[ ] a a a ... a vector = offset / displacement 1 2 n - PowerPoint PPT Presentation

University of British Columbia News CPSC 314 Computer Graphics • usual lecture order switched Jan-Apr 2013 • math review today (Jan 2) Tamara Munzner • James Gregson, TA • usually second lecture Math Review • course intro and overview Friday (Jan 4) guest lecture: James Gregson • Tamara Munzner, instructor • usually first lecture Week 1, Wed Jan 2 http://www.ugrad.cs.ubc.ca/~cs314/Vjan2013 2 Notation: Scalars, Vectors, Matrices Vectors a • scalar • arrow: length and direction • (lower case, italic) • oriented segment in nD space [ ] a a a ... a • vector = • offset / displacement 1 2 n • (lower case, bold) • location if given origin • matrix a a a & # • (upper case, bold) 11 12 13 $ ! A a a a = $ ! 21 22 23 $ a a a ! % " 31 32 33 3 4

Column vs. Row Vectors Vector-Vector Addition • row vectors [ ] • add: vector + vector = vector a a a ... a = row 1 2 n • parallelogram rule a & # • tail to head, complete the triangle 1 • column vectors $ ! a 2 geometric algebraic $ ! a = col ... u + v $ ! u v + & # v 1 1 $ ! $ ! a u v u v + = + % " n $ 2 2 ! u • switch back and forth with transpose u v $ ! + % " 3 3 T ( 3 , 2 ) ( 6 , 4 ) ( 9 , 6 ) a a + = = examples: col row ( 2 , 5 , 1 ) ( 3 , 1 , 1 ) ( 5 , 6 , 0 ) + − = 5 6 Vector-Vector Subtraction Vector-Vector Subtraction • subtract: vector - vector = vector • subtract: vector - vector = vector u v u v − − & # & # 1 1 1 1 $ ! $ ! u v u v u v u v − = − − = − 2 2 2 2 $ ! $ ! v v u v u v $ − ! $ − ! % " % " 3 3 3 3 u u v v − − ( 3 , 2 ) ( 6 , 4 ) ( 3 , 2 ) ( 3 , 2 ) ( 6 , 4 ) ( 3 , 2 ) − = − − − = − − ( 2 , 5 , 1 ) ( 3 , 1 , 1 ) ( 1 , 4 , 2 ) ( 2 , 5 , 1 ) ( 3 , 1 , 1 ) ( 1 , 4 , 2 ) u − v − − = − u − v − − = − = = u ( v ) u ( v ) + − + − v u + v u argument reversal v v − u u 7 8

Scalar-Vector Multiplication Vector-Vector Multiplication • multiply: scalar * vector = vector • multiply: vector * vector = scalar u • v • vector is scaled • dot product, aka inner product u v ' $ ' $ 1 1 % " % " a * u ( ) ( ) ( ) u v u v u v u v • = ∗ + ∗ + ∗ a * u ( a * u , a * u , a * u ) = 2 2 1 1 2 2 3 3 % " % " 1 2 3 u v % " % " & # & # 3 3 u 2 * ( 3 , 2 ) ( 6 , 4 ) = . 5 * ( 2 , 5 , 1 ) ( 1 , 2 . 5 ,. 5 ) = 9 10 Vector-Vector Multiplication Vector-Vector Multiplication • multiply: vector * vector = scalar • multiply: vector * vector = scalar u • v u • v • dot product, aka inner product • dot product, aka inner product u u v v " % " % u 1 v 1 ' ' $ $ ' ' $ $ 1 1 1 1 $ ' $ ' % % " " % % " " ( ( ) ( ) ( ) ( ) ( ) ) u u v v u u v v u u v v u u v v ( ) + u 2 ∗ v 2 ( ) + u 3 ∗ v 3 ( ) u 2 v 2 = u 1 ∗ v 1 • • = = ∗ ∗ + + ∗ ∗ + + ∗ ∗ • 2 2 2 2 1 1 1 1 2 2 2 2 3 3 3 3 $ ' $ ' % % " " % % " " u u v v $ ' $ ' % % " " % % " " u 3 v 3 # & # & & & # # & & # # 3 3 3 3 u v u v cos • = θ • geometric interpretation u • lengths, angles • can find angle between two θ vectors v 11 12

Dot Product Geometry Dot Product Example • can find length of projection of u onto v " % " % u 1 v 1 $ ' $ ' u u v u v cos ( ) + u 2 ∗ v 2 ( ) + u 3 ∗ v 3 ( ) u 2 v 2 = u 1 ∗ v 1 • • = θ $ ' $ ' $ ' $ ' u 3 v 3 # & # & θ u v • v u cos θ = v u cos 6 1 θ & # & # u • v 0 → $ ! $ ! • as lines become perpendicular, 1 7 ( 6 * 1 ) ( 1 * 7 ) ( 2 * 3 ) 6 7 6 19 • = + + = + + = $ ! $ ! 2 3 $ ! $ ! % " % " 13 14 Vector-Vector Multiplication, The Sequel Vector-Vector Multiplication, The Sequel • multiply: vector * vector = vector • multiply: vector * vector = vector • cross product u v • cross product u v u v u v u v u v & # & # − & # & # − & # & # 1 1 2 3 3 2 1 1 2 3 3 2 • algebraic $ ! $ ! $ ! • algebraic $ ! $ ! $ ! u v u v u v u v u v u v × = − × = − 2 2 3 1 1 3 2 2 3 1 1 3 $ ! $ ! $ ! $ ! $ ! $ ! u v u v u v u v u v u v $ ! $ ! $ ! $ ! $ ! $ ! − − % " % " % " % " % " % " 1 2 2 1 1 2 2 1 3 3 3 3 15 16

