Geometric Transformations

Transcription

1 CS INTRODUCTION TO COMPUTER GRAPHICS Geometric Transformations D and D Andries an Dam 9/9/7 /46

2 CS INTRODUCTION TO COMPUTER GRAPHICS How do we use Geometric Transformations? (/) Objects in a scene at the lowest leel are a collection of ertices These objects hae location, orientation, size Correspond to transformations: Translation (T), Rotation (R), and Scaling (S) Andries an Dam 9/9/7 /46

3 CS INTRODUCTION TO COMPUTER GRAPHICS How do we use Geometric Transformations? (/) A scene has a camera/iew point from which the scene is iewed The camera has some location and some orientation in D-space These correspond to Translation and Rotation transformations Need other tpes of iewing transformations as well we ll learn about them soon! Andries an Dam 9/9/7 /46

4 CS INTRODUCTION TO COMPUTER GRAPHICS Some Linear Algebra Concepts... D coordinate geometr Vectors in D space and D space Dot product and cross product definitions and uses Vector and matri notation and algebra Identit matri Multiplicatie associatiit for matrices E.g. A(BC) = (AB)C Matri transpose and inerse definition, use, and calculation Homogeneous coordinates (,, z, w) You will need to understand these concepts! Linear algebra help session notes: Andries an Dam 9/9/7 4/46

5 CS INTRODUCTION TO COMPUTER GRAPHICS Linear Transformations (/4) We represent ectors as bold-italic letters () and scalars as italic letters (c) Vector addition: head to tail, add components Multipling a ector b a scalar changes the ector s magnitude Recall that a basis for a ector space is a set of ectors with the following properties:. The ectors in the set are linearl independent. An ector in the ector space can be epressed as a linear combination of the basis ectors: V = c V + c V a b = a + b Andries an Dam 9/9/7 5/46

6 CS INTRODUCTION TO COMPUTER GRAPHICS Linear Transformations (/4) Definition of a linear function/map f - straight line in D, hperplane in n-d Vector addition: f(+w) = f() + f(w) for all and w in the domain of f Scalar multiplication: f(c) = cf() for all scalars c and elements in the domain Linear functions can be epressed as (a +b, c +d ) Eample: f() = f(, ) := ( +, - +4 ) Now let and w be two elements in the domain of f f(+w) = f( +w, +w ) // b the rule of ector addition = (( +w )+( +w ), -( +w )+4( +w )) = ( +, - +4 ) + (w +w, -w +4w ) = f() + f(w) Let = e, w = e We can check the second propert the same wa Andries an Dam 9/9/7 6/46 e f(e ) f(e +e ) e +e e f(e )

7 CS INTRODUCTION TO COMPUTER GRAPHICS Linear Transformations (/4) Linear functions can be represented as matrices! The function from the preious slide, f() = f(, ) := ( +, - +4 ) can be represented as: f() = M, where M = 4 and = Properties of linear functions: Leaes origin inariant Maps parallelograms to (possibl distorted) parallelograms Let M be a matri representing a linear transformation. If M is inertible, then there is a sequence of rotations, scales, and/or shears that can perform the mapping represented b that linear transformation Andries an Dam 9/9/7 7/46

8 CS INTRODUCTION TO COMPUTER GRAPHICS Linear Transformations (4/4) Graphical use: transformations of points around the origin (leaes the origin inariant) These include Scaling and Rotations Translation is not a linear function (moes the origin) An linear transformation of a point will result in another point in the same coordinate sstem, transformed about the origin Aside: How do we know the origin is inariant from the definition of linearit? Andries an Dam 9/9/7 8/46

9 CS INTRODUCTION TO COMPUTER GRAPHICS Andries an Dam 9/9/7 9/46 Linear Transformations as Matrices (/) Linear transformations can be represented as inertible (non-singular) matrices Let s start with D transformations. These can be represented b matrices: A transformation of an arbitrar column ector = has the form: T = a b c d é ë ê ù û ú d c b a d c b a T

