Mathematical Foundations Sections A-1 to A-5 Some of the material in these slides may have been adapted from university of Virginia, MIT and Åbo Akademi University

2 Mathematical Foundations ● I’ll give a brief, informal review of some of the mathematical tools we’ll employ ■ Geometry (2D, 3D) ■ Trigonometry ■ Vector and affine spaces ○ Points, vectors, and coordinates ■ Dot and cross products ■ Linear transforms and matrices ● I know that everyone knows that…

3 2D Geometry ● Know your high-school geometry: ■ Total angle around a circle is 360° or 2π radians ■ When two lines cross: ○ Opposite angles are equivalent ○ Angles along line sum to 180° ■ Similar triangles: ○ All corresponding angles are equivalent ○ Corresponding pairs of sides have the same length ratio and are separated by equivalent angles ○ Any corresponding pairs of sides have same length ratio

4 Trigonometry ● Sine: “opposite over hypotenuse” ● Cosine: “adjacent over hypotenuse” ● Tangent: “opposite over adjacent” ● Unit circle definitions: ■ sin (  ) = y/z ■ cos (  ) = x/z ■ tan (  ) = y/x ■ Z = sqrt(x 2 + y 2 ) ■ Degrees vs Radians ■ Etc… y x Z 

5 3D Geometry ● To model, animate, and render 3D scenes, we must specify: ■ Location ■ Displacement from arbitrary locations ■ Orientation ● We’ll look at two types of spaces: ■ Vector spaces ■ Affine spaces ● We will often be sloppy about the distinction

6 Co-ordinates ● Cartesian ■ 3 Perpendicular Axes ■ Right Hand Rule ● Curvilinear ■ Any non-cartesian ■ Formed by the intersection of any 3 surfaces ■ Each Surface is constructed by fixinging one co-ordinate ■ May be orthogonal y x z

7 Cylindrical Co-ordinate  z Related to Cartesian A surface is formed by fixing one co-ordinate Plane including the z-axis is formed by fixing  Plane intersecting z-axis is formed by fixing z Cylinder is formed by fixing   x-axis y-axis ● How to convert from Cylindrical to Cartesian

8 Polar Co-ordinates 3 co-ordinates:  Surface of constant  is sphere Surface of constant  is a cone Surface of a constant  is a plane including z- axis  z  x-axis y-axis  ● How to convert from Polar to Cartesian

9 Vector Spaces ● A space is basically a set of elements with operations defined on the elements ● Two types of elements: ■ Scalars (real numbers):  … ■ Vectors (n-tuples): u, v, w, … ● Supports two operations: ■ Addition operation u + v, with: ○ Identity 0 v + 0 = v ○ Inverse - v + (- v ) = 0 ■ Scalar multiplication: ○ Distributive rule:  ( u + v ) =  ( u ) +  ( v ) (  +  ) u =  u +  u

10 Vector Spaces ● A linear combination of vectors results in a new vector: v =  1 v 1 +  2 v 2 + … +  n v n ● If the only set of scalars such that  1 v 1 +  2 v 2 + … +  n v n = 0 is  1 =  2 = … =  3 = 0 then we say the vectors are linearly independent ● The dimension of a space is the greatest number of linearly independent vectors possible in a vector set ● For a vector space of dimension n, any set of n linearly independent vectors form a basis

11 Vector Spaces: A Familiar Example ● Our common notion of vectors in a 2D plane is (you guessed it) a vector space: ■ Vectors are “arrows” rooted at the origin ■ Scalar multiplication “streches” the arrow, changing its length (magnitude) but not its direction ■ Addition can be constructed using parallelogram ( “trapezoid rule”) u+v y x u v

12 Vector Spaces: Basis Vectors ● Given a basis for a vector space: ■ Each vector in the space is a unique linear combination of the basis vectors ■ The coordinates of a vector are the scalars from this linear combination ■ Orthonormal Basis ○ Orthogonal u k.u j = 0, j  k ○ Unit length u k.u k = 1 ■ Best-known example: Cartesian coordinates ○ Draw example on the board ○ (i, j, k) ■ Note that a given vector v will have different coordinates for different bases

13 Vectors And Points ● We commonly use vectors to represent: ■ Direction (i.e., orientation) ■ Points in space (i.e., location) ■ Displacements from point to point ● But we want points and directions to behave differently ■ Ex: To translate something means to move it without changing its orientation ■ Translation of a point = different point ■ Translation of a direction = same direction

14 Affine Spaces ● An affine space is what is left of a vector space after you've forgotten which point is the origin. ● Points support these operations ■ Point-point subtraction: Q - P = v ○ Result is a vector pointing from P to Q ■ Vector-point addition: P + v = Q ○ Result is a new point ○ P + 0 = P ■ Note that the addition of two points is undefined ■ Scaler times point is undefined P Q v

15 Affine Spaces ● Points, like vectors, can be expressed in coordinates ■ The definition uses an affine combination ■ Net effect is same: expressing a point in terms of a basis ● Thus the common practice of representing points as vectors with coordinates ● Be careful to avoid nonsensical operations ■ Point + point ■ Scalar * point

16 Affine Lines: An Aside ● Parametric representation of a line with a direction vector d and a point P 1 on the line: P(  ) = P origin +  d ● Restricting 0   produces a ray in the direction of the vector d ● Setting d to P - Q and restricting 1    0 produces a line segment between P and Q

17 Dot Product ● The dot product or, more generally, inner product of two vectors is a scalar: v 1 v 2 = x 1 x 2 + y 1 y 2 + z 1 z 2 (in 3D) ● Useful for many purposes ■ Computing the length of a vector: length(v) = sqrt(v v) ■ Normalizing a vector, making it unit-length ■ Computing the angle between two vectors: u v = |u| |v| cos(θ) ■ Checking two vectors for orthogonality ■ Projecting one vector onto another θ u v

18 Cross Product ● The cross product or vector product of two vectors is a vector: ● v 1 xv 2 = u|v 1 ||v 2 |sin  ● Cross product of two vectors is orthogonal to both ● Right-hand rule dictates direction of cross product ● Cross product is handy for finding surface orientation ■ Lighting ■ Visibility ● Not Associative ● Distributive

19 Linear Transformations ● A linear transformation: ■ Maps one vector to another ■ Preserves linear combinations ● Thus behavior of linear transformation is completely determined by what it does to a basis ● Turns out any linear transform can be represented by a matrix

20 Matrices ● By convention, matrix element M rc is located at row r and column c: ● By convention (mathematical), vectors are columns:

21 Matrices ● Matrix-vector multiplication applies a linear transformation to a vector: ● Recall how to do matrix multiplication ● Dot product ● Cross product is the determinant of a matrix

22 Matrices ● Matrix multiplication is NOT commutative ● Matrix multiplication is associative ● Linear transformations can be represented as matrix multiplications ● When we talk about transformations, we will see that all linear transformations can be represented by matrices

23 Matrix Transformations ● A sequence or composition of linear transformations corresponds to the product of the corresponding matrices ■ Note: the matrices to the right affect vector first ■ Note: order of matrices matters! ● The identity matrix I has no effect in multiplication ● Some (not all) matrices have an inverse: