CS 450: COMPUTER GRAPHICS 2D TRANSFORMATIONS SPRING 2016 DR. MICHAEL J. REALE

Transcription

1 CS 45: COMUTER GRAHICS 2D TRANSFORMATIONS SRING 26 DR. MICHAEL J. REALE

2 INTRODUCTION Now that we hae some linear algebra under our resectie belts, we can start ug it in grahics! So far, for each rimitie, we hae a descrition that includes the ositions of the ertices model coordinates Eamles: A square with ertices (-,-), (,-), (,), (-,) A teaot with its ertices relatie to its bottom We would like to be able to lace this rimitie in the world whereer we want (as well as rotate it and/or scale it) use a model transform Recall: Model coordinates (relatie to model) MODEL TRANSFORM World coordinates We would also like to be able to get the world coordinates relatie to the camera use iew transform Recall: World coordinates VIEW TRANSFORM Camera (Ee) Coordinates

3 TRANSFORMATIONS Transformations or transforms = oerations that change osition, orientation, and/or size of geometric object Use matrices to erform transformations on ectors/oints Eamles: model transforms, the iew transform, rojection transform, etc.

4 OVERVIEW Three most basic transformations: Rotation Scaling (i.e., resizing the object) Translation (i.e., moing/lacing the object) Other transformations: shear, reflection, etc. We re going to start with 2D transformations first Then, later, we ll moe on to 3D transformations Kee in mind, we will be building simle transformations (with certain imlicit assumtions) Howeer, we will combine these to make more owerful transformations

5 ROTATION

6 ROTATION: INTRODUCTION To rotate an object, we need: Rotation angle how much to rotate Counterclockwise in lane we re rotating Rotation ais what we re rotating around In 2D, just use z ais WARNING: Rotation erformed around ORIGIN Origin (,) = rotation oint (or iot oint) (We ll talk later about how to rotate around an arbitrar rotation oint)

7 ROTATION: DERIVING Recall: ( A B) A B A B ( A B) A B A B = (,) original oint = (, ) transformed oint ϕ = original angle of oint (,) from ais θ = difference in angle between old and new oint So, our original oint (,) and transformed oint (, ) in olar coordinates are as follows r r r ( ) r ( ) r r r r After substitution, we can eress the transformed oint in terms of θ onl:

8 ROTATION MATRIX We hae: If our rotation matri transform is R, then: Therefore, our 2D rotation matri is: R R R

9 SCALING

10 SCALING: INTRODUCTION Scaling an object = altering the size of an object The scaling we will be doing here siml multiling each coordinate b a scaling factor: The corresonding scaling matri transformation s s s s

11 SCALING: FACTORS Scaling factor >. enlarge Scaling factor <. shrink Scaling factor < negatie scaling resizes AND reflects object Uniform scaling = scaling factors are all the same (e.g., s = s ) Otherwise, called differential scaling

12 SCALING: ASSUMTIONS WARNING: Because of the wa we are doing scaling: Onl scaling in X or Y direction (or both), but NOT in arbitrar direction! Scaling relatie to ORIGIN! ORIGIN = fied oint (oint unaffected b scaling) (We ll talk later about how to use a different fied oint)

13 TRANSLATION

14 TRANSLATION: INTRODUCTION Translation = moing a oint b a certain distance (t, t ) (t, t ) = translation distances = translation ector = shift ector If we re stuck with 22 matrices and 2 ectors, we hae to add ectors to erform a translation: t t t t t t T

15 TRANSLATION: ROBLEM At some oint, we would like to be able to combine multile transformations into a gle matri: This means we can multil all our transformations together first (M), and then al it to each oint we want HOWEVER, because translation is handled as addition, we need to comute intermediate stes: M A B C D E F A B C D E F ) ( T R ) (

16 HOMOGENEOUS COORDINATES

17 HOMOGENEOUS COORDINATES To fi this, we will etend our 22 matrices (and our 2 ectors) to 33 matrices (and to 3 ectors) Homogeneous coordinates = for 2D coordinates, etension to ( h, h, h) h = homogeneous arameter nonzero alue such that: Often just set h = (, ) becomes (,, ) Often use w instead of h (eseciall for 3D ectors (,,z,w) ) As we ll see, this allows us to reresent translation as a matri multilication! h h h h

18 TRANSLATION MATRIX WITH HOMOGENEOUS COORDINATES The 2D translation matri is sometimes reresented as T(t, t ) t t t t T

19 ROTATION MATRIX WITH HOMOGENEOUS COORDINATES The 2D rotation matri is sometimes reresented as R(θ)

20 SCALING MATRIX WITH HOMOGENEOUS COORDINATES The 2D scaling matri is sometimes reresented as S(s, s ) s s

21 ATTERN WITH HOMOGENEOUS COORDINATE MATRICES With the translation matri, we urosel use the additional elements of the matri (in this case, the etra column): T For rotation, scaling, and shear matrices (discussed later), the original matri is augmented with an etra row and column of zeros (ecet for the last (row,column) osition, which is set to ): M m m m m M t t m m m 2 m m m 2 m m m

22 HOMOGENEOUS COORDINATES: OINTS VS. VECTORS Recall: a ector can also be interreted as: Location (w = ) Direction (w = ) in sace Note: sometimes, location called oint and direction called ector Deending on how we want to interret the ector, we will set a different alue for w

