2D and 3D Coordinate Systems and Transformations

Transcription

1 Graphics & Visualization Chapter 3 2D and 3D Coordinate Systems and Transformations Graphics & Visualization: Principles & Algorithms

2 Introduction In computer graphics is often necessary to change: the form of the objects the coordinate system Examples: In a model of a scene, the digitized form of a car may be used in several instances, positioned at various points & directions and in different sizes In animation, an object may be transformed from frame to frame As objects traverse the graphics pipeline, they change their coordinate system: object coordinates world coordinates world coordinates eye coordinates Coordinates transformations: - tools of change - the most important & classic topic in computer graphics 2

3 Introduction (2) Points in 3D Euclidean space E 3 : 3 1 column vectors (here) p p p p x y z Linear transformations: matrices - they are post-multiplied by a point to produce another point p m m m p p m m m p p x y z m m m p x y z 3

4 Introduction (3) If points were represented by 1 3 row vector p [ px py pz ] the linear transformations would be pre-multiplied by the point p p p p p p m m m x y z x y z m m m m m m Right-handed three-dimensional coordinate systems are used: 4

5 Affine Transformations - Combinations Mathematics: Transformations: mappings whose domain & range are the same set (e.g. E 3 to E 3 ) Computer graphics & visualization: Affine Transformations: transformations which preserve important geometric properties of the objects being transformed Affine transformations preserve affine combinations Examples of Affine combinations: line segments, convex polygons, triangles, tetrahedra the building blocks of our models 5

6 3 3 An affine combination of points p,p,...,p E is a point p E : where Affine Combinations a : affine coordinates of p with respect to, a1,..., a p n,p 1,...,pn i n a i Convex affine combination: 1 p i 1 n if all a i, i =,1,,n The affine combination p is within the convex hull of the original points p,...,p E.g.1: Line segment between points p 1 and p 2 is the set of points p: p = a 1 p 1 + a 2 p 2 with a 1 1 and a 2 = 1 a 1 E.g. 2: Triangle with vertices p 1,p 2,p 3 is the set of points p: n a i p = a 1 p 1 + a 2 p 2 + a 3 p 3 with a 1, a 2, a 3 1 and a 1 + a 2 + a 3 = 1 p i n 6

7 Affine transformation: transformation which preserves affine combinations it retains the inter-relationship of the points 3 3 A transformation : E E is affine if n where p a p : an affine combination The result of the application of an affine transformation onto the result p of an affine combination should equal the affine combination of the result of performing the affine transformation on the defining points with the same weights a i E.g. Affine Transformations i i i n ( p) a ( p ) i i i 7

8 Affine Transformations (2) Practical consequence: Internal points need not be transformed It suffices to transform the defining points E.g. to perform an affine transformation on a triangle: Transform its three vertices only, not its (infinite) interior points General affine transformation Mappings of the form ( p) A p t are affine transformations in E 3. (1) where A is a 3 3 matrix Proof: we shall show that (1) preserves affine combinations n n n n ( a p ) A( a p ) t a Ap a t i i i i i i i i i i i i n a( A p t) a ( p ). i i i n i i t is a 3 1 matrix 8

9 2D Affine Transformations 2D results generalize to 3D Four basic affine transformations: translation scaling rotation shearing 9

10 2D Translation Defines a movement by a certain distance in a certain direction, both specified by the translation vector T The translation of a 2D point p=[x,y] by a vector d [ d, ] T x dy gives another T point p'=[x',y'] : p p d Instantiation of ( ) where (I is the 2 2 identity matrix) p A p t and A I t d 1

11 2D Scaling Changes the size of objects Scaling factor: specifies the change in each direction 2D: s x & s y are the scaling factors which are multiplied by the respective coordinates of p=[x, y] T Scaling of 2D point p=[x, y] T to give p =[x, y ] T : p S( s, s ) p where S( s, s ) x y x y Instantiation of ( p) A p t where A S( s, s ) and t s x s x y y 11

12 2D Scaling (2) Not possible to observe the effect on a single point If scaling factor < 1 reduce the object s size scaling factor > 1 increase the object s size Translation side-effect (proportional to the scaling factor): The object has moved toward the origin of the x-axis (s x < 1) The object has moved away from the origin of the y-axis (s y > 1) Isotropic scaling: If all the scaling factors are equal (in 2D s x = s y ) Preserves the similarity of objects (angles) Mirroring: Special case, using a -1 scaling factor About x-axis: S(1,-1) and y-axis: S(-1,1) 12

13 2D Rotation Turns the objects about the origin The distance from the origin does not change, only the orientation changes Counterclockwise rotation is positive p [ x, y ] T p [ xy, ] T We can estimate from : x l cos( ) l(cos cos sin sin ) x cos y sin y l sin( ) l(cos sin sin cos ) x s in y cos Thus: p R( ) p 13

14 2D Rotation (2) where: R() cos sin sin cos Rotation is an instantiation of the general affine transformation ( p) A p t where A R() and t 14

15 2D Shear Increases one of the object s coordinates by an amount equal to the other coordinate times a shearing factor Physical example: cards placed flat on a table and then tilted by a hard book. 15

16 2D Shear (2) p [ x, y ] T p [ xy, ] T We can estimate from : shear along x axis x shear along y axis x x x, ay, y y bx y y where α and b are the respective shear factors In matrix form: shear on x axis shear on y axis p SH ( a) p p SH ( b) p x y, where, where SH x ( a) SH y () b 1 a 1 1 b 1 Shear is an instantiation of the general affine transformation ( p) A p t where A SH ( a) or A SH () b and t x y 16

17 Composite Transformations Useful transformations in computer graphics and visualization rarely consist of a single basic affine transformation All transformations must be applied to all objects of a scene Objects are defined by thousands or even millions of vertices EXAMPLE: Rotate a 2D object by 45 o and then isotropically scale it by a factor of 2 Apply the rotation matrix: R(45 ) Then apply the scaling matrix: S(2,2)

18 Composite Transformations (2) It is possible to apply the matrices sequentially to every vertex p: S(2, 2) (R(45 o ) p) NOT EFFICIENT Another way is to exploit the associative property of matrix multiplication and apply the pre-computed composite to the vertices: (S(2, 2) R(45 o ) )p MORE EFFICIENT Thus the composite transformation is only computed once and the composite matrix is applied to the vertices Matrix multiplication is not commutative the order of multiplying the transformation matrices is important Having chosen the column representation of points transformation matrices are right-multiplied by the points write the matrix composition in the reverse of the order of application 18

19 Composite Transformations (3) To apply the sequence of transformations T1,T2,...,Tm, we compute the composite matrix Tm... T2 T1 Problem with the translation transformation: Translation cannot be described by a linear transformation matrix such as: x ax by y cx dy Thus translation cannot be included in a composite transformation Solution to the problem homogeneous coordinates 19

20 Homogeneous Coordinates Homogeneous Coordinates use one additional dimension than the space we want to represent x 2D space: y, where w is the new coordinate that corresponds to w the extra dimension; w Fixing w=1 maintains our original dimensionality by taking slice w=1 In 2D we use the plane w=1 instead of the xy-plane 2

21 Homogeneous Coordinates (2) Points whose homogeneous coordinates are multiples of each other are equivalent: e.g., [1,2,3] T and [2,4,6] T represent the same point The actual point that they represent is given by their unique basic representation, which has w = 1 and is obtained by dividing all coordinates by w: [x/w, y/w, w/w] T = [x/w, y/w, 1] T In general, we use the basic representation for points, and we ensure that our transformations preserve this property 21

22 Homogeneous Coordinates (3) How do homogeneous coordinates treat the translation problem? Exploit the fact that points have w = 1, in order to represent the translation of a point p = [x, y, w] T by a vector d [ d, ], as a x dy linear transformation: x 1x y d w x d y x 1y d w y d w x y 1w 1 x y x y Transformation on the w-coordinate ensures that the resulting point p [ x, y, w ] T has w 1 22

23 Homogeneous Coordinates (4) The above linear expressions can be encapsulated in matrix form, thus treating translation in the same way as the other basic affine transformations In non-homogeneous transformations, the origin [, ] T is a fixed point: a c b d A positive effect of homogeneous coordinates is that there is no fixed point under homogeneous affine transformations 23

24 Homogeneous Coordinates (4) The 2D origin is now [,, 1] T which is not a fixed point The point [,, ] T is outside w=1 plane disallowed since it has w= From here on points will be represented by their homogeneous coordinates: 2D [x, y, 1] T 3D [x, y, z, 1] T 24

25 2D Homogeneous Affine Transformations 2D homogeneous translation matrix: Translation is treated like the other basic affine transformations: p T( d) p The last row of a homogeneous transformation matrix is always [,, 1] in order to preserve the unit value of the w-coordinate 2D homogeneous scaling matrix: 1 Td ( ) 1 d d 1 x y S( s, s ) s s x y y x 1 25

26 2D Homogeneous Affine Transformations (2) 2D homogeneous rotation matrix: 2D homogeneous shear matrices: cos sin R( ) sin cos 1 shear on x axis shear on y axis 1 a SH x ( a) SH y ( b) b

27 2D Homogeneous Inverse Transformations Often necessary to reverse a transformation 2D inverse homogeneous translation matrix: 1-1 T ( d) T( -d) 1 1 2D inverse homogeneous scaling matrix: -1 S 1 s d d x y ( sx, sy) S(, ) s s s x y y x 1 27

28 2D Homogeneous Inverse Transformations (2) 2D inverse homogeneous rotation matrix: -1 R 2D inverse homogeneous shear matrix: shear on x axis SH shear on y axis SH cos sin ( ) R( ) sin cos -1 x -1 y 1 1 a ( a) SH ( a) 1 x 1 1 ( b) SH ( b) b 1 y 1 28

29 2D Homogeneous Inverse Transformations Applying an object transformation on an object is equivalent to the application of the inverse transformation on the coordinate system (axis transformation) EXAMPLE: Isotropically scaling an object by a factor of 2 is equivalent to isotropically scaling the coordinate system axis by a factor of 1/2 (shrinking) 29

30 Properties of Homogeneous Matrices Some properties of homogeneous affine transformation matrices: T( d1) T( d2) T( d2) T( d1) T( d1 d2) S( s, s ) S( s, s ) S( s, s ) S( s, s ) S( s s, s s ) x1 y1 x2 y2 x2 y2 x1 y1 x1 x2 y1 y2 R( 1) R( 2) R( 2) R( 1) R( 1 2) S( s, s ) R( ) R( ) S( s, s ) for isotropic scaling only (s x =s y ) x y x y 3

31 2D Transformation Example 1 EXAMPLE 1: Rotation about an arbitrary point Determine the transformation matrix R(θ, p) required to perform rotation about an arbitrary point p by an angle θ SOLUTION Step 1: Translate by p, T( p) Step 2: Rotate by θ, R(θ) Translate by p, T( p) 1 p cos sin 1 p cos sin p p cos p sin x x x x y R(, p) 1 p sin cos 1 p sin cos p p sin p cos y y y x y

32 2D Transformation Example 2 EXAMPLE 2: Rotation of a triangle about a point Rotate the triangle a=[,]t, b=[1,1] T and c=[5,2] T SOLUTION abc by 45 o about the point p=[-1,-1] T, where The triangle can by represented by the matrix We shall apply R(θ, p) [Ex. 1] to the triangle: T R(45,[ 1, 1] ) T The rotated triangle is abcwhere a =[-1, 2 1] T, b =[-1, 2 2 1] T 3 9 and c =[ 2 1, 2 1 ] T 2 2 T 32

33 2D Transformation Example 3 EXAMPLE 3: Scaling about an arbitrary point Determine the transformation matrix S(s x,s y,p) required to perform scaling by s x and s y about an arbitrary point p SOLUTION Step 1: Translate by p, T( p) Step 2: Scale by s x and s y, S(s x,s y ) p, T( p) Step 3: Translate by to undo the initial translation 1 p s 1 p s p p s x x x x x x x S( s, s, p) 1 p s 1 p s p p s x y y y y y y y y

34 2D Transformation Example 4 EXAMPLE 4: Scaling of a triangle about a point Double the lengths of the sides of triangle keeping its vertex c fixed. The coordinates of its vertices are a=[,]t, b=[1,1] T, c=[5,2] T SOLUTION abc The triangle can by represented by the matrix T [Ex. 2] We shall apply the matrix S(s x,s y,p) [Ex. 3] to the triangle, setting the scaling factor equal to 2 and p=c T S(2,2,[5,2,1] ) T The scaled triangle is abc where a =[-5,-2] T, b =[-3,] T and c =[5,2] T 34

35 2D Transformation Example 5 EXAMPLE 5: Axis transformation Suppose that the coordinate system is translated by the vector v [ v, v ] T. Determine the matrix that describes this effect SOLUTION The required transformation matrix must produce the coordinates of the objects with respect to the new coordinate system. This is achieved by applying the inverse translation to the objects: x y 1 T( v) 1 v v 1 x y Similar argument holds for any other axis transformation. Its effect is encapsulated by applying the inverse transformation to the objects 35

36 2D Transformation Example 6 EXAMPLE 6: Mirroring about an arbitrary axis Determine the transformation matrix required to perform mirroring about an axis specified by a point p=[p x,p y ] T and a direction vector v [ v, ] T x vy SOLUTION Step 1: Translate by p, T( p) Step 2: Rotate by θ (negative as it is clockwise), R(-θ) θ forms between x-axis and vector sin v v y v 2 2 x y and: The two previous steps make the general axis coincide with the x-axis Step 3: Perform mirroring about the x-axis, S(1, -1) Step 4: Rotate by θ, R(θ) v cos v v x v 2 2 x y 36

37 2D Transformation Example 6 (2) Step 5: Translate by p, T( p) Summarizing the previous steps we have: M SYM 1 p cos sin 1 cos sin 1 x 1 p sin cos 1 sin cos 1 y cos sin 2sin cos px px(cos sin ) 2 py sin cos sin cos sin cos py py (sin cos ) 2 px sin cos 1 p p x y 37

38 2D Transformation Example 7 EXAMPLE 7: Mirror polygon Given a polygon, determine its mirror polygon with respect to (a) the line y=2 and (b) the axis specified by the point p=[,2] T and the vector The polygon is given by its vertices a=[-1,] T, b=[,-2] T, c=[1,] T and d=[,2] T SOLUTION The polygon can be represented by matrix In case (a) p=[,2] T and v [1,] T thus θ= ο, sinθ=, cosθ=1 and we have: MSYM Π Π v [1,1] T 38

39 2D Transformation Example 7 (2) v [1,1] T In case (b) p=[,2] T and thus sinθ = cosθ = 1 2 and we have: MSYM Π where M SYM is the matrix of [Ex. 6] 39

40 2D Transformation Example 8 EXAMPLE 8: Window-to-Viewport transformation The contents of a 2D window must be transferred to a 2D viewport. Both the window and viewport are rectangular parallelograms with sides parallel to the x- and y-axis. Determine the window to viewport transformation matrix. Also determine how objects are deformed by this transformation. SOLUTION Suppose that the window and the viewport are defined by two opposite T T T T vertices [ wxmin, wymin ],[ wxmax, w ymax ] and [ vxmin, vymin ],[ vxmax, vymax ] Step 1: Translate [ w, w ] T to the origin, using T( w ) where w min [ w, w ] T xmin ymin xmin ymin min 4

41 2D Transformation Example 8 (2) Step 2: Scale the window to the size of the viewport, using S(s x, s y ) where vxmax v v xmin ymax vymin sx sy w w w w Step 3: Translate to the minimum viewport vertex [ v, v ] T, using xmax xmin Tv where [ v, v ] T ( ) min v min xmin ymin Summarizing the previous steps we have: M T( v ) S( s, s ) T( w ) WV min x y min ymax ymin vxmax vxmin wxmax wxmin 1 vxmin 1 w vymax vymin 1 vymin 1 w wymax wymin xmin xmin ymin ymin 41

42 2D Transformation Example 8 (3) Finally : M WV v v v v vxmin wxmin w w w w xmax xmin xmax xmin xmax xmin xmax xmin v v v v v w w w w w ymax ymin ymax ymin ymin ymin ymax ymin ymax ymin 1 42

43 2D Transformation Example 8 (4) M WV contains non-isotropic scaling (s x s y ) objects will be deformed. A circle will become an ellipse and a square will become a rectangular parallelogram. The aspect ratios of the window and the viewport are defined as the ratios of their x- to their y-sizes: a w w w v v, av w w v v xmax xmin xmax xmin ymax ymin ymax ymin If a w a v then objects are deformed. To avoid this, is to use the largest part of the viewport with the same aspect ratio as the window. For example we can change the v xmax or the v ymax boundary of the viewport in the following manner: if (a v >a w ) then v xmax = v xmin + a w *(v ymax v ymin ) else if (a v <a w ) then ( vxmax vxmin ) vymax vymin a w 43

44 2D Transformation Example 9 EXAMPLE 9: Window-to-Viewport transformation instances Determine the window-to-viewport transformation from the window: [ w, w ] T [1,1] T,[ w, w ] T [3,5] T xmin ymin xmax ymax [ v, ] T [,] T,[, ] T [1,1] T xmin vymin vxmax vymax to the viewport : If there is deformation, how can it be corrected? SOLUTION Direct application of the M WV [Ex. 8] For the window and viewport pair gives 1 1 Now aw and av, so there is distortion since ( av aw) 2. It can 1 be corrected by reducing the size of the viewport by setting: 1 vxmax vxmin aw *( vymax vymin ) 2 44 M WV

45 2D Transformation Example 1 EXAMPLE 1: Tilted window-to-viewport transformation Suppose that the window is tilted and given by its four vertices a=[1,1] T, b=[5,3] T, c=[4,5] T and d=[,3] T. Determine the transformation TILT M WV SOLUTION that maps it to the viewport [ v, v ] [,],[ v, v ] [1,1] T T T T xmin ymin xmax ymax Step 1: Rotate the window by angle θ about a point a. For this we shall use matrix R(θ, p) [Ex. 1], instantiating it as R(-θ, a) where 1 2 sin cos 5 5 Step 2: Apply the window to viewport transformation M WV [Ex. 8] 45

46 Before Step 2 we must determine the maximum x- and y- coordinates of the rotated window by computing: Thus 2D Transformation Example 1 (2) c R(, a) c 1 5 T [ w, w ] a,[ w, w ] xmin ymin xmax ymax T 1 c, and we have: TILT MWV MWV R(, a)

47 3D Homogeneous Affine Transformations 3D homogeneous coordinates are similar to 2D Add an extra coordinate to create [x, y, z, w] T where w corresponds to the extra dimension Points whose homogeneous coordinates are multiples are equivalent e.g. [1, 2, 3, 2] T and [2, 4, 6, 4] T Basic representation of a point is unique has w=1 is obtained by dividing by w : [x/w, y/w, z/w, w/w] T = [x/w, y/w, z/w, 1] T, w Example: T T [,,, ] T [,,, ] [,1,,1] Obtain a 3D projection of 4D space by setting w=1 Points: 4 1 vectors. Transformations: 4 4 matrices 47

