?

T

b

a a a b a b a b a b

b

a b a b

x

x y z y x x y y z z

z

*

0

0

0

a a b

a a b

a a b

z y x

a b A b z x y

y x z

Dual matrix of vector a


Linear Independence

• A set of vectors v1, v2, …, vn is linearly independent if

1v1+2v2+.. nvn=0 iff 1=2=…=0

• If a set of vectors is linearly independent, we cannot

represent one in terms of the others

• If a set of vectors is linearly dependent, as least one can

be written in terms of the others

• AB ≠ BA

• ABC ≠ CBA

• So order matters!

COSC4328/5327 Computer Graphics 61 62 COSC4328/5327 Computer Graphics

Dimension

• In a vector space, the maximum number of

linearly independent vectors is fixed and is called

the dimension of the space

• In an n-dimensional space, any set of n linearly

independent vectors form a basis for the space

• Given a basis v1, v2,…., vn, any vector v can be

written as

v=1v1+ 2v2 +….+nvn

where the {i} are unique


Representation

• Until now we have been able to work with geometric

entities without using any frame of reference, such as a

coordinate system

• Need a frame of reference to relate points and objects to

our physical world.

• For example, where is a point? Can’t answer without a reference

system

• World coordinates

• Camera coordinates


Coordinate Systems

• Consider a basis v1, v2,…., vn

• A vector is written v=1v1+ 2v2 +….+nvn

• The list of scalars {1, 2, …. n}is the

representation of v with respect to the given basis

• We can write the representation as a row or

column array of scalars

a=[1 2 …. n]T

=

n

2

1

.


Example

• v=2v1+3v2-4v3

• a=[2 3 –4]T

• Note that this representation is with respect to a particular

basis

• For example, in OpenGL we start by representing vectors

using the object basis but later the system needs a

representation in terms of the camera or eye basis


Coordinate Systems

• Which is correct?

• Both are because vectors have no fixed location

v

v

67

Co-ordinate Systems

X

Y

Z

Right-Handed System

(Z comes out of the screen)

X

Y

Z

Left-Handed System

(Z goes in to the screen)


Frames

• A coordinate system is insufficient to represent points

• If we work in an affine space we can add a single point, the

origin, to the basis vectors to form a frame

P0

v1

v2

v3


Representation in a Frame

• Frame determined by (P0, v1, v2, v3)

• Within this frame, every vector can be written as

v=1v1+ 2v2 +….+nvn

• Every point can be written as

P = P0 + b1v1+ b2v2 +….+bnvn


Confusing Points and Vectors

Consider the point and the vector

P = P0 + b1v1+ b2v2 +….+bnvn

v=1v1+ 2v2 +….+nvn

They appear to have the similar representations

p=[b1 b2 b3] v=[1 2 3]

which confuses the point with the vector

A vector has no position v

p

v

Vector can be placed anywhere

point: fixed


A Single Representation

If we define 0•P = 0 and 1•P =P then we can write

v=1v1+ 2v2 +3v3 = [1 2 3 0 ] [v1 v2 v3 P0]

T

P = P0 + b1v1+ b2v2 +b3v3= [b1 b2 b3 1 ] [v1 v2 v3 P0]

T

Thus we obtain the four-dimensional

homogeneous coordinate representation

v = [1 2 3 0 ] T

p = [b1 b2 b3 1 ] T


Homogeneous Coordinates

The homogeneous coordinates form for a three dimensional point [x y z] is given as

p =[x’ y’ z’ w] T =[wx wy wz w] T

We return to a three dimensional point (for w0) by

xx’/w

yy’/w

zz’/w

If w=0, the representation is that of a vector

Note that homogeneous coordinates replaces points in three dimensions by lines through the origin in four dimensions

For w=1, the representation of a point is [x y z 1]


Homogeneous Coordinates and

Computer Graphics

• Homogeneous coordinates are key to all computer

graphics systems

• All standard transformations (rotation, translation, scaling) can be

implemented with matrix multiplications using 4 x 4 matrices

• Hardware pipeline works with 4 dimensional representations

• For orthographic viewing, we can maintain w=0 for vectors and w=1

for points

• For perspective we need a perspective division


Change of Coordinate Systems

• Consider two representations of a the same vector with

respect to two different bases. The representations are

v=1v1+ 2v2 +3v3 = [1 2 3] [v1 v2 v3]

T

=b1u1+ b2u2 +b3u3 = [b1 b2 b3] [u1 u2 u3]

T

a=[1 2 3 ]

b=[b1 b2 b3]

where


Representing second basis in terms of

first Each of the basis vectors, u1,u2, u3, are vectors

that can be represented in terms of the first basis

u1 = g11v1+g12v2+g13v3

u2 = g21v1+g22v2+g23v3

u3 = g31v1+g32v2+g33v3

v


Matrix Form

The coefficients define a 3 x 3 matrix

and the bases can be related by

see text for numerical examples

a=MTb

ggg

ggg

ggg

33

M =


Change of Frames

• We can apply a similar process in homogeneous coordinates to the representations of both points and vectors

• Any point or vector can be represented in either frame

• We can represent Q0, u1, u2, u3 in terms of P0, v1, v2, v3

Consider two frames:

(P0, v1, v2, v3)

(Q0, u1, u2, u3) P0 v1

v2

v3

Q0

u1 u2

u3


Representing One Frame in Terms of the Other

u1 = g11v1+g12v2+g13v3

u2 = g21v1+g22v2+g23v3

u3 = g31v1+g32v2+g33v3

Q0 = g41v1+g42v2+g43v3 +g44P0

Extending what we did with change of bases

defining a 4 x 4 matrix

ggg

ggg

ggg

ggg

M =


Working with Representations

Within the two frames any point or vector has a representation of the same form

a=[1 2 3 4 ] in the first frame b=[b1 b2 b3 b4 ] in the second frame

where 4 b4 for points and 4 b4 for vectors and

The matrix M is 4 x 4 and specifies an affine transformation in homogeneous coordinates

a=MTb


Affine Transformations

• Every linear transformation is equivalent to a change in

frames

• Every affine transformation preserves lines

• However, an affine transformation has only 12 degrees of

freedom because 4 of the elements in the matrix are fixed

and are a subset of all possible 4 x 4 linear transformations


The World and Camera Frames

• When we work with representations, we work

with n-tuples or arrays of scalars

• Changes in frame are then defined by 4 x 4

matrices

• In OpenGL, the base frame that we start with is

the world frame

• Eventually we represent entities in the camera

frame by changing the world representation using

the model-view matrix

• Initially these frames are the same (M=I)


Moving the Camera

If objects are on both sides of z=0, we must move

camera frame

1000

d100

0010

0001

M =

geometry - - texas a&m university corpus christi slides modified from angel book 6e and from...

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