1

Graphics

CSCI 343, Fall 2015Lecture 9

Geometric Objects

2

Geometric Objects1. A vector space contains vectors and scalars.

A vector has direction and magnitude (but not position).Vectors are denoted by u, v, w (lower case).

A scalar is a real number. Scalars are denoted by

• An affine space is an extension of the vector space to include points (positions in space).

Points are denoted by P, Q, R (upper case)3. A Euclidean space extends the linear vector space to add

a measure of size or distance.

3

Combining entities in affine space

1. Vector-scalar multiplication|v| = || |v|, where |v| is the magnitude of vThe direction of v is the same as the direction of v

v 2v?

2. Point-vector additionAdding a vector to a point results in another point.

Q

PQ + v = Pvv = P - Q

-0.5v?

4

Vector additionu = (x1, y1) (x1 is horizontal component of u, y1 is vertical component of u.)

v = (x2, y2)

u+v = (x1+x2, y1+y2)

u

v

Place the tail of v at the head of u. Draw a vector from the tail of u to the head of v.

5

A parametric line

P() = P0 + v P0

P()

v

A line in space:

Affine sums:P() = Q + v defines a line.

From this we show that:P = 1R + 2Q where 1 + 2 = 1

If 0 <= <= 1, then all P lie on the line between Q and R.

Q

R = 0

= 1

6

Affine sums for more pointsThe affine sum for three points:

P = 1P1 + 2P2 + 3P3, where 1 + 2 + 3 = 1, i >=0defines all points inside triangle P1P2P3.

P1P2

P3

P = 1P1 + 2P2 + ... + nPn, where 1 + 2 + ...+ n = 1, i >=0defines all points inside convex hull around P1P2...Pn.

The affine sum for n points:

A convex hull is like shrink-wraparound all n points.

7

The dot product

The dot product (or inner product) of two vectors is defined as follows:

where represents the angle between the two vectors.

u

v

Projection, w, of u onto v:

ww = ?

8

The cross productThe cross product of two linearly independent (non-parallel) vectors is a third vector that is orthogonal to both of them.

u v

n

The direction of n is defined by the right handed coordinate system.

9

A parametric planeA plane in affine space can be defined in terms of two linearly independent vectors as follows:

u

vP0

uv P

10

Defining a coordinate systemAny 3D vector, w, can be defined in terms of 3 linearly independent vectors, v1, v2, v3:

v1

v2

v3

w

1, 2 and 3 are components of w with respect to the basis (v1, v2, v3).

11

Matrix representation of vectors

We can represent w as a column matrix:

12

Adding a point of referenceBecause vectors have no position, we must add a reference point to specify a coordinate frame.

Choose P0 as reference.

Vectors can be written as:

Points can be written as:

v1

v2

v3

P

P0