Mathematical Foundations: Intro Graphics relies on 3 basic objects: 1. Scalars 2. Vectors 3. Points Mathematically defined in terms of spaces: 1. Vector space 2. Affine space 3. Euclidean space Math required: 1. Algebra 2. Trigonometry 3. Geometry 4. Linear algebra 5. Calculus 1
Consists of 1. Scalars 2. Vectors Scalars 1. Real numbers 2. Operators: +, * 3. Laws that hold: Mathematical Foundations: Vector Spaces - Scalars (a) Associative (b) Commutative (c) Distributive 4. Every scalar has additive and multiplicative inverse: (a) α + α 1 = 0 (b) α α 1 = 1 2
Mathematical Foundations: Vector Spaces - Vectors Consist of magnitude and direction Represented as directed arrow Symbolized as v, v Magnitude symbolized as v, or v Operations 1. Vector addition (a) Parallelogram rule (b) Special vector 0 (additive identity): v + 0 = v (c) Additive inverse: For every v there exists a vector w such that i. v + w = 0 ii. w = v (d) Properties: i. Commutative ii. Associative 2. Scalar multiplication Properties: (a) αβv = α(βv) (b) (α + β)v = αv + βv (c) α(v + w) = αv + αw 3. Vector/Cartesian/Cross/Outer product v w = u u is cross product of v and w u is normal to v and w Magnitude of u ( u )= v w sinθ, where θ is angle from v to w Direction of u determined by right-hand rule v and w are parallel if v w = 0 3
Mathematical Foundations: Vector Spaces - Linear Combinations Linear combination of vectors v 1, v 2,, v n is a vector of form v = α 1 v 1 + α 2 v 2 + + α n v n Given a set of vectors, the span of the set is the set of all linear combinations of the vectors Linear dependence Vectors v 1, v 2,, v n are linearly dependent if α 1 v 1 + α 2 v 2 + + α n v n = 0 and at least one α i 0 If v 1,, v n are linearly dependent, then one vector lies within the span of the rest. Proof: 1. α 1 v 1 + α 2 v 2 + + α n v n = 0 2. Assume α 1 0 3. Then v 1 = (1/α 1 )α 2 v 2 + (1/α 1 )α 3 v 3 + + (1/α 1 )α n v n 4. Thus, by definition, v 1 is within span of v 2,, v n Linear independence Vectors v 1, v 2,, v n are linearly independent if v = 0 only when α 1 = α 2 = = α n = 0 The dimension of a space is the size of the minimum spanning set of vectors (smallest number of linearly independent vectors needed to define the space) Subspace Given vector space V and non-empty subset of V called S, S is a linear subspace of V if, whenever v and w are in S, so are v + w and αv The smallest number of linearly independent vectors that can define a space determine the dimension of the space 4
Mathematical Foundations: Vector Spaces - Bases For a space of dimension n, n linearly independent vectors define a basis The set of coefficients α 1,, α n are the representation of v with respect to the basis v 1,, v n Any vector v can be uniquely represented in terms of the basis as Proof: v = α 1 v 1 + α 2 v 2 + + α n v n 1. Given: space V and basis B = v 1, v 2,, v n 2. Then every vector v in V can be written as α 1 v 1 + α 2 v 2 + + α n v n 3. Suppose there is another representation for v: β 1 v 1 + β 2 v 2 + + β n v n 4. Then v v = 0 = (β 1 α 1 )v 1 + + (β n α n )v n 5. Given the assumption that the 2 representations are different, at least one of the coefficients (β i α i ) must be non-zero 6. But if this is true, B cannot be linearly independent, and the assumption that the 2 representations are different must be erroneous 7. Hence, β i = α i for all i, and thus are unique Given a basis, can use matrices to represent vectors: v = α 1 α 2 α n A basis B represents a minimal span Proof: 1. Given space V and basis B = v 1, v 2,, v n 2. Assume B is not a minimal span 3. Then, some v i is linearly dependent on the remaing n 1 v j 4. Let v i = v 1 ; then, v 1 = α 2 v 2 + α 3 v 3 + + α n v n 5. Consider v = 1v 1 + α 2 v 2 + α 3 v 3 + + α n v n 6. Then v = (α 2 v 2 + α 3 v 3 + + α n v n ) + (α 2 v 2 + α 3 v 3 + + α n v n ) = (α 2 α 2 )v 2 + (α 3 α 3 )v 3 + + (α n α n )v n = 0 7. But, this means B cannot be linearly independent 5
Mathematical Foundations: Vector Spaces - Vector Matrix Operations Given v = α 1 α 2 α n w = 1. Scalar multiplication γv = 2. Addition v + w = γ α 1 γ α 2 γ α n 3. Cross product: v w = α 1 + β 1 α 2 + β 2 α n + β n α 11 α 12 α 13 β 1 β 2 β n where the α ij are defined by the determinant i j k α 1 α 2 α 3 β 1 β 2 β 3 6