10 CS INTRODUCTION TO COMPUTER GRAPHICS Andries an Dam 9/9/7 /46 Linear Transformations as Matrices (/) Let e and e be the standard basis ectors: Now substitute each basis ector for to get: Notice that the columns of the matri representation of our transformation matri T are precisel T applied to e and e : This gies us a strateg for deriing transformation matrices! We can derie the columns of a transformation matri one b one b considering how our desired transformation affects each of the standard unit ectors c a d c b a T d b d c b a T, e e T(e ) = a c é ë ê ù û ú, T(e ) = b d é ë ê ù û ú

11 CS INTRODUCTION TO COMPUTER GRAPHICS Andries an Dam 9/9/7 /46 Scaling in D (/) Side effect: House shifts position relatie to origin Scale b, b (S =, S = ) = (original erte); = (new erte) = S Derie S b determining how e and e should be transformed (scale in X b S ) (scale in Y b S ) Thus we obtain ' ' s s S 6 9 é ë ê ù û ú

12 CS INTRODUCTION TO COMPUTER GRAPHICS Scaling in D (/) S is a diagonal matri; we can quickl check using matri multiplication that our deriation is correct: S s s s s ' ' S multiplies each coordinate of b the appropriate scaling factor, as epected In general, the i th entr of D, where D is diagonal, is (D[i,i] * [i]) Properties of scaling to look out for: Does not presere angles between lines in the plane (ecept when scaling is uniform, i.e. s = s ) If the object doesn t start at the origin, scaling will moe it closer to or farther from the origin (often not desired) Andries an Dam 9/9/7 /46

13 CS INTRODUCTION TO COMPUTER GRAPHICS Rotation in D (/) Rotate b θ about the origin = R θ, where ' ' = (original erte) = (new erte) Derie R θ b determining how e and e should be transformed: é e = ù é ê ú cosq ù ê ú ë û ë sinq û (first column of R θ ) é e = ù é ê ú -sinq ù ê ú(second column of R θ ) ë û ë cosq û é Þ R q = ê ë cosq -sinq ù ú sinq cosq û Andries an Dam 9/9/7 /46

14 CS INTRODUCTION TO COMPUTER GRAPHICS Rotation in D (/) Let s tr matri-ector multiplication R θ = cos sin sin cos cos sin sin cos ' ' ' '= cosq - sinq '= sinq + cosq Other properties of rotation: Preseres lengths in objects and angles between parts of objects (rigid-bod rotation) For objects not centered at the origin, an unwanted translation might be introduced (rotation is alwas about the origin) Andries an Dam 9/9/7 4/46

15 CS INTRODUCTION TO COMPUTER GRAPHICS What about translation? Translation is not a linear transformation (the origin is not inariant) Therefore, it can t be represented as a inertible matri Question: Is there another solution? Answer: Yes, = + t, where t = d d Howeer, using ector addition is not consistent with our method of treating transformations as matrices If we could treat all transformations in a consistent manner, i.e., with matri representation, then could combine transformations b composing their matrices Let s tr using a matri again How? Homogeneous Coordinates: add an additional dimension, the w- ais, and an etra coordinate, the w- component Thus D -> D (effectiel the hperspace for embedding D space) Andries an Dam 9/9/7 5/46

16 CS INTRODUCTION TO COMPUTER GRAPHICS Homogeneous Coordinates (/) Allow epression of all three D transformations as matrices We start with the point P d on the plane and appl a mapping to bring it to the w- plane in the hperspace P d (,) P h (w, w, w), w The resulting (, ) coordinates in our new point P h are different from the original (,), since = w, = w P h (,, w), w Andries an Dam 9/9/7 6/46

