程序代写代做代考 AI cse3431-lecture2.key

cse3431-lecture2.key

Vectors
n-tuple:

v 2 0 b · c = 0 b · c < 0 Perpendicular vectors Definition Vectors b and c are perpendicular iff b·c = 0 Also called normal or orthogonal It is easy to see that the standard unit vectors form an orthogonal basis: i·j = 0, j·k = 0, i·k = 0 Cross product Defined only for 3D Vectors and with respect to the standard unit vectors Definition Properties of the cross product 1. i⇥ j = k, i⇥ j = k, i⇥ j = k. 2. Antisymmetry: a⇥ b = �b⇥ a. 3. Linearity: a⇥ (b+ c) = a⇥ b+ a⇥ c. 4. Homogeneity: (sa)⇥ b = s(a⇥ b). 5. The cross product is normal to both vectors: (a⇥ b) · a = 0 and (a⇥ b) · b = 0. 6.|a⇥ b| = |a||b|sin(✓). Geometric interpretation of the cross product Clarification for the figure: a and b need not be perpendicular Recap Vector spaces Operations with vectors Representing vectors through a basis v = a1b1+…anbn , vB = (a1,…,an) Standard unit vectors Dot product Perpendicularity Cross product Normal to both vectors Points vs Vectors What is the difference? Points vs Vectors What is the difference? Points have location but no size or direction. Vectors have size and direction but no location. Problem: we represent both as triplets! Relationship between points and vectors A difference between two points is a vector: Q – P = v We can consider a point as a point plus an offset Q = P + v v Q P Coordinate systems Defined by: (a,b,c,θ) The homogeneous representation of points and vectors Switching coordinates Normal to homogeneous: • Vector: append as fourth 
 coordinate 0 • Point: append as fourth 
 coordinate 1 Switching coordinates Homogeneous to normal: • Vector: remove fourth 
 coordinate (0) • Point: remove fourth 
 coordinate (1) Does the homogeneous representation support operations? Operations : • v + w = (v1,v2,v3,0) +(w1,w2,w3,0)= 
 (v1+w1, v2+w2, v3+w3, 0) Vector! • av = a(v1,v2,v3,0) = (av1, av2, av3, 0), Vector! • av + bw = a(v1,v2,v3,0) +b(w1,w2,w3,0)=
 (av1+bw1, av2+bw2, av3+bw3, 0) Vector! • P+v = (p1,p2,p3,1) +(v1,v2,v3,0)= 
 = (p1+v1, p2+v2, p3+v3, 1) Point! Linear combination of points Points P, R scalars f,g: fP+gR = f(p1,p2,p3,1) +g(r1,r2,r3,1) 
 = (fp1+gr1, fp2+gr2, fp3+gr3, f+g) What is it? Linear combination of points Points P, R scalars f,g: fP+gR = f(p1,p2,p3,1) +g(r1,r2,r3,1) 
 = (fp1+gr1, fp2+gr2, fp3+gr3, f+g) What is it? • If (f+g) = 0 then vector! • If (f+g) = 1 then point! Affine combinations of points Definition: Points Pi: i = 1,…,n Scalars fi: i = 1,…,n f1P1+ … + fnPn iff f1+ …+fn = 1 Example: 0.5P1 + 0.5P2 Geometric explanation Recap Vector spaces Dot product Cross product Coordinate systems (mostly orthonormal) Homogeneous representations of points and vectors Matrices Rectangular arrangement of elements: Special square matrices Symmetric: (Aij )n x n= (Aji)n x n Zero: Aij = 0, for all i,j Iii = 1, for all i Identity: In = Iij = 0 for i ≠ j Operations with matrices Addition: Properties: Definition: A few properties: Matrix Multiplication Cm⇥r = Am⇥nBn⇥r (Cij) = ( nX k aikbkj) 1. Not commutative: AB 6= BA. 2. Associative: A(BC) = (AB)C. 3. Compatible with Scalar multiplication: f(AB) = (fA)B and (AB)f = A(Bf). 4. Distributive: A(B + C) = AB +AC, and (B + C)A = BA+ CA. 5. (AB)T = BTAT . Inverse of a square matrix Definition MM-1 = M-1M = I Important property (square matrices only) (AB)-1= B-1 A-1 Convention Vectors and points are represented as column matrices However, always keep track of the base, i.e. the corresponding coordinate system Dot product as a matrix multiplication A vector is a column matrix Lines and Planes Usually defined by an appropriate number of points (vertices) Lines Line (in 2D) • Explicit • Implicit • Parametric (extends to 3D) Planes Plane equations Implicit Parametric Explicit Exercises Orthogonal projection of a vector on another vector. Orthogonal projection of a point on a plane. Z-buffer Graphics Pipeline Modeling transformation Viewing transformation Projection transformation Perspective division Viewport transformation OCS WCS VCS CCS NDCS DCS Rasterization Transformations (2D) P Q = T(P) T Why Transformations? x y T Affine Transformations (2D) Linear in the coordinates Matrix Form of Affine Transformations Transformation as a matrix multiplication Transforming Points and Vectors