CS 428: Fall Introduction to. Geometric Transformations. Andrew Nealen, Rutgers, /15/2010 1

Transcription

1 CS 428: Fall 21 Introduction to Comuter Grahics Geometric Transformations Andrew Nealen, Rutgers, 21 9/15/21 1

2 Toic overview Image formation and OenGL (last week) Modeling the image formation rocess OenGL rimitives, OenGL state machine Transformations and viewing Polygons and olygon meshes Programmable ielines Modeling and animation Rendering Andrew Nealen, Rutgers, 21 9/15/21 2

3 Toic overview Image formation and OenGL Transformations and viewing (next weeks) Linear algebra review, Homogeneous coordinates Geometric + rojective transformations Viewing, Vieworts, Cliing Polygons and olygon meshes Programmable ielines Modeling and animation Rendering Andrew Nealen, Rutgers, 21 9/15/21 3

4 Transformations in CG Secify lacement of objects in the world relative to the configuration in which they are defined Allow for reuse of objects in different laces, sizes Secify the camera osition Secify the camera model (rojection) Andrew Nealen, Rutgers, 21 9/15/21 4

5 Transformations in CG The where is secified by translations and rotations (= rigid body motions) Shae changes include For now we will only use linear deformations Linear algebra! Andrew Nealen, Rutgers, 21 9/15/21 5

6 Reresentations in CG Comutations should not deend on coordinate system (such as midoint/origin) Need careful accounting of oints and vectors 3 Both R (3 tules of floating oint values) Vectors Dislacements, velocities, directions, trajectories, surface normals, etc. Points Locations! Andrew Nealen, Rutgers, 21 9/15/21 6

7 Vector/oint oerations Vector + vector = vector Point + vector = oint Point + oint = invalid! Street address analogy Point oint = vector Works! Andrew Nealen, Rutgers, 21 9/15/21 7

8 Vector review [ + q] i = i + q i [s ] i = s i = sqrt[ ( i ) 2 ] addition scalar multilication length Andrew Nealen, Rutgers, 21 9/15/21 8

9 Vector review q = Σ i q i Normalization = ˆ dot roduct q cosθ θ q Andrew Nealen, Rutgers, 21 9/15/21 9

10 Perendicular vectors < vw, >= v = ( x, y ) v = ± ( y, x ) v v v v v In 2D only! v Andrew Nealen, Rutgers, 21 9/15/21 1

11 Linear combination + Basis Linear combination λ 1 v 1 + λ 2 v λ n v n with λ i R Linear indeendence of vectors v 1,, v n λ 1 v λ n v n = only when λ i = = λ n = Basisof n-dimensions is a set of n linearly indeendent vectors Every vector in R n has a unique set of λ s to reresent it Cartesian coordinates Andrew Nealen, Rutgers, 21 9/15/21 11

12 Inner (dot) roduct Defined for vectors: < v, w>= v w cos θ L θ w v L cosθ = w < v, w> L= v Projection of wonto v Andrew Nealen, Rutgers, 21 9/15/21 12

13 Distance between oint and line Pythagoras : q l (1) L + dist( q, q ) = q < q, v> (2) L= v dist( q, q ) = q L = < q, v> = v 2 2 q. 2 v q L 9/15/21 Andrew Nealen, Rutgers, 21 13

14 Reresentation of a lane in 3D sace A lane πis defined by a normal nand one oint in the lane. A oint q lane <q, n>= The normal nis erendicular to all vectors in the lane n q π Andrew Nealen, Rutgers, 21 9/15/21 14

15 Distance between oint and lane Geometric way: Project (q - )onto n! dist= = < q, n> n q n π Andrew Nealen, Rutgers, 21 9/15/21 15

16 Coordinates Connect drawing lane/sace with R 2 or R 3 Coordinate origin and axes are roblem secific Examle: orthogonal coordinates in the lower corner of this room Affine saces have No fixed origin No fixed axes (which is not the case in linear saces) Andrew Nealen, Rutgers, 21 9/15/21 16

17 Coordinates Affine sace An affine sace is a vector sace that's forgotten its origin John Baez In R 3, the origin, lines and lanes through the origin and the whole sace are linear oints, lines and lanes in general as well as the whole sace are the affine subsaces. Andrew Nealen, Rutgers, 21 Linear subsace Affine subsace 9/15/21 17