Mathematical Foundations: Euclidean Spaces - Scalar/Dot/Inner Product Concept of Euclidean space Denoted as v w (also as v, w ) Result is a scalar v w = v w cosθ, where θ is angle from v to w When θ = 90, v w = 0 Orthogonal (perpendicular) vectors have dot product of 0 Properties: 1. Commutative 2. Non-degenerate: v v = 0 only when v = 0 3. Bilinear: v (u + αw) = v u + α(v w) 4. If v is a unit vector, then v w is the projection of w onto v Proof: (a) v w = v w cosθ (b) Since v = 1, cos(θ) = v w w (c) Using trig, cos(θ) = αv w (d) Substituting, αv w = v w w (e) And hence v w = αv, where αv is the projection of v onto w In terms of matrix representations: v w = Σ n i=1(α i β i ) Magnitude of a vector = v v Normalized vector Vector of magnitude = 1 in same direction of original Denoted v v = v v 7
Mathematical Foundations: Bases and Dot Products Basis for which every pair of vectors is orthogonal is an orthonormal basis Given basis v 1, v 2,, v n v i v j = 0 when i j v i v j = 1 when i = j Coefficient i of vector v can be found by α i = v v i To convert a general basis to an orthonormal one, use Graham-Schmidt process Result is a set of orthonormal unit vectors Process iteractively takes a vector not considered so far, and makes it orthogonal to those already considered. Algorithm: 1. 2. 3. v 1 = v 1 w 1 = v 1 v 1 v 2 = v 2 (v 2 w 1 )w 1 4. w 2 = v 2 v 2 8
5. Mathematical Foundations: Bases and Dot Products (2) v 3 = v 3 (v 3 w 1 )w 1 (v 3 w 2 )w 2 6. w 3 = v 3 v 3 Alternative for w 3 : w 3 = ±w 1 w 2, where sign of w 3 = v 3 (w 1 w 2 ) Standard basis in Cartesian system: e 1 = i = e 2 = j = e 3 = k = 1 0 0 0 1 0 0 0 1 9
Mathematical Foundations: Transformations Between Bases Given basis B = b 1, b 2,, b n, and a set of vectors A = a 1, a 2,, a n What is the transformation matrix that takes B A? Consider the standard basis e 1, e 2,, e n Multiplying e i by any n n matrix Q will result in extracting the i th column from Q To generate a set of vectors v i from e i, create matrix Q so that each column consists of one of the v i To find matrix T that transforms B A: 1. Let T A be the matrix that transforms E to A 2. Let T B be the matrix that transforms E to B 3. If T B transforms E to B, then TB 1 transforms B to E 4. Then, T A TB 1 b i converts b 1 e i a i 5. So, T = T A TB 1 10
Mathematical Foundations: Change of Representation Given 1. Basis U = u 1, u 2,, u n 2. Basis V = v 1, v 2,, v n v 1 v 3. Vector w = b 2 where b = [ ] β 1 β 2 β n v n 4. Know representation of u i in terms of v i Want to find w s representation in terms of u i w s representation in terms of U can be determined by the following algorithm 1. Let w s representation in terms of U be a 2. Each u i can be represented in terms of V 3. Let u i = γ i1 v 1 + γ i2 v 2 + + γ in v n, or u 1 u 2 u n = A where A = v 1 v 2 v n γ 11 γ 12 γ 1n γ 21 γ 22 γ 2n γ n1 γ n2 γ nn u 1 u 2 u n where a = [ α 1 α 2 α n ] 11
4. If w = a Mathematical Foundations: Change of Representation (2) u 1 u 2 u n then w = a A 5. And since w = b, v 1 v 2 v n v 1 v 2 v n, (a) b = a A, and (b) b T = A T a T, where a T and b T are w s column representations in U and V, respectively 6. Then, a T = (A T ) 1 b T 12
Mathematical Foundations: Direction Cosines Often represent direction of vector in terms of its direction cosine Let v = αi + βj, or α β Consider 1. 2. In terms of dot products: 1. v i = v i cosθ cosθ = αi v = α v cosφ = βj v = β v 2. Since i = 1, v i is projection of v onto i, which is α i 3. So α i = v cosθ 4. And cosθ = αi v = α v 13
Mathematical Foundations: Direction Cosines (2) Another way of interpreting direction cosines is: ˆv = cosθi + cosφj cosθ and cosφ are the representation of the unit vector in the same direction as v ˆv i = α v β v ˆv = 1 0, which is the projection of v onto i (and vice-versa) α v β v = α v = v i cosθ 14
Affine space consists of 1. Vector space 2. Set of points Operations: 1. Those of vector space 2. Subtraction Mathematical Foundations: Affine Spaces Given points P and Q, v = P Q i.e., subtraction of any 2 points defines a vector Rewriting: P = v + Q i.e., a point can be defined as the sum of a point and a vector Vector addition can be defined in terms of 3 points: (P R) = (P Q) + (Q R) 15
