6/16/2015©Zachary Wartell 2D Coordinate Systems, Change of Coordinates, and Matrices Revision: 1/8/2008 6:14:11 PM Copyright Zachary Wartell, University of North Carolina All Rights Reserved Textbook: Chapter 5 and Appendix

6/16/2015©Zachary Wartell Overview points, vectors, alignments & coordinates thereof coordinate systems & coordinates thereof change of coordinates for pt’s, vt’s

6/16/2015©Zachary Wartell Geometric Point Point – a location in space

6/16/2015©Zachary Wartell Consider points in 1D space, drawn as a dot We also have 1D distance, drawn as a line segment –Note, any line segment of the right length is a valid drawing of a given distance Distance (in 1D space) 0 1 …. d=2 l 0 =[0,2] 0 1 …. l 1 =[4.5,6.5] d=2 l 0 =[0,2] 0 1 …. l 1 =[4.5,6.5] d=2 l 2 =[5.5,7.5]

6/16/2015©Zachary Wartell Displacement in 1D space is a “signed” distance, drawn as an directed line segment (also an ‘arrow’) –Note, any directed line segment of the correct length and direction is a valid drawing of a given displacement Displacement (in 1D space) d=2 1/3 d=-1.75 l 2 =[6,4.25]l 2 =[0,2 1/3] d=1.75 l 1 =[2,3.75] 0 1 d=-1.75 l 2 =[6,4.25] d=-1.75 l 1 =[3.75,2] 0 1 d=-1.75 l 3 =[2.25,1.5]

6/16/2015©Zachary Wartell What about 2D Displacements? (Step #1) Step #1, need to define an alignment. –“Parallel lines all share a common alignment” or –“An alignment, is what is common in a set of parallel lines” An alignment is drawn by drawing any line parallel to the alignment W/E alignment ?/? alignment NE/SW alignmentN/S alignment 3 different lines but all have the same “alignment”

6/16/2015©Zachary Wartell geometric vectors are “2D displacements” or “direction with sign and magnitude” -- drawn by an (2D) directed line segment (‘arrow’) –Note, any arrow of the right length and alignment is valid drawing of a given geometric vector 2D Displacements are called ‘geometric vectors’ Length=5 “Move North 5 meters” Length=7 “Move NW 7 meters”“Move SE 7 meters” Length=7 3 different arrows but all representative of the same vector

6/16/2015©Zachary Wartell Geometric Vector versus Alignment ∙ a geometric vector gives more information than an alignment Alignment is N/S Vector is N at magnitude l l this vector has same alignment as this one but different magnitude

6/16/2015©Zachary Wartell Please draw point (5,3) ?

6/16/2015©Zachary Wartell Point (5,3) A (5,3) A “regular” recti-linear coordinate system (syn. Cartesian coordinate system)

6/16/2015©Zachary Wartell Point (5,3) B (5,3)

6/16/2015©Zachary Wartell (5,3) Point (5,3) C (5,3)

6/16/2015©Zachary Wartell Point (5,3) D (5,3) oblique (!) recti-linear coordinate system

6/16/2015©Zachary Wartell Please draw vector (3,2) ?

6/16/2015©Zachary Wartell Vector (3,2) A (3,2)

6/16/2015©Zachary Wartell Vector (3,2) B (3,2)

6/16/2015©Zachary Wartell Review: Points, Vectors, Arrows, Alignments p: (5,3) v: (1,3) a 0 : ((0,0),(1,3)) a 1 : ((2,1),(3,4)) a 3 : ((-2,-1),(-1,2)) al 0 : (2k,1k),  k ∈ R, k≠0 all lines with slope ½ have alignment al 0

6/16/2015©Zachary Wartell DOF’s of 2D Elements ElementDegrees of freedom Point2 DOF Vector2 DOF Arrow4 DOF Alignment1 DOF

