1. Basic Entities Scalars - real numbers sizes/lengths/angles Vectors - typically 2D, 3D, 4D directions Points - typically 2D, 3D, 4D locations Basic Geometry Review

2. 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

3. 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 +a 1 v 1 + a 2 v 2 + … + a n v n Affine spaces

4. A Euclidean space is an affine space with a distance metric based on inner product: a =(a 1, a 2,… a n ) b =(b 1, b 2,… b n ) Cartesian space: Euclidean space with a standard orthonormal frame: Basis vectors have length 1 Basis vectors are pairwise perpendicular (orthogonal) Euclidean /Cartesian spaces

5. Addition u + v = (u 1 +v 1, u 2 +v 2,…, u n +v n ) Multiplication by a scalar av = (av 1, av 2,… av n ) Dot product (scalar product) uv = u 1 v 1 + u 2 v 2 +…+ u n v n Operations on Vectors u =(u 1, u 2,… u n ) v =(v 1, v 2,… v n )

6. Cross product (vector product) u= (u x, u y, u z ) v= (v x, v y, v z ) u x v = x(u y v z - u z v y )+y(u z v x - u x v z )+z(u x v y - u y v x ) (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

7. Vector size/length/norm: Unit vector = pure direction, |v| =1 General vector =size *direction Vector normalization: v = v/|v| Operations on Vectors

8. 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 > 2 points. Operations on Points

9. 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 + sD 1 +tD 2 Where, t, s are real numbers Lines and Planes O D O D1D1 D2D2