18 Primitives Points Andrew Nealen, Rutgers, 21 9/15/21 18

19 Primitives Lines Andrew Nealen, Rutgers, 21 9/15/21 19

20 Primitives Triangles Andrew Nealen, Rutgers, 21 9/15/21 2

21 Primitives Shaes Andrew Nealen, Rutgers, 21 9/15/21 21

22 Primitives Shaes are tessellated Andrew Nealen, Rutgers, 21 9/15/21 22

23 Primitives Positioning Absolute coordinates? Andrew Nealen, Rutgers, 21 9/15/21 23

24 Primitives Positioning Transformation + relative coordinates Translation Rotation Scaling Shearing Affine mas / Transformations! Andrew Nealen, Rutgers, 21 9/15/21 24

25 The set v n n v= λi vi, mit λi = 1 i= i= V Affine mas Affine combinations is an affine combination of vectors v i (or of oints i ). 1 v v 1 v Andrew Nealen, Rutgers, 21 9/15/21 25

26 Affine mas Barycentric coordinates Given and affine sace Awith coordinate system B={, n } For a oint i 1with = i= λ the λ i are known as barycentriccoordinates Physical interretation: n n i= λ = 1 Points i have mass λ i istthe centroid(= center of mass) Andrew Nealen, Rutgers, 21 1 λ +λ 1 1= λ = i 9/15/21 26 λ =

27 1 2 ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ,,,,,,,, A A A A = = λ λ Affine mas Barycentric coordinates ( ) ( ) ( ) ( ) ( ) ( ) ,,,,,, A A A = λ = λ λ λ ( ) ( ) ( ) ( ) ,, A = 9/15/21 27 Andrew Nealen, Rutgers, 21

28 The set co { } Affine mas Convex hull n n = =,..., n λi i, λi = 1, undλi, i=,..., n i= i= is the convex hull co{,..., n } of oints,..., n The convex hull contains all convex combinations of the oints Convex combinations = 5 affine combinations /w barycentric coordinates 2 greater/equal to zero Andrew Nealen, Rutgers, 21 9/15/21 28

29 A ma Φ: R n R m is affine Affine mas as linear mas when Φcan be reresented as Φ(v)=A(v)+b where Ais a linear ma and b R m Affine mas have a linear art (multilication) and a translation (additive) x' y' z' a = a a 1 2 a a a a a a Linear transformation x y + z x y z Translation Andrew Nealen, Rutgers, 21 9/15/21 29

30 Affine transformations Preserve arallel lines lines lines, lanes lanes Might not reserve length and angles But do reserve relative length along lines If they do reserve length and angles then the transformation is an isometry Affine = linear + translation Andrew Nealen, Rutgers, 21 9/15/21 3

31 Affine mas as linear mas Leads to the use of rojective geometry 2D oints andvectors reresented as (x, y, w) homogeneous coordinates w = 1 w = oint vector Point (,, ) not allowed, so domain R 3 {(,, )} If w (,1] then (x, y, w) (x/w, y/w, 1) A oint Andrew Nealen, Rutgers, 21 9/15/21 31

32 What is w? 2D case! A kind of a tye Points + oints at infinity Points at infinity are not affected by translation Infinite # of oints corresond to (x, y, 1) {(tx, ty, t) t } Line through origin {origin} Andrew Nealen, Rutgers, 21 9/15/21 32

33 Homogeneous coordinates Works nicely for oints and vectors Adding and scaling works too More in rojections, where w [,1] Andrew Nealen, Rutgers, 21 9/15/21 33

34 Linear transformation Purely linear transformation Origin does not move New coordinate axes are lin. comb. of old ones Andrew Nealen, Rutgers, 21 9/15/21 34

35 Linear transformation Purely linear transformation Andrew Nealen, Rutgers, 21 9/15/21 35

36 Affine transformation as a linear transformation + translation in n dimensions Origin moves translation Andrew Nealen, Rutgers, 21 9/15/21 36

37 Origin moves translation Affine transformation as a linear transformation in n+1 dimensions Andrew Nealen, Rutgers, 21 9/15/21 37

38 What is so great about this? Easy to imlement Checks for errors in the imlementation Can always check the w coordinate to make sure that oints and vectors remain unchanged Unified reresentationfor linear + translation Can comose many transformations into a single matrix through concatenation M = M rot M scale M translate Andrew Nealen, Rutgers, 21 9/15/21 38