Physically Based Rendering ( ) Geometry and Transformations

Transcription

1 Phsicall Based Rendering (6.657) Geometr and Transformations

2 3D Point Specifies a location Origin

3 3D Point Specifies a location Represented b three coordinates Infinitel small class Point3D { public: Coordinate ; Coordinate ; Coordinate z; }; (,,z) Origin

4 3D Vector Specifies a direction and a magnitude

5 3D Vector Specifies a direction and a magnitude Represented b three coordinates Magnitude V = sqrt(d d + d d + dz dz) Has no location class Vector3D { public: Coordinate d; Coordinate d; Coordinate dz; }; (d,d,dz)

6 3D Vector Specifies a direction and a magnitude Represented b three coordinates Magnitude V = sqrt(d d + d d + dz dz) Has no location class Vector3D { public: Coordinate d; Coordinate d; Coordinate dz; }; Dot product of to 3D vectors V V = d d + d d + dz dz V V = V V cos(q) Q (d,d,dz ) (d,d,dz )

7 3D Vector Specifies a direction and a magnitude Represented b three coordinates Magnitude V = sqrt(d d + d d + dz dz) Has no location class Vector3D { public: Coordinate d; Coordinate d; Coordinate dz; }; Cross product of to 3D vectors V V = Vector normal to plane V, V V V = V V sin(q) Q (d,d,dz ) (d,d,dz )

8 Cross Product: Revie Let U = V W: U = V W z - V z W U = V z W - V W z U z = V W - V W V W = - W V (remember right-hand rule) We can do similar derivations to sho: V V = V V sin(q)n, here n is unit vector normal to V and V V V =

9 3D Normal Specifies a differential patch b the area and perpendicular direction. n t t

10 3D Ra Line segment ith one endpoint at infinit Parametric representation: P = P + t V, ( <= t < ) class Ra3D { public: Point3D P; Vector3D V; }; V P Origin

11 Simple D Transformations Translation p T p d d P P

12 Simple D Transformations Translation p T p d d

13 Simple D Transformations Scale p S p s s z

14 Simple D Transformation Rotation p R p cos sin sin cos z

15 Matri Representation Represent D transformation b a matri a c b d Multipl matri b column vector appl transformation to point a b c d a c b d

16 Matri Representation Transformations combined b multiplication a c be d g f i h k j l Matrices are a convenient and efficient a to represent a sequence of transformations!

17 Matrices What tpes of transformations can be represented ith a matri? a c b d Onl linear D transformations can be represented ith a matri

18 Linear Transformations Linear transformations are combinations of Scale, and Rotation a c b d Properties of linear transformations: Satisfies: T( sp sp) st ( p) st ( p) Origin maps to origin Lines map to lines Parallel lines remain parallel Closed under composition

19 Linear Transformations Linear transformations are combinations of Scale, and Rotation a c b d Properties of linear transformations: Satisfies: T( sp sp) st ( p) st ( p) Origin maps to origin Lines map to lines Parallel lines remain parallel Closed under composition Translations do not map the origin to the origin

20 D Translation D translation represented b a 33 matri Point represented ith homogeneous coordinates t t t t

21 D Translation D translation represented b a 33 matri Point represented ith homogeneous coordinates t t t t

22 Homogeneous Coordinates Add a 3rd coordinate to ever D point (,, ) represents a point at location (/, /) (,, ) represents a point at infinit (,, ) is not alloed (,,) or (4,,) or (6,3,3) Convenient coordinate sstem to represent man useful transformations

23 Basic D Transformations Basic D transformations as 33 matrices Q Q Q Q cos sin sin cos t t Translate Rotate s s Scale

24 Affine Transformations Affine transformations are combinations of Linear transformations, and Translations Properties of affine transformations: Origin does not necessaril map to origin Lines map to lines Parallel lines remain parallel Closed under composition f e d c b a

25 Projective Transformations Projective transformations Affine transformations, and Projective arps Properties of projective transformations: Origin does not necessaril map to origin Lines map to lines Parallel lines do not necessaril remain parallel Closed under composition i h g f e d c b a

26 Matri Composition Matrices are a convenient and efficient a to represent a sequence of transformations General purpose representation Hardare matri multipl Efficienc ith pre-multiplication Matri multiplication is associative p = (T * (R * (S*p) ) ) p = (T*R*S) * p

27 3D Transformations Same idea as D transformations Homogeneous coordinates: (,,z,) 44 transformation matrices z p o n m l k j i h g f e d c b a z

28 Basic 3D Transformations z z z tz t t z z sz s s z Identit Scale Translation

29 Basic 3D Transformations Pitch-Roll-Ya Convention: An rotation can be epressed as the combination of a rotation about the -, the -, and the z-ais. Q Q Q Q z z cos sin sin cos Rotate around Z ais: Q Q Q Q z z cos sin sin cos Rotate around Y ais: Q Q Q Q z z cos sin sin cos Rotate around X ais:

30 Homogenous Coordinates In 3D e can represent points b the 4-tuple: (,,z,) under the relationship that the position is unchanged b scale: (,,z,) = (,, z, ) for all non-zero values.

