Basic Entities  Scalars - real numbers

•sizes/lengths/angles Vectors - typically 2D, 3D, 4D

•directions Points - typically 2D, 3D, 4D

•locations

Basic Geometry Review

Scalar field:

formed by scalars and the operations between them (+,*). Vector space:

formed by vectors, scalars, and the operations between them. Note: a purely abstract vector space has no notion of: distance, size, angle, or point!

Spaces

An affine space adds the notion of point to the vector space: A =(P, V), where V is a vector space and P is a set of points:  for every point p and vector v, p+v P for every two points p, q: p - q V. By choosing a basis and designating an origin point, we define an affine frame: p = a +a1v1+ a2v2+ … + anvn

Affine spaces

A Euclidean space is an affine space with a distance metric based on inner product: a =(a1, a2,… an) b =(b1, b2,… bn)

 Cartesian space: Euclidean space with a standard orthonormal frame: Basis vectors have length 1 Basis vectors are pairwise perpendicular (orthogonal)

Euclidean /Cartesian spaces

n

iii babababa

1

2,

Addition u + v = (u1+v1, u2+v2,…, un+vn)

Multiplication by a scalar av = (av1, av2,… avn)  Dot product (scalar product) uv = u1v1+ u2v2 +…+ unvn

Operations on Vectors

u =(u1, u2,… un) v =(v1, v2,… vn)

Cross product (vector product) u= (ux, uy, uz) v= (vx, vy, vz)

u x v = x(uy vz- uz vy)+y(uz vx- ux vz)+z(ux vy- uy vx)  

(x, y, z) is a right-handed orthonormal basis. This can be written in a shorthand notation that takes the form of a determinant ..

Here u x v is always perpendicular to both u and v, with the orientation determined by the right-hand rule

What is the geometric meaning of these operations?

Operations on Vectors

zyx

zyx

vvv

uuu

zyx

vu

xyyx

zxxz

yzzy

vuvu

vuvu

vuvu

vu

Vector size/length/norm:

 

Unit vector = pure direction, |v| =1 

General vector =size *direction 

Vector normalization: v = v/|v|

Operations on Vectors

222

21 nvvvv

Operations on Points Can’t add points… but can add a vector to a point: a+v = b also can subtract points v = b - a Can’t multiply a point by a scalar… but can take the affine combination of two points:  c = sa + t b where s + t =1 can also take the affine combination of n > 2points. 

Operations on Points

A line in 2D can be defined by: all affine combinations of two points implicit equation: f(x,y) = ax+by+c = 0 parametric equation: L(t) = O + tD A plane can be defined by: all affine combinations of three points implicit equation: f(x,y,z) = ax+by+cz+d = 0 parametric equation: P(s,t) = O + sD1 +tD2

Where, t, s are real numbers 

Lines and Planes

O

D

O

D1

D2