6/16/2015©Zachary Wartell Operation on Points & Vectors (2D or 3D) Points: p i - p j = v p i ± v = p j (p + 0 = p) t p i + (1- t )p j = p k - “affine combination” Vectors: v i ±v j = v k, v j = a ∙ v k (a  R ) or a v i ± b v j = v k - ”linear combination” length(v), |v| = normalize(v), v'= (1/|v|) ∙ v dot product: v i  v j =

6/16/2015©Zachary Wartell Notation (5,3) A A,B,C – capital script letter is a coordinate system ( x,y ) B or ( x B,y B ) – is a coordinate with respect to coordinate system B A Coordinate System is a 3-tuple, ( o,x,y ) o : a point called the origin x : a vector called x-basis vector y : a vector called y-basis vector ox y

6/16/2015©Zachary Wartell Different Point/Coordinate & Same Coordinate/Point So far we’ve looked at different points with the same coordinate but conversely we can considered the same point with different coordinates!

6/16/2015©Zachary Wartell (3,3) B, (1.33,0.33) A Coordinates of point p B A p

6/16/2015©Zachary Wartell Coordinates of Coordinate System A relative to B B A (2.5,1.5) B (1/√2,1/√2) B (-1/√2,1/√2) B Equivalent Questions: What is M B ← A ? -- “ A to B" Where is A (as measured) in B ? How do I superimpose B onto A ? (3,3) B, (1.33,0.33) A p

6/16/2015©Zachary Wartell What is M inches ← feet ? p with coordinates (0.4) feet or(4.8) inches 0 inches unit vector feet unit vector Where is Feet in Inches ? How to superimpose Inches onto Feet ?

6/16/2015©Zachary Wartell Where is Celsius in Fahrenheit ? How to superimpose Fahrenheit onto Celsius ? p with coordinates (-8.81) cel or (16) far 0 far fahrenheit unit vector & origin celsius unit vector & origin 0 cel, 32 far scale by 9/5 add 32 ‘s What is M far ← cel ?

6/16/2015©Zachary Wartell Review: Matrices (1)

6/16/2015©Zachary Wartell Review: Matrices (2)

6/16/2015©Zachary Wartell Review: Matrices (3)

6/16/2015©Zachary Wartell Review: Matrices (4)

6/16/2015©Zachary Wartell Where is A in B ? A.x = ( 1/√2, 1/√2) A.y = (-1/√2, 1/√2) A. o = (2.5, 1.5) Where is A in B ? = What is M B ← A ? (1,2) A, (1.79,3.62) B p for “point”

6/16/2015©Zachary Wartell What about the vectors? Where is A in B ? A.x = ( 1/√2, 1/√2) A.y = (-1/√2, 1/√2) A. o = (2.5, 1.5) Where is A in B ? = What is M B ← A ? - vectors V for “vector”

6/16/2015©Zachary Wartell More Compact: (1.79,3.62) Where is A in B ? A.y = ( 1/√2, 1/√2) A.x = (-1/√2, 1/√2) A. o = (2.5, 1.5) Compact Representation

6/16/2015©Zachary Wartell Generalization In General: If I were being absurdly pedantic what sub-sub-script should go under the sub-scripts x,y,z? Answer:

6/16/2015©Zachary Wartell Where is B in A ? = What is M A ← B ? More Compact: Where is B in A ? B.x = ( 1/√2, -1/√2) B.y = ( 1/√2, 1/√2) B. o = (-2.76, 1.73)

6/16/2015©Zachary Wartell What is relationship of M A ← B and M B ← A ? -generally inverse computation requires expensive Gaussian elimination. But in various specific cases (translate, scale, rotate) inverse computation can be quick!

6/16/2015©Zachary Wartell Revisions Added slide 13 2.Added Overview Slide 3.Added explicit slide referencing “ITCS D Transforms.ppt” 4.re-ordered slides to better suit insertion of “ITCS D Transforms.ppt” 5.added 1D change of coordinates slides 1.2 -typos 1.3 -added Matrix: Review slides 1.4 -moved affine transform slides to other.ppt file -added DOF slide -added ‘superimpose’ terminology to coordinate transform slides

6/16/2015©Zachary Wartell Alternative Convention M B ← A and M A ← B