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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

Q

V

P

2, 3, 9

VQ-P

y

y

z

Right-hand system

Left-hand system

x

x

z

Vector-scalar multiplication

- Vector permits two fundamental operations add

them, multiply them with real number

- Multiplication gives a vector
- Has the same direction as that of

2.5v

v

-v

Vector addition

v1

v1-v2

- Sum of two vectors
- Subtraction of two vectors
- Adding and subtraction of corresponding

components of two vectors gives a new vector

v2

v1v2

v1v2

v1

v1-v2

v1

v1

v2

v2

v2

Linear Combination and Affine Combination

- A linear combination of vectors
- W a1v1a2v2 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

va1v1 a2v2 (1-a1-a2)v3

v3

v2

v(1-a)v1 av2 V1a(V2-V1)

v2

a(V2-V1)

v1

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

Point-vector addition

- Subtraction of two points P and Q gives a vector

v vP-Q - Adding a vector v to point Q gives a point K

KQu

P

v

Q

u

K

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.

v

u

The Angle between Two Vectors

- Hence, dot product varies as the cosine of the

angle from u to v.

v

u

v

v

u

u

v

u

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

v x u

u

v

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 points location as an offset from the

origin by a certain amount

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.

B

B

B

C

C

C

Parametric Representation Line

- When t varies, the point P trace out all of the

points on the line defined by C and B.

tgt1

B

L

t1

C

t0

tlt0

Parametric Plane

- Heavy use of polygons in 3D graphics, planes seem

to appear everywhere. - Three point form
- Parametric representation
- Point normal form

Parametric form

- Equation
- Given any values of s and t, we can identify the

corresponding point on the plane.

b

B

a

C

A

Three point form of Plane

- Given three points A, B, and C. We rewrite the

parametric form - Affine combination of points

Point Normal form

n

- 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

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 ilt n i)
- Point p
- P Tween(Ai, Bi, t)
- if ( i 0 ) moveTo (Px, Py)
- else lineTo (Px, Py)

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