16RHS 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
17Basis VectorsTake any two vectors that are linearly independent (nonzero and nonparallel)can use linear combination of these to define any other vector:
18Orthonormal Basis Vectors If basis vectors are orthonormal (orthogonal (mutually perpendicular) and unit length)we have Cartesian coordinate systemfamiliar Pythagorean definition of distanceOrthonormal algebraic properties
19Basis 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.
20Working 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.
23Working with FramesF1F1p = (3,-1)F2F2p = (-1.5,2)
24Working with FramesF1F1p = (3,-1)F2F3F2p = (-1.5,2)F3
25Working with Frames F1 p = (3,-1) F2 p = (-1.5,2) F3 p = (1,2) F1 F2
26Named 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, ...
27Lines Slope-intercept form Implicit form y = mx + b y – mx – b = 0 Ax + By + C = 0f(x,y) = 0
28Implicit 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
29Implicit 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
30Parametric Curves Parameter: index that changes continuously (x,y): point on curvet: parameterVector form
312D Parametric Linesstart at point p0, go towards p1, according to parameter tp(0) = p0, p(1) = p1
32Linear 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
40Matrix-Matrix Multiplication row by columnNon commutative: AB != BA
41Matrix-Vector Multiplication points as column vectors: post-multiply
42Matrices Transpose Identity Inverse Note: Not all matrices are invertible
43Matrices and Linear Systems Linear system of n equations, n unknownsMatrix form Ax=b
44Exercise 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