48 3D Homogeneous Translation Specified by a 3-dimensional vector Matrix form: 1 1 Td ( ) 1 d d d 1 x y z d [ d, d, d ] x y z T Td ( ) can be combined with other affine transformation matrices by matrix multiplication Inverse translation: T 1 ( d) T( d) 48

49 3D Homogeneous Scaling Three scaling factors: s x, s y, s z If scaling factor < 1 the object s size is reduced in the respective dimension scaling factor > 1 the object s size is increased Matrix form: S( s, s, s ) x y z s x s y s 1 z Scaling has a translation side-effect, proportional to the scaling factor 49

50 3D Homogeneous Scaling (2) Isotropic scaling: if s x = s y = s z preserves the similarity of objects (angles) Mirroring: about one of the major planes (xy, xz, yz) using a -1 scaling factor e.g. mirroring about the xy-plane is S(1, 1, -1) Inverse scaling: S ( sx, sy, sz ) S(,, ) s s s x y z 5

51 3D Homogeneous Rotation Different from 2D rotation: rotate about an axis, not a point Basic rotation transformations: rotate about the 3 main axes x, y, z Rotation about an arbitrary axis: by combining basic rotations Positive rotation about an axis α: in right-handed coordinate system is the one in the counterclockwise direction when looking from the positive part of an axis toward the origin Positive rotation about y-axis 51

52 3D Homogeneous Rotation(2) The distance from the axis of rotation does not change Rotation does not affect the coordinate that corresponds to the axis of rotation Rotation matrices about the main axes: R x ( ) 1 cos sin sin cos 1 R y ( ) cos sin 1 sin cos 1 R z ( ) cos sin sin cos 1 1 Inverse rotation: R ( ) R ( ), R ( ) R ( ) and R ( ) R ( ), x x y y z z Rotations can also be expressed using quaternions 52

53 3D Homogeneous Shear Shears object along one of the major planes Increases 2 coordinates by an amount equal to the 3 rd coordinate times the respective shearing factors 3 cases in 3D shear: xy, xz, yz xy shear: increases x by increases y by SH xy ( ab, ) 1 a 1 b 1 1 SH xz a z coordinate b z coordinate ( ab, ) 1 a 1 b 1 1 SH yz Similar for xz & yz : ( ab, ) 1 a b Inverse shear: SH SH SH -1 xy -1 xz -1 yz ( a, b) SH ( a, b), xy ( a, b) SH ( a, b), xz ( a, b) SH ( a, b) yz 53

54 EXAMPLE 11: Composite Rotation - Bending Compute the bending matrix: rotation about the x-axis by θ x rotation about the y-axis by θ y Does the order of the rotations matter? SOLUTION 1. 3D Transformation Example Compute the reverse order cos sin 1 cos sin sin cos sin y y y x y x y 1 cos x sin x cos x sin x MBEND Ry ( y) Rx( x) sin cos sin cos sin sin cos cos cos y y x x y x y x y cos sin cos sin y y y y cos x sin x 1 sin x sin y cos x sin x cos y MBEND Rx( x) Ry ( y) sin cos sin cos cos sin sin cos cos x x y y x y x x y M BEND M' BEND so the order of the rotations matters 54

55 3D Transformation Example 12 EXAMPLE 12: Alignment of Vector with Axis Determine the transformation v [ abc,, ] T SOLUTION with the unit vector Av ( ) that aligns a given vector ˆk along the positive z-axis. Using two rotations: v Initial Step1 Step 2 Step 1: Rotate about x by θ 1 so that v is mapped onto v 1 which lies on the xz-plane, R x (θ 1 ) Step 2: Rotate about y by θ 2 so that it coincides with ˆk, R y (θ 2 ) v 1 55

56 3D Transformation Example 12 (2) Alignment matrix ( ) : Compute angle θ 1 : Av A v Ry 2 Rx 1 ( ) ( ) ( ) θ 1 is equal to the angle formed between the projection of v onto the yz-plane and the z-axis For the tip p of v we have p=[a, b, c] T The tip of its projection on yz is p =[, b, c] T Assuming that b, c are not both we get: sin b c, cos b c b c Thus, R x ( ) 1 1 cos sin 1 1 sin cos c b b c b c b c b c b c 1 56

57 Apply R x (θ 1 ) to v, to get its xz projection : Note that: Compute θ 2 : sin Thus 3D Transformation Example 12 (3) R y a a b v R ( ) v R ( ) c b c x 1 x v v a b c a 2 2, cos b c a b c a b c ( ) v b c a cos 2 sin 2 a b c a b c 1 1 sin cos a b c a b c a b c 1 57