23 HOMOGENEOUS COORDINATES: OINTS VS. VECTORS oints all transformations should hae an effect (translation, rotation, scaling, etc.) w set to Direction translation has no meaning (other transformation should work though) w set to t t t t T t t T

24 INVERSE TRANSFORMATIONS

25 INVERSE TRANSFORMATIONS Fortunatel, the inerses of the translation, rotation, and scaling matrices can be comuted directl: Aling an inerse transformation does the oosite transformation T - translate object (-t, -t ) s s S R t t T

26 QUICK ASIDE: INVERSE OF ROTATION MATRIX We comuted the inerse directl b ug the negatie angle (-θ) onl e was affected b this It turns out, an rotation matri is ORTHOGONAL inerse = transose = swaing rows and columns Also note: R = 2 θ + 2 θ = R R R T

27 CHANGING COORDINATE SYSTEMS

28 CHANGING 2D COORDINATE SYSTEMS Gien a oint, man times we want to get the coordinates of this oint relatie to some OTHER coordinate basis ectors and origin oint Eamle: iewing transform getting coordinates with resect to camera s iew (camera s basis ectors) and camera s starting oint (camera s origin) As with all our slides thus far, we ll stick to 2D for now. For a 2D coordinate sstem, we need: An origin oint (, ) Two aes and

29 CHANGING 2D COORDINATE SYSTEMS To change 2D coordinate sstems: Translate so that the new origin is at (,) T(-, - ) Rotate the ais onto the ais need to get angle θ between and R(-θ) Final transformation: R(-θ) T(-, - )

30 COMUTING NEW COORDINATE AXES If we onl know one ector V (and we want to oint in the same direction): Alternatiel, we might be gien (origin) and (endoint of ais): ), ( V V

31 CHANGING 2D COORDINATE AXES ONLY If our new coordinate aes are u = (u, u ) and = (, ), then we can construct the ROTATION matri directl: When multiling b a oint, will be rojected on each ais ( dotted with each ais): u u R u u u u u R

32 CHANGING 2D COORDINATE AXES ONLY To drie the oint home: ) ( 9 9 ) 9 ( 9 9 ) 9 ( u u R u u B A B A B A B A B A B A ) ( ) ( Recall:

33 UNDOING A 2D COORDINATE AXIS CHANGE To go back to the standard basis, we need to get the original coordinates back To get the coordinate * u + * * To get the coordinate * u + * * The matri form becomes: u u u u R

34 UNDOING A 2D COORDINATE AXIS CHANGE Note: If coordinate aes are orthonormal matri R is orthogonal R - = R T T u u R R u u R

35 COMOSITE TRANSFORMATIONS

36 COMOSITE TRANSFORMATIONS Comosite transformation matri = roduct of indiidual transformations rocess of forming one is called concatenation or comosition of matrices Eamle: al M then M 2 M = M 2 M NOTE THE ORDER! Must multil from RIGHT to LEFT!!! M 2 M M

37 WHY USE COMOSITE TRANSFORMATIONS? There are a LOT of different things ou can do with these; we ll talk about a few er common eamles: Rotation around an arbitrar iot oint Scaling around an arbitrar fied oint Scaling in an arbitrar direction

38 ROTATION AROUND AN ARBITRARY IVOT OINT Let s sa we want to rotate around an iot oint (, ) Basic idea: Translate (-, - ) (, ) is now at origin Rotate oints Translate back to (, ) ), ( ) ( ), ( T R T

39 SCALING AROUND AN ARBITRARY FIXED OINT Sa we want to scale an object relatie a fied oint (, ) Basic idea: Translate (-, - ) (, ) is now at origin Scale oints Translate back to (, ) ), ( ), ( ), ( s s T s s S T

40 SCALING IN AN ARBITRARY DIRECTION Let s assume our fied oint is alread at the origin Basic idea: Rotate our arbitrar ais u into the ais Scale along Rotate back If u is our ais, then our ais = = (-u, u ): ), ( u u s s u u R s s S R

41 ORDER MATTERS! What order ou al our matrices will affect what transformations ou erform!!! Eamle: rotation then scale s. scale then rotation First transformation RIGHT-most matri when multiling!

42 RIGID-BODY TRANSFORMATIONS

43 RIGID-BODY TRANSFORMATIONS Rigid-bod transformation = onl includes translation and/or rotation All angles and distances between coordinate ositions resered Intuitiel, object doesn t change shae onl changes osition and/or orientation Also called a rigid-motion transformation In 2D, has the general form: r r r r tr tr WARNING: (tr, tr )!= (t, t ) from earlier! (Deends on order of transformations) Uer-left 22 matri orthonormal ector set

44 OTHER TRANSFORMATIONS (REFLECTION AND SHEAR)

45 REFLECTION TRANSFORMATION Reflection transformation = transformation that roduces a mirror image of object In 2D 8 rotation about ais of reflection ais Reflection about line = ( ais): Reflection about line = ( ais): Reflection about origin:

46 REFLECTION ACROSS DIAGONALS Reflection across = : Reflection across = -:

47 SHEAR TRANSFORMATION Shear transformation = distorts shae of object like as if the object is comosed of laers sliding oer each other Shear in -direction: Shear in -direction: h h H h h H

48 SHEAR TRANSFORMATION WARNING: B default, assumes reference line is either the ais or the ais (for the -shear and - shear, resectiel)