17 CS INTRODUCTION TO COMPUTER GRAPHICS Homogeneous Coordinates (/) Once we hae this point, we can appl a homogenized ersion of our T, R and S transformation matrices (net slides) to get a new point in the hperspace Finall, we want to obtain the corresponding point in D-space, so perform the inerse of the preious mapping (diide all entries b w) The erte = is now represented as = Andries an Dam 9/9/7 7/46

18 CS INTRODUCTION TO COMPUTER GRAPHICS Andries an Dam 9/9/7 8/46 Homogeneous Coordinates (/) The transformations we use will alwas map points in the hperplane defined b w = to other such points. (That wa, we don t hae to diide b w to get our equialent point in D) In other words, we want our transformations T to map points = to points = How do we achiee this with the matrices we hae alread deried? For linear transformations (i.e. scaling and rotation), embed the eisting matri in the upper-left of a new matri: ' ' d c b a

19 CS INTRODUCTION TO COMPUTER GRAPHICS Andries an Dam 9/9/7 9/46 Back to Translation Our translation matri (T) can now be represented b embedding the translation ector in the right column: To erif that this is the right matri, multipl it b our homogenized point: Coordinates hae been translated, and is still homogeneous d d T T ' d d d d

20 CS INTRODUCTION TO COMPUTER GRAPHICS Transformations Homogenized Let s homogenize our all matrices! Doesn t affect linearit of scaling and rotation Our new transformation matrices look like this Transformation Scaling Rotation Translation Matri s s Note: These transformations are called affine transformations, which means the presere ratios of distances between points on a straight line (but not necessaril (, ) ) é ê ê ê ëê ù ú ú ú ûú é cosq -sinq ê ê sinq cosq ê ë é ê ê ê ë d d ù ú ú ú û ù ú ú ú û Andries an Dam 9/9/7 /46

21 CS INTRODUCTION TO COMPUTER GRAPHICS Eamples Scaling: Scale b 5 in the direction, 7 in the 5 7 Rotation: Rotate b o cos() sin() sin() cos() Translation: Translate b -6 in the direction, +8 in the é ê ê ê ë -6 8 ù ú ú ú û Andries an Dam 9/9/7 /46

22 CS INTRODUCTION TO COMPUTER GRAPHICS Before we continue! Vectors s. Points Up until now, we e onl used the notion of a point in our D space We now present a distinction between points and ectors We used homogeneous coordinates to more conenientl represent translation; hence points are represented as (,, ) T A ector can be rotated/scaled, but not translated (can think of it as alwas starting at origin), so don t use the homogeneous coordinate: (,, ) T That wa, the translation matri won t hae an affect on our ectors. For now, let s focus on just our points (tpicall ertices) Andries an Dam 9/9/7 /46

23 CS INTRODUCTION TO COMPUTER GRAPHICS Inerses When we want to undo a transformation, we ll need to find the matri s inerse. Thanks to homogenization, the are all inertible! Transformation Matri Inerse Does it make sense? Scaling If ou scale something b factor a, the inerse is scaling b /a Rotation Translation Inerse of rotation b θ is rotation b θ. The properties sin(-θ) = -sin(θ) and cos(-θ) = cos(θ) gie this matri. Also, the matri is orthonormal, so inerse is just the transpose (see net slide). If ou translate b, the inerse is translation b - Andries an Dam 9/9/7 /46

24 CS INTRODUCTION TO COMPUTER GRAPHICS Andries an Dam 9/9/7 4/46 A moment of appreciation for linear algebra The inerse of a rotation matri M is just its transpose M T! That s reall conenient, so let s understand how it works using orthonormal matrices Take a rotation matri M = [ ] (where each i is a ector) First note some properties of M The columns are orthogonal to each other: i j = (i j) Columns hae unit length: i = Let s see what multipling M T and M produces: Using the properties aboe, we see that this is the identit matri, so M T = M - z z z z z z