58 3D Transformation Example 12 (4) Compute matrix Av ( ): A( v) R ( ) R ( ) y 2 x 1 ab ac v v v c b a b c v v v 1 where v a b c and b c Compute the inverse matrix Av ( ) 1 (useful in next example): 1 A ( v) ( R ( ) R ( )) R ( ) R ( ) R ( ) R ( ) y 2 x 1 x 1 y 2 x 1 y 2 If b=c= then v coincides with the x-axis rotate about y by 9 or -9 depending on the sign of α A v R y 2 ( ) ( ) a v v a a 1 a a 1 ab c b v v ac b c v v 1 58

59 3D Transformation Example 13 EXAMPLE 13: Rotation about an Arbitrary Axis using 2 Translations & 5 Rotations Find the transformation which performs a rotation by an angle θ about an arbitrary axis specified by a vector v and a point p. SOLUTION Av ( ) transformation [Ex. 12] : aligns an arbitrary vector with z-axis use it to reduce the problem to rotation around z Step 1: Translate p to the origin, T( p) Step 2: Align v with the z-axis using Av ( ) matrix Step 3: Rotate about the z-axis by the desired angle θ, R z () -1 Step 4: Undo the alignment, A ( v) Step 5: Undo the translation, Tp ( ) -1 M T( p) A ( v) R ( ) A( v) T( p) ROT-AXIS z 59

60 3D Transformation Example 14 EXAMPLE 14: Coordinate System Transformation using 1 Translation&3 Rotations Determine the transformation M ALIGN required to align a given 3D coordinate system with basis vectors ( ˆl, mˆ, nˆ) with the xyz coordinate system with basis vectors ( ˆˆ i, j, kˆ ). The origin of the 1 st coordinate system relative to xyz is O lmn. SOLUTION Axis transformation: aligning the ( ˆl, mˆ, nˆ) basis to ( ˆˆ i, j, kˆ ) the basis changing an object s system from ( ˆˆ i, j, kˆ ) to The solution is an extension of Av ( )[Ex. 12] ( ˆl, mˆ, nˆ) 6

61 3D Transformation Example 14 (2) Step 1: Translate by -O lmn to make the 2 origins coincide, Step 2: Align the ˆn basis vector with the ˆk basis vector, using of [Ex. 12], An ( ˆ) T( O lmn ) Av ( ) Step 3: Rotate by φ around the z-axis to align the other 2 axes, R z ( ) M ˆ ALIGN Rz( ) A( n) T( Olmn) Necessary to transform the ˆl or the ˆm vector by An ( ˆ) to estimate φ. e.g. m A( nˆ) mˆ : the sinφ and cosφ values required for the rotation are then the x and y components of m 61

62 3D Transformation Example 14 (3) CONCRETE EXAMPLE : The orthonormal basis vectors of the 2 coordinate systems are: and the origins of the 2 coordinate systems coincide ( O lmn = [..] T ). Find the transformation. 1 ˆi, ˆj 1, kˆ ˆ l, mˆ, nˆ M ALIGN The basis vectors of the 2 nd system are expressed in terms of the 1 st From the coordinates of ˆn [Ex. 12]: a, b, c and λ b c ( ) ( ) ;

63 3D Transformation Example 14 (4) Compute An ( ˆ) : Compute m : An ( ˆ) mˆ A( nˆ ) mˆ A( nˆ ) So sin and cos

64 3D Transformation Example 14 (5) Compute ( ) : R z R z ( ) Compute T( O lmn ) : the origins of the 2 coordinate systems coincide so So, T( O ) ID lmn M R ( ) A( nˆ ) T( O ) ALIGN z lmn M ( ) ( ˆ ALIGN Rz A n) ID %

65 3D Transformation Example 15 EXAMPLE 15: Change of Basis Determine the transformation M BASIS required to change the orthonormal basis of a coordinate system with from B1 ( ˆi ˆ ˆ 1, j 1, k 1 ) to B2 ( ˆi ˆ ˆ 2, j 2, k 2 ) and vice versa. SOLUTION Let the coordinates of the same vector in the 2 bases be vb 1 and vb2. If the coordinates of the ˆi, ˆj,kˆ basis vectors in B1 are: a d p ˆi b ˆj e kˆ q 2, B1 2, B1 2, B1 c f r then (from linear algebra): v a d p b e q v B1 B2 c f r 65

66 3D Transformation Example 15 (2) Thus, 1 M BASIS a d p b e q c f r B2 is an orthonormal basis M 1 BASIS is an orthonormal matrix a b c M BASIS 1 ( M ) BASIS Τ d e f p q r Homogeneous form: M BASIS a b c d e f p q r 1 66

