Vector Calculus

Mengxia Zhu

Fall 2007

Objective

Review vector arithmetic Distinguish points and vectors Relate geometric concepts to their

algebraic representation Describe point, line, and planes Exploit the dot product and cross product

and their applications in Graphics

Basic Entities Coordinate system: has an origin and some mutually

perpendicular axes emanating from the origin.

Point P: a location in space

Vector v : with length and direction, physical entities, such as force, and velocity. Vector has no fixed location, seen as points displacement

x

y

z

z

x

y

Left-hand systemRight-hand system

P

Q

V

V=Q-P{2, 3, 9}

Vector-scalar multiplication

Multiplication gives a vector Has the same direction as that of

vv

v 2.5v

-v

Vector permits two fundamental operations: add them, multiply them with real number

Vector addition Sum of two vectors Subtraction of two vectors Adding and subtraction of

corresponding components of two vectors gives a new vector

v1

v1+v2

v2

v1

v2

v1-v2

v1

v1+v2

v2

v1

v2

v1-v2

Linear Combination and Affine Combination A linear combination of vectors

W = a1v1+a2v2 + a3v3 +…+anvn: all weights are scalars.

A linear combination is affine combination of vectors if the sum of all coefficients add up to unity.

A convex combination poses a further restriction on affine combination. Not only must the coefficients sum to unit, but each must also be non-negative

Linear Combination of Vectors

The combination is Convex if the coefficients sum to 1, and are not negative. Partition of unit

1

m

i ii

v a v

v1

v2 v=(1-a)v1 + av2

= V1+a(V2-V1)

v1v2

v3

v=a1v1 + a2v2 +(1-a1-a2)v3

a(V2-V1)

Normalize a vector

v is represented by n-tuple ( v1,v2,…vn) Magnitude (length): the distance from the tail to the

head.

Normalization: Scale a vector to have a unity length, unit vector

2 2 21 2 ... nv v v v

~ vv

v

Point-vector addition

Subtraction of two points P and Q gives a vector v: v=P-Q

Adding a vector v to point Q gives a point K: K=Q+u

PQ

uK

v

Dot product Dot product between vector v and vector u gives a scalar If u and v are orthogonal, the dot product equals zero. (a1,a2) dot (b1,b2) = a1xb1 + a2xb2

The most important application of the dot product is to find the angle between two vectors or between two intersecting lines.

vu

v

u

The Angle between Two Vectors Hence, dot product varies as the cosine of

the angle from u to v.

cosu v u v v v v

u

v

vu

0u v

v u 0u v

v u 0u v

Cross Product

a = (a1, a2, a3), b=(b1, b2, b3)

a x b = ( a2b3 –a3b2), (a3b1-a1b3), (a1b2 – a2b1)

Cross Product Cross product between vector v and u gives a

vector

n is a unit vector perpendicular to both u and v. Follow the right-hand rule

u and v are parallel if

The length of the cross product equals the area of the parallelogram determined by u and v

~

sinv u n v u

0v u v

u

v x u

u x v

Operation Calculation

Operation Calculation

Homogeneous representation Revisit coordinate frame system

Ordinary coordinate system, points and vectors are represented the same. However, they are quite different.

Points have location, no direction and size. Vector has no location, but with direction and size.

Homogeneous representation cont Represent both points and

vector using the same set of basic underlying objects,

One of the hallmarks of computer graphics, keep straight the distinction between points and vectors with a compact notation. Easy to program…

We view point’s location as an offset from the origin by a certain amount , , ,a b c

1

21 2 3

3

1

21 2 3

3

, , ,

0

( , , , )

1

v

vv av bv cv a b c

v

p

pP ap bp cp a b c

p

Homogeneous representation

.

OpenGL uses 4D homogeneous coordinates for all its vertices.

To go from ordinary to homogenous coordinates, if the object is a point, append a 1, if the object is a vector, append a 0.

Point: (x,y,z,1) Vector: (x,y,z,0)

Line

A line is defined by two points. It is infinite in length and extending forever in both directions.

A line segment is defined by two endpoints. A ray is semi-infinite, specified by a starting point

and a direction.

C

B

C

B

CB

Parametric Representation Line When t varies, the

point P trace out all of the points on the line defined by C and B.

( )L t C bt

b B C

C

B

t<0

t=0

t=1

t>1

L

Parametric Plane

Heavy use of polygons in 3D graphics, planes seem to appear everywhere. Three point formParametric representationPoint normal form

Parametric form

Equation

Given any values of s and t, we can identify the corresponding point on the plane.

P(s,t) = C + as + bt

C

b

aa = A-C, b = B-C

B

A

Three point form of Plane

Given three points A, B, and C. We rewrite the parametric form:

Affine combination of points:

P(s,t) = C + s (A-C) + t(B-C)

P(s,t) = sA+tB+(1-s-t)C

Point Normal form

Plane can be specified by a single point B, that lies within it and direction n, normal to the plane.

Normal n is perpendicular to any line lying in the plane.

R is any point on the plane, it satisfies:

B

n

( ) 0n R B

Tweening for Art and Animation

The affine combination of points P = A(1-t) + Bt, performs a linear

interpolation between the points A and B. Interesting animation can be created that

show one figure being “tweened” into another. The procedure if simplest if the two figures are polylines based on the same number of points.

Tweening

void drawTween ( Point A[], Point B[], int n, float t){

for (int i= 0; i< n; i++){

Point p; P = Tween(A[i], B[i], t);

if ( i == 0 ) moveTo (Px, Py);else lineTo (Px, Py);

}}