Vector-Vector Multiplication, The Sequel Vector-Vector Multiplication, The Sequel • multiply: vector * vector = vector • multiply: vector * vector = vector • cross product u v • cross product u v u v u v u v u v − − & # & # & # & # 3 & # & # 1 1 2 3 3 2 1 1 2 3 3 2 • algebraic $ ! $ ! $ ! • algebraic $ ! $ ! $ ! u v u v u v u v u v u v × = − × = − 1 2 2 3 1 1 3 2 2 3 1 1 3 $ ! $ ! $ ! $ ! $ ! $ ! u v u v u v u v u v u v $ ! $ ! $ ! $ ! $ ! $ ! − − 2 % " % " % " % " % " % " 3 3 1 2 2 1 3 3 1 2 2 1 • geometric a × b a × b = a b sin θ • parallelogram a × b b area θ � • perpendicular a × b to parallelogram a blah blah 17 18 RHS vs. LHS Coordinate Systems Basis Vectors • right-handed coordinate system convention • take any two vectors that are linearly independent (nonzero and nonparallel) z right hand rule: index finger x, second finger y; • can use linear combination of these to define right thumb points up y x any other vector: z x y = × • left-handed coordinate system c w a w b b = + 1 2 c left hand rule: index finger x, second finger y; c = 2 a + 0.5 b a left thumb points down x y z x y 0.5 b = × z 2 a 19 20

Orthonormal Basis Vectors Basis Vectors and Origins • if basis vectors are orthonormal (orthogonal • coordinate system: just basis vectors (mutually perpendicular) and unit length) • can only specify offset: vectors • we have Cartesian coordinate system • coordinate frame: basis vectors and origin • familiar Pythagorean definition of distance • can specify location as well as offset: points orthonormal algebraic properties y p p o x + i y j = + x x y 1 , = = j o x y 0 y • = 5 . 0 i + x 2 = c 0.5 y 2 x 21 22 Working with Frames Working with Frames p o x + i y j p o x + i y j = + = + j j F 1 F 1 o o p p i i F 1 F 1 p = (3,-1) 23 24

Working with Frames Working with Frames F 1 p o x + i y j p o x + i y j = + = + j j F 1 F 1 o o p p i i F 1 p = (3,-1) F 1 p = (3,-1) F 2 j o i F 2 25 26 Working with Frames Working with Frames p o x + i y j p o x + i y j F 2 = + = + j j F 1 F 1 o o p p i i F 1 p = (3,-1) F 1 p = (3,-1) F 2 F 2 j j o o i i F 2 F 2 p = (-1.5,2) p = (-1.5,2) 27 28

Working with Frames Working with Frames p o x + i y j p o x + i y j = + = + j j F 1 F 1 o o p p i i F 1 p = (3,-1) F 1 p = (3,-1) F 2 F 2 j j o o i i F 2 F 2 p = (-1.5,2) p = (-1.5,2) j j i i o o F 3 F 3 F 3 F 3 p = (1,2) 29 30 Working with Frames Named Coordinate Frames p o a x b y c z • origin and basis vectors = + + + F 3 • pick canonical frame of reference p o x + i y j = + • then don’t have to store origin, basis vectors j F 1 p ( a , b , c ) • just = o p i • convention: Cartesian orthonormal one on F 1 p = (3,-1) F 2 previous slide j o i • handy to specify others as needed F 2 p = (-1.5,2) j i • airplane nose, looking over your shoulder, ... o F 3 F 3 p = (1,2) • really common ones given names in CG • object, world, camera, screen, ... 31 32

Lines Implicit Functions • slope-intercept form • find where function is 0 f y f(x,y)=0 • y = mx + b • plug in (x,y), check if • implicit form • 0: on line f(x,y)=0 y=b • < 0: inside • y – mx – b = 0 y x x=a x • > 0: outside • Ax + By + C = 0 f(x,y) = y - mx - b • analogy: terrain y • f(x,y) = 0 m = -b/a • sea level: f=0 f(x,y)=0 • altitude: function value x • topo map: equal-value contours (level sets) 33 34 Implicit Circles Parametric Curves 2 2 2 f ( x , y ) ( x x ) ( y y ) r • = − + − − c c • parameter: index that changes continuously • circle is points (x,y) where f(x,y) = 0 • (x,y): point on curve x g ( t ) & # & # 2 = • p ( x , y ), c ( x , y ) : ( p c ) ( p c ) r 0 = = − • − − = • t: parameter $ ! $ ! c c y h ( t ) % " % " • points p on circle have property that vector • vector form from c to p dotted with itself has value r 2 p f ( t ) • = 2 2 • p c r 0 − − = • points points p on the circle have property that squared distance from c to p is r 2 p c r 0 • − − = • points p on circle are those a distance r from center point c 35 36