67 3D Transformation Example 16 EXAMPLE 16: Coordinate System Transformation using Change of Basis Use the change-of-basis result of Ex. 15 to align a given 3D coordinate system with basis vectors ( ˆl, mˆ, nˆ) with the xyz-coordinate system with basis vectors ( ˆˆ i, j, kˆ ). The origin of the 1 st coordinate system relative to xyz is O lmn. SOLUTION This is an axis transformation: ( ˆˆ i, j, kˆ ) changing an object s coordinate system from to ( ˆl, mˆ, nˆ) Change-of-basis replaces the 3 rotational transformations in Ex. 14 Step 1: Translate by O lmn to make the 2 origins coincide, Step 2: Use to change the basis from ( ˆˆ i, j, kˆ ) to ( ˆl, mˆ, nˆ) M BASIS T( O ) lmn 67

68 3D Transformation Example 16 (2) M M T( O ) ALIGN2 BASIS lmn a b c ( a o b o c o ) d e f ( d o e o f o ) p q r ( p o q o r o ) 1 x y z x y z x y z Where the basis vectors (l,m,n) ˆ ˆ ˆ expressed in the basis (i, ˆˆj,k) ˆ are: ˆl [ a, b, c] T, mˆ [ d, e, f ] T, nˆ [ p, q, r] T and O lmn [ ox, oy, oz] CONCRETE EXAMPLE [of Ex 14]: No translation because the 2 origins coincide. M BASIS with homogeneous form M BASIS

69 EXAMPLE 17: Rotation about an Arbitrary Axis using Change of Basis Use the change-of-basis result of Ex. 15 to find an alternative transformation which performs a rotation by an angle θ about an arbitrary axis specified by a vector v and a point p. SOLUTION Let 3D Transformation Example 17 a v b and p c Equation of plane perpendicular to v through p: α(x-x p )+b(y-y p )+c(z-z p )= Let: q a point on that plane, such that q p and m q p and l m v Normalize the vectors l,m, v to define a coordinate system basis (l,m,n) ˆ ˆ ˆ with one axis being v and the other 2 axes on the given plane M BASIS Use transformation to align it with the xyz-coordinate system Perform the desired rotation by θ around the z-axis x y z p p p 69

70 3D Transformation Example 17 (2) Step 1: Translate p to the origin, T( p) Step 2: Align the (l,m, ˆ ˆ v) ˆ basis with the (i, ˆˆj,k) ˆ basis, Step 3: Rotate about the z-axis by desired angle θ, R z () Step 4: Undo alignment, -1 M BASIS Step 5: Undo the translation, Tp ( ) M BASIS 1 M T( p) M R ( ) M T( p) ROT AXIS2 BASIS z BASIS The algebraic derivation of the MROT AXIS2 matrix is simpler than M ROT AXIS 7

71 3D Transformation Example 18 EXAMPLE 18: Rotation of a Pyramid T T T Rotate the pyramid defined by the vertices a [,,], b [1,,], c [,1, ] d [,,1] T 45 T SOLUTION The pyramid is represented by a matrix P : and by about the axis defined by c and the vector v [,1,1]. P a b c d

72 3D Transformation Example 18 (2) Rotate the pyramid : use the matrix. -1 M T( p) A ( v) R ( ) A( v) T( p) The submatrices: M ROT-AXIS ROT-AXIS z T( c) A( v) Rz (45 ) A ( v) Tc ()

73 3D Transformation Example 18 (3) Combine the submatrices to get: M ROT AXIS Compute the rotated pyramid: The vertices of the rotated pyramid are: P MROT AXIS P a [,, ], b [,, ], c [,1,] and d [1,, ] T T T T 73

74 Quaternions Used as an alternative way to express rotation Useful for animating rotations A quaternion consists of 4 real numbers: q = (s, x, y, z) s scalar part of quaternion q v ( x, y, z) vector part of quaternion q Thus an alternative representation of quaternion q is: Can be viewed as an extension of complex numbers in 4D: Using imaginary units i, j and k such that: i 2 =j 2 =k 2 =-1 & ij=k, ji=-k and so on by cyclic permutation, quaternion q may be written as: q = s+ xi+ yj+ zk A real number u corresponds to the quaternion: q = v An ordinary vector corresponds to the quaternion: q = (, ) A point p corresponds to the quaternion: q = (, p) q ( u, ) ( s, v) v 74

75 Quaternions (2) Addition between quaternions: q q ( s, v ) ( s, v ) ( s s, v v ) Multiplication between quaternions: q q ( s s v v, s v s v v v ) ( s s x x y y z z, s x x s y z z y, s1 y2 y1s2 z1x2 x1z 2, s1z 2 z1s2 x1 y2 y1x2 ) Multiplication is associative Multiplication is not commutative The conjugate quaternion of q is defined as: It holds that: q q q q The norm of q is defined as: q q q q q s v s x y z q ( s, v) 75

76 Quaternions (3) It holds that: q1 q2 q1 q2 A unit quaternion is one whose norm: q 1 The inverse quaternion of q is defined as: q 1 1 q 2 q It holds that: q q q q q 1 If then q 1 q 76

77 Expressing rotation using quaternions Consider a rotation by an angle θ about an axis through the origin whose direction is specified by a unit vector ˆn. The rotation can be expressed by the unit quaternion: The above unit quaternion can be applied to a point p represented by 1 the quaternion p = (, p) as: Thus: where q (cos,sin nˆ ) 2 2 p q p q q p q 2 p,( s v v ) p 2 v( v p ) 2 s( v p). ( ) s cos and v sin nˆ 2 2 Notice that quaternion p represents a point p since its scalar part is Point p is exactly the image of the original point p after rotation by angle θ about the given axis 77

78 Expressing rotation using quaternions (2) Expressing 2 consecutive rotations: q ( q p q ) q ( q q ) p ( q q ) ( q q ) p ( q q ) The composite rotation is represented by the unit quaternion: q = q 2 q 1 Quaternion multiplication is simpler, requires fewer operations and is numerically more stable than rotation matrix multiplication Proof of quaternion rotation: Consider a unit vector ˆv, a rotation axis ˆn and the images ˆv, ˆv 2 of ˆv 1 after 2 consecutive rotations by θ/2 around ˆn The respective quaternions are: p (, vˆ ), p (, vˆ ), p (, vˆ ) We observe : Thus we write: q Similarly: q cos vˆ vˆ and sin nˆ vˆ vˆ 2 2 ( vˆ vˆ, vˆ vˆ ) p p p p

79 Expressing rotation using quaternions (3) Then: q p q ( p p ) p ( p p ) ( p p ) p p p p p p p since p p ( 1,) 1 because 1 1 and also p vˆ ( 1) (, ) (, vˆ ) p v Thus q p q results in the rotation of ˆv by angle θ about ˆn Using similar arguments, q p1 q results in the same rotation for ˆv 1, whereas q (, nˆ ) q yields ˆn, which agrees with the fact that ˆn is the axis of rotation 79

80 Expressing rotation using quaternions (4) Generalizing the above for an arbitrary vector: ˆv, ˆv, ˆn are linearly independent. Therefore a vector may be 1 p written as a linear combination p vˆ ˆ ˆ 1v 1 n Then: q (, p) q q (, vˆ vˆ nˆ ) q 1 1 q (, vˆ ) q q (, vˆ ) q q (, nˆ ) q 1 1 ( q (, vˆ ) q) ( q (, vˆ ) q) ( q (, nˆ ) q) 1 1 which is exactly a quaternion with scalar part and a vector part made up of the rotated components of p 8

81 Conversion between Quaternions and Rotation Matrices The rotation matrix corresponding to a rotation represented by a unit quaternion q = (s, x, y, z) is : R q If the following matrix m m m R y 2z 2xy 2sz 2xz 2sy 2 2 2xy 2sz 1 2x 2z 2yz 2sx 2 2 2xz 2sy 2yz 2sx 1 2x 2y m m m m m m represents a rotation then the corresponding quaternion q = (s, x, y, z) may be computed as follows 1 81

82 Conversion between Quaternions and Rotation Matrices (2) Step 1: 1 s m m11 m Step 2: m21 m12 m m m m x, y, z 4s 4s 4s If s = (or is a number near ), use a different set of relations: then 1 m m x m m11 m22 y 2 4x m2 m2 m21 m12 z, s 4x 4x 1 1 1,, 82

83 An Example EXAMPLE 19: Rotation of a pyramid Re- work example 18, using quaternions. SOLUTION Step 1: Translation by c, T( c) so that the axis passes through the origin Step 2: perform the rotation using R q. The quaternion that expresses the rotation by 45 o about an axis with direction ˆv is: sin 22.5 sin 22.5 q ( cos,sin vˆ ) (cos 22.5,,, ) where 1 cos cos cos 22.5,sin

84 An Example (2) Therefore: R q Step 3: Translate by c, T( c) The final transformation is: M 3 T() c R T( c) M ROT AXIS q ROT AXIS [Ex. 18] 84

85 Geometric Properties Affine transformations preserve important geometric features of objects. That s why they are used in Computer Graphics and Visualization For example let Φ be an affine transformation and p, q points, then: ( p (1 ) q) ( p) (1 ) ( q ), [,1] states that the affine transformation of a line segment under Φ is another line segment Ratios of distances on the line segment ( λ / (1-λ) ) are preserved Affine transformation subclasses: linear, similitudes, rigid 85

86 Geometric Properties (2) Linear transformation can be presented by a matrix A Linear: all homogeneous affine transformations Similitudes preserve the similarity of objects. The result is identical to the initial object, except for its size Similitudes: rotation, translation, isotropic scaling Rigid transformations preserve all the geometric features of objects Rigid: rotations and translations 86