6/10/2015©Zachary Wartell Points, Vectors, Alignments, Affine Coordinate Systems and Affine Transformations Textbook: Section 5

6/10/2015©Zachary Wartell Geometric Point Point – a location in space

6/10/2015©Zachary Wartell Alignment Alignment – a set of parallel lines have a common alignment Alignment is “North/South” Alignment is “East/West”

6/10/2015©Zachary Wartell Geometric Vector 3 different arrows but all representative of the same vector 3 different lines but all the same “alignment” Vector – a direction with magnitude in space -not really an arrow, but rather a set of arrows -more information than an “alignment” Alignment is N/S Direction is N Vector is N at “speed” l l

6/10/2015©Zachary Wartell Please draw point (5,3) ?

6/10/2015©Zachary Wartell Point (5,3) A (5,3) A

6/10/2015©Zachary Wartell Point (5,3) B (5,3) A B

6/10/2015©Zachary Wartell (5,3) Point (5,3) C (5,3) B C

6/10/2015©Zachary Wartell Point (5,3) D (5,3) B C D

6/10/2015©Zachary Wartell Please draw vector (3,2) ?

6/10/2015©Zachary Wartell Vector (3,2) A (3,2) A

6/10/2015©Zachary Wartell Vector (3,2) B (3,2) …. etc ….. B A

6/10/2015©Zachary Wartell Alignments (1 k,2 k ) and (0 k,1 k ) al 0 : (1k,2k),  k ∈ R all lines with slope ½ have alignment al 0 al 1 : (0k,1k),  k ∈ R slope=∞

6/10/2015©Zachary Wartell Arrows versus Vectors and Alignments v: (1,3) a 0 : ((0,0),(1,3)) a 1 : ((2,1),(3,4)) a 3 : ((-2,-1),(-1,2)) al 0 : (1k,2k),  k ∈ R all lines with slope ½ have alignment al 0

6/10/2015©Zachary Wartell Affine Transformations ● translate (T) – preserve area, length, angles, orientation ● rotate (R) – preserve area, length, angles ● scale (S) – preserve perpendicularity ● rigid-body/congruency – (T · R) preserve area, length, angles ● uniform scale/”similarity” – preserve angles, orientation ● generally all affine transformations: ●preserve lines ●preserve parallelism ● preserve distance ratio (→equal spacing) ●map points to points & vectors to vectors Start:

6/10/2015©Zachary Wartell ' Preserve Lines l l'l' M transforming entire grid l: p = p 0 + t ( p 1 - p 0 ) → l': M(p) =M( p 0 ) + t (M( p 1 )-M( p 0 )) p0p0 p1p1 p p0p0 p p1p1 ' '

6/10/2015©Zachary Wartell Parallelism lala lblb la'la' lb'lb' M transforming entire grid l a || l b → M(l a ) || M(l b ) = l' a || l' b

6/10/2015©Zachary Wartell Distance Ratios l l'l' M transforming entire grid ab/bc = a'b'/b'c' a b c a'a' b'b' c'