25 CS INTRODUCTION TO COMPUTER GRAPHICS More with Homogeneous Coordinates Some uses we ll see later: Placing sub-objects in parent s coordinate sstem to construct hierarchical scene graph Transforming primities in their own coordinate sstems View olume normalization Mapping arbitrar iew olume into canonical iew olume along z-ais Parallel (orthographic, oblique) and perspectie projections Perspectie transformation (turn iewing pramid into a cuboid to turn perspectie projection into parallel projection) after perspectie foreshortening Andries an Dam 9/9/7 5/46

26 CS INTRODUCTION TO COMPUTER GRAPHICS Composition of Transformations (D) (/) We now hae a number of tools at our disposal; we can combine them! An object in a scene uses man transformations in sequence. How do we represent this in terms of functions? Transformation is a function; b associatiit, we can compose functions: (f o g)( i ) This is the same as first appling g, then appling f: f( g( i ) ) Consider our functions f and g as matrices (M and M ) and our input as a ector Our composition is equialent to M M Andries an Dam 9/9/7 6/46

27 CS INTRODUCTION TO COMPUTER GRAPHICS Composition of Transformations (D) (/) We can now form compositions of transformation matrices to form a more comple transformation For eample, TRS, which scales a point, then rotates it, then translates it: dcos sin s d sin cos s Note that we appl the matrices in sequence right to left. We can use associatiit to compose them first; it is often useful to be able to appl a single matri if, for eample, we want to use it to transform man points at once. Important: order matters! Matri multiplication is NOT commutatie. Be sure to check out the Transformation Game at applets/transformationgame/transformation_game_guide.html Let s see an eample Andries an Dam 9/9/7 7/46

28 CS INTRODUCTION TO COMPUTER GRAPHICS Not commutatie 6 Translate b = 6, =, then rotate b 45 o Rotate b 45 o, then translate b = 6, = Y X Andries an Dam 9/9/7 8/46

29 CS INTRODUCTION TO COMPUTER GRAPHICS Andries an Dam Start: Goal: Important concept: make the problem simpler Translate object to origin first, scale, rotate, and translate back: Appl to all ertices 9/9/7 9/46 Composition (an eample) (D) (/) 4 4 cos9 sin 9 sin 9 cos9 RST T -Rotate 9 o -Uniform scale 4 -Both around object s center, not the origin

30 CS INTRODUCTION TO COMPUTER GRAPHICS Andries an Dam 9/9/7 /46 Composition (an eample) (D) (/) T - RST But what if we mied up the order? Let s tr RT - ST: Oops! We scaled properl, but when we rotated the object, it s center was not at the origin, so its position was shifted. Order matters! 4 4 cos9 sin 9 sin 9 cos9

31 CS INTRODUCTION TO COMPUTER GRAPHICS Aside: Skewing/shearing Skew an object to the side, like shearing a card deck b displacing each card relatie to the preious one What phsical situations mirror this behaior? Squares become parallelograms; -coordinates skew to right, stas the same Y Skew tan D non-homogeneous X Notice that the base of the house (at = ) remains horizontal but shifts right. Wh? tan Skew D homogeneous Andries an Dam 9/9/7 /46

32 CS INTRODUCTION TO COMPUTER GRAPHICS Andries an Dam 9/9/7 /46 Inerses Reisited What is the inerse of a sequence of transformations? (M M M n ) - = M n - M n- - M - Inerse of a sequence of transformations is the composition of the inerses of each transformation in reerse order (wh?) Sa we want to do the opposite transformation of the eample on slide 7 What will our sequence look like? (T - RST) - = T - S - R - T We still translate to the origin first, then translate back at the end! cos9 sin 9 sin 9 cos9 / / T R S T - - -