Mathematical Foundations: Parametric Equations Parametric equation of a line Since P = v + Q, A linear set of points is defined by P (α) = αv + Q, where α This is parametric equation of a line Let v = R Q Then P (α) = α(r Q) + Q = αr + (1 α)q This is called an affine sum If 0 α 1, the above defines the line segment from Q to R Above often generalized as P (α) = α 1 R + α 2 Q, where α 1 + α 2 = 1 16
Mathematical Foundations: Parametric Equations (2) Planes Given 3 points P, Q, R Let S(α) = αq + (1 α)p Let T (β) = βr + (1 β)s(α) Then T (α, β) = (1 β)[αq + (1 α)p ] + βr = α(1 β)q α(1 β)p + P + βr βp = P + α(1 β)(q P ) + β(r P ) = P + ωv + δw 17
Let n = Mathematical Foundations: Dot Products and Plane Equations n x n y n z 0, P 0 = x 0 y 0 z 0 1 The plane defined by n that goes through P 0 is the set of points P such that n (P P 0 ) = 0 Expanding, n (P P 0 ) = n x (x x 0 ) + n y (y y 0 ) + n z (z z 0 ) = n x x + n y y + n z z + (n x ( x 0 ) + n y ( y 0 ) + n z ( z 0 )) = 0 The standard equation for a plane is Ax + By + Cz + D = 0 Comparing this to the above, A = n x, B = n y, C = n z, D = (n x x 0 + n y y 0 + n z z 0 ) Hence, the representation of the normal to a plane can be extracted directly from the standard equation for the plane: n = A B C 0 18
Mathematical Foundations: Frames Frame is a set consisting of a point and a basis The point can be thought of as an origin from which measurements are made CG uses many frames, and need to be able to transform between frames Given basis B = v 1, v 2,, v n A vector v has a unique representation in terms of the basis: v = α 1 v 1 + α 2 v 2 + + α n v n Within a given frame F = {P, B}, Point Q also has a unique representation in terms of some vector v: Q = P + α 1 v 1 + α 2 v 2 + + α n v n The representation in both cases is This is problematic α 1 α 2 α n 19
Mathematical Foundations: Relation Between Affine and Vector Spaces Every vector space can be converted into an affine space 1. The points of the affine space are the set of points obtained by the affine sums of the vectors with the origin 2. The associated vector space is the original set of vectors 3. Point difference is equivalent to vector difference of vectors that define points 4. Affine sum is vector sum of vector of vector space with vector that defines a point Affine subspaces A non-empty set S of vector space V is an affine subspace if S = {u v u, v S} is a linear subspace of V S is an affine space in its own right S is the associated vector space of S The difference between 2 points lies in S The sum of a vector of S and a point in S is a point in S S is not a vector space The standard affine spaces A standard affine space uses an additional dimension to represent affine and vector spaces This dimension called h The set of points of form (p 1, p 2,, 1) represents an affine subspace of R n+1 The difference of any 2 points in this space is a vector of form (a 1, a 2,, a n, 0) This subspace is known as the standard affine n-space in R n+1 20
Mathematical Foundations: Relation Between Affine and Vector Spaces (2) The standard affine 2-space in R 3 Points of the affine space are in plane defined by h = 1 Associated vector space consists of plane defined by h = 0 Points defined by vectors in 3-space [x, y, h] t anchored at origin Vectors in vector space derived by vector subtraction of vectors defining points in affine space Point addition is meaningless, as point addition results in object that lies one unit above the affine plane Span of points in affine space Set of all affine combinations of points in the set, OR Let v 1 = P 2 P 1,, v n 1 = P n P 1 1. Find span S of the v i 2. The span of the affine space is the set of points of form P 1 + v, where v S A set of points in an affine space is dependent if one is in affine span of the others 21
Mathematical Foundations: Relation Between Affine and Vector Spaces (3) Coordinate systems Coordinate system is a set of independent points in an affine space whose span is the entire space If P 1, P 2,, P n is a coordinate system, every point in the space can be written uniquely as an affine combination of the P i The coefficients are the coordinates of the point with respect to the coordinate system P 1, P 2,, P n 22