31 Homogenous Coordinates To cases:. : The 4-tuple represents a point (,,z,)=(/,/,z/,). =: The 4-tuple represents a (unit) vector (,,z,)=(,,z,)/ + + z

32 Appling a Transformation Position Direction Normal i h g f e d c b a tz t t tz i h g t f e d t c b a M Affine Translate Linear M T M L

33 Appling a Transformation Position Appl the full affine transformation: p = M(p) = (M T M L )(p) = M(p,p,p z,) i h g f e d c b a tz t t tz i h g t f e d t c b a M Affine Translate Linear M T M L

34 Appling a Transformation Direction Appl the linear component of the transformation: p = M L (p) = M(p,p,p z,) i h g f e d c b a tz t t tz i h g t f e d t c b a M Affine Translate Linear M T M L

35 Appling a Transformation Direction Appl the linear component of the transformation: p = M L (p) = M(p,p,p z,) A direction vector v is defined as the difference beteen to positional vectors p and q: v=p-q. p q v

36 Appling a Transformation Direction Appl the linear component of the transformation: p = M L (p) = M(p,p,p z,) A direction vector v is defined as the difference beteen to positional vectors p and q: v=p-q. Appling the transformation M, e compute the transformed direction as the distance beteen the transformed positions: v =M(p)-M(q). p M(p) q v M(q) v

37 Appling a Transformation Direction Appl the linear component of the transformation: p = M L (p) = M(p,p,p z,) A direction vector v is defined as the difference beteen to positional vectors p and q: v=p-q. Appling the transformation M, e compute the transformed direction as the distance beteen the transformed positions: v =M(p)-M(q). The translation terms cancel out! q v p M(q) v M(p)

38 Appling a Transformation Normal p =? i h g f e d c b a tz t t tz i h g t f e d t c b a M M T M L Affine Translate Linear

39 Normal Transformation D Eample: Translate Scale M M T M L

40 Normal Transformation D Eample: If v is a direction in D, and n is a vector perpendicular to v, e ant the transformed n to be perpendicular to the transformed v: ), (, n v M n v L Translate Scale M M T M L

41 Normal Transformation D Eample: Sa v Translate Scale M M T M L (,) v

42 Normal Transformation D Eample: Sa then v n, n v Translate Scale M M T M L.5.5, n (,) v

43 Normal Transformation D Eample: Sa then Transforming, v n M L (v), n v Translate Scale M M T M L (,4) ) ( v M L.5.5, n (,) v

44 Normal Transformation D Eample: Sa v (,) then Transforming, Translate M M T M L n.5,.5 M L ( v ) (,4) ( n) and M L Scale.5, n v M L (n) M L (v) v, n M ( v ), M ( n) L L

45 Normal Transformation D Eample: M M T M L Simpl appling n the directional.5,.5 part of the transformation M ( v ) (,4) and M ( n) L to n does not result in L a vector that is perpendicular to the transformed v. Sa v (,) then Transforming, Translate M n v L (n) Scale M L (v).5, v, n M ( v ), M ( n) L L

46 Recall Transposes: The transpose of a matri M is the matri M t hose (i,j)-th coeff. is the (j,i)-th coeff. of M: m M m m 3 m m m 3 m m m M t m m m 3 m m m 3 m m m

47 Recall Transposes: The transpose of a matri M is the matri M t hose (i,j)-th coeff. is the (j,i)-th coeff. of M: m M m m 3 m m m 3 m m m M t m m m If M and N are to matrices, then the transpose of the product is the inverted product of the transposes: t t t MN N M 3 m m m 3 m m m

48 Recall Dot-Products: The dot product of to vectors v=(v,v,v z ) and =(,, z ) is obtained b summing the product of the coefficients: v, v v v z z

49 Recall Dot-Products: The dot product of to vectors v=(v,v,v z ) and =(,, z ) is obtained b summing the product of the coefficients: We can also epress this as a matri product: z z v v v v, z z t v v v v v,

50 Recall Transposes and Dot-Products: If M is a matri, the dot product of v ith