33 CS INTRODUCTION TO COMPUTER GRAPHICS Andries an Dam Windowing transformation maps contents of D clip rectangle ( window ) to a iewport rectangle on the screen, e.g., interior canas ( client area ) of a window manager s window; also called window-to-iewport transformation Sends rectangle with bounding coordinates (u, ), (u, ) to (, ), (, ) The transformation matri here is: 9/9/7 /46 Aside: Windowing Transformations (CG terminolog) ) ) /( ( ) ) /( ( ) ) /( ( ) ) /( ( u u u u u u

34 CS INTRODUCTION TO COMPUTER GRAPHICS Aside: Transforming Coordinate Aes We understand linear transformations as changing the position of ertices relatie to the standard aes Can also think of transforming the coordinate aes themseles Rotation Scaling Translation Just as in matri composition, be careful of which order ou modif our coordinate sstem! Andries an Dam 9/9/7 4/46

35 CS INTRODUCTION TO COMPUTER GRAPHICS Dimension++ (D!) How should we treat geometric transformations in D? Just add one more coordinate/ais! A point is represented as z A matri for a linear transformation T can be represented as [ T(e ) T(e ) T(e )] where e is the standard basis ector along the z-ais, But remember to use homogeneous coordinates! Embed scale and rotation matrices as upper left submatrices and translation ectors as upper right subectors of the right column Andries an Dam 9/9/7 5/46

36 CS INTRODUCTION TO COMPUTER GRAPHICS Transformations in D Transformation Matri Comments Scaling s s Looks just like the D ersion. We just added an s z term. Rotation (see net slide) In D, onl one ais of rotation; now there are infinitel man! Must take all into account. See net slide... s z Translation d d dz Similar to the D ersion, just with one more entr dz, representing change in the z- direction. Andries an Dam 9/9/7 6/46

37 CS INTRODUCTION TO COMPUTER GRAPHICS Rodrigues s Formula Rotation b angle θ around ector u = [u u u z ] T Note: this is an arbitrar unit ector u in z-space Here s a not so friendl-looking rotation matri u u cos u u u z ( cos ) ( cos ) u ( cos ) u z sin sin u u ( cos ) u cos u ( cos ) ( cos ) u sin sin This is called the coordinate form of Rodrigues s formula Let s tr a different approach u u z z u u u z ( cos ) u sin uz ( cos ) u sin cos u ( cos ) z Andries an Dam 9/9/7 7/46

38 CS INTRODUCTION TO COMPUTER GRAPHICS Rotating ais b ais (/) Eer rotation can be represented as the composition of different angles of counter-clockwise rotation around aes, namel ais in the z plane b ψ; ais in the z plane b θ; z ais in the plane b ϕ Also known as Euler angles, make problem of rotation much easier R z (ψ): rotation about ais b ψ é ê ê ê ëê cos -sin sin cos ù ú ú ú ûú R z (θ): rotation about ais b θ é cosq sinq ù ê ú ê ú ê -sinq cosq ú ê ë ú û R (ϕ): rotation about z ais b ϕ é cosf -sinf ù ê ú ê sinf cosf ú ê ú ëê ûú Note these differ onl in how the submatri is embedded in the homogeneous matri, but the row-column order is different for R z You can compose these matrices to form a composite rotation matri Andries an Dam 9/9/7 8/46

39 CS INTRODUCTION TO COMPUTER GRAPHICS Rotating ais b ais (/) It would still be difficult to find the angles to rotate b, gien arbitrar ais u and specified angle ψ Solution? Make the problem easier b mapping u to one of the principal aes Step : Find a θ to rotate around ais to put u in the plane Step : Then find a ϕ to rotate around z the z ais to align u with the ais (Now that u is in a conenient alignment, we can do our transformation rotation for erte!) Step : Rotate b ψ around ais (which is coincident with u ais) Step 4: Finall, undo the alignment rotations (inerse). The onl rotation we e presered is the one around ais u b ψ, which was our goal Rotation matri: M = R - z (θ)r - (ϕ)r z (ψ)r (ϕ)r z (θ) ϕ θ u u ψ Andries an Dam 9/9/7 9/46

