16 RHS vs. LHS Coordinate Systems Right-handed coordinate systemLeft-handed coordinate systemright hand rule:index finger x, second finger y;right thumb points upleft hand rule:index finger x, second finger y;left thumb points down
17 Basis VectorsTake any two vectors that are linearly independent (nonzero and nonparallel)can use linear combination of these to define any other vector:
18 Orthonormal Basis Vectors If basis vectors are orthonormal (orthogonal (mutually perpendicular) and unit length)we have Cartesian coordinate systemfamiliar Pythagorean definition of distanceOrthonormal algebraic properties
19 Basis Vectors and Origins Coordinate system: just basis vectorscan only specify offset: vectorsCoordinate frame (or Frame of Reference): basis vectors and origincan specify location as well as offset: pointsIt helps in analyzing the vectors and solving the kinematics.
20 Working with FramesF1F1Coefficient of i: Start from O, move along i and count until the line passing through P parallel to j.Coefficient of j: Start from O, move along j and count until the line passing through P parallel to i.Coefficient of i or j will be –ve if the movement is opposite to the direction of i or j, resp.
23 Working with FramesF1F1p = (3,-1)F2F2p = (-1.5,2)
24 Working with FramesF1F1p = (3,-1)F2F3F2p = (-1.5,2)F3
25 Working with Frames F1 p = (3,-1) F2 p = (-1.5,2) F3 p = (1,2) F1 F2
26 Named Coordinate Frames Origin and basis vectorsPick canonical frame of referencethen don’t have to store origin, basis vectorsjustconvention: Cartesian orthonormal one on previous slideHandy to specify others as neededairplane nose, looking over your shoulder, ...really common ones given names in CGobject, world, camera, screen, ...
27 Lines Slope-intercept form Implicit form y = mx + b y – mx – b = 0 Ax + By + C = 0f(x,y) = 0
28 Implicit Functions Find where function is 0 An implicit function f(x,y) = 0 can be thought of as a height field where f is the height (top). A path where the height is zero is the implicit curve (bottom).Find where function is 0plug in (x,y), check if= 0 on line< 0 inside> 0 outside
29 Implicit Circles circle is points (x,y) where f(x,y) = 0 points p on circle have property that vector from c to p dotted with itself has value r2points p on the circle have property that squared distance from c to p is r2points p on circle are those a distance r from center point c
30 Parametric Curves Parameter: index that changes continuously (x,y): point on curvet: parameterVector form
31 2D Parametric Linesstart at point p0, go towards p1, according to parameter tp(0) = p0, p(1) = p1
32 Linear Interpolation Parametric line is example of general concept p goes through a at t = 0p goes through b at t = 1Linearweights t, (1-t) are linear polynomials in t
40 Matrix-Matrix Multiplication row by columnNon commutative: AB != BA
41 Matrix-Vector Multiplication points as column vectors: post-multiply
42 Matrices Transpose Identity Inverse Note: Not all matrices are invertible
43 Matrices and Linear Systems Linear system of n equations, n unknownsMatrix form Ax=b
44 Exercise 1Q 1: If the vectors, u and v in figure are coplanar with the sheet of paper, does the cross product v x u extend towards the reader or away assuming right-handed coordinate system?For u = [1, 0, 0] & v = [2, 2, 1]. FindQ 2: The length of vQ 3: length of projection of u onto vQ 4: v x uQ 5: Cos t, if t is the angle between u & v