40 CS INTRODUCTION TO COMPUTER GRAPHICS Andries an Dam Inerses are once again parallel to their D ersions Composition works eactl the same wa 9/9/7 4/46 Inerses and Composition in D! Transformation Inerse Matri Scaling Rotation R z - (ψ) R z - (θ) R - (ϕ) Translation / / / z s s s dz d d cos sin sin cos, cos sin sin cos, cos sin sin cos

41 CS INTRODUCTION TO COMPUTER GRAPHICS Andries an Dam 9/9/7 4/46 Eample in D! Let s take some D object, sa a cube centered at (,,) Rotate clockwise in object s space b o around ais, 6 o around, and 9 o around z Scale in object space b in the, in the, and in the z Translate b (,,4) in world space Transformation sequence: TT - S R R z R z T o, where T translates to (,): cos sin sin cos cos6 sin 6 sin 6 cos6 cos9 sin 9 sin 9 cos9 4 T T - S R R z R z T

42 CS INTRODUCTION TO COMPUTER GRAPHICS Transformations and the scene graph (/5) Objects are tpicall composites: Scene Graph ROBOT upper bod lower bod stanchion base head trunk arm arm D scenes are often stored in a directed acclic graph (DAG) called a scene graph WPF (Windows Presentation Foundation) Open Scene Graph (used in the Cae) XD (VRML was a precursor to XD) most game engines Tpical scene graph format: objects (cubes, sphere, cone, polhedra etc.): stored as nodes (default: unit size at origin) attributes (color, teture map, etc.): stored as separate nodes Transformations: also nodes Eample scenegraph from a game engine Andries an Dam 9/9/7 4/46

43 CS INTRODUCTION TO COMPUTER GRAPHICS Transformations and the scene graph (/5) For our assignments use simplified format: Attributes stored as a components of each object node (no separate attribute node) A transform node affects its subtree Onl leaf nodes are graphical objects. All internal nodes that are not transform nodes are object group nodes Andries an Dam 9/9/7 4/46

44 CS INTRODUCTION TO COMPUTER GRAPHICS Transformations and the scene graph (/5) Step : Various transformations are applied to each of the leaes (object primities head, base, etc.) Step : Transformations are then applied to groups of objects (form upper and lower bod, etc ) Represents a transformation This format means that instead of designing new primities for eer single shape we need, we can just appl transformations to a smaller set of primities to form comple composite D shapes. Together the aboe hierarch of transformations forms the robot scene as a whole Andries an Dam 9/9/7 44/46

45 CS INTRODUCTION TO COMPUTER GRAPHICS Transformations and the scene graph (4/5) A cumulatie transformation matri (CTM) builds as ou moe up the tree. Note that higher leel transformation matrices are appended to the front of the sequence Eample: M M M M 4 For object (o), CTM = M M 5 For o, CTM = M M For o, CTM = M M 4 M 5 For a erte in o, position in world coordinate sstem is CTM = (M M 4 M 5 ) object nodes (geometr) transformation nodes group nodes Andries an Dam 9/9/7 45/46

46 CS INTRODUCTION TO COMPUTER GRAPHICS Transformations and the scene graph (5/5) You can easil reuse groups of objects (sub-trees in the scene graph) if the hae been defined alread This might occur if ou hae multiple similar components to our scene. For eample, the robot s arms Here, group has been used twice. Transformations defined within group itself do not change; there are different CTMs for each use of group as a whole obj root T group T T group obj group T T 4 obj group group T 5 T 6 obj4 T T s. T T T 4 Andries an Dam 9/9/7 46/46

47 CS INTRODUCTION TO COMPUTER GRAPHICS Further Reading More information can be found in the tetbook Read Chapters and for more detail and adanced topics on transformations Chapter 7 discusses the D/D coordinate space geometr Chapter goes into more detail about some of the linear algebra inoled in graphics Andries an Dam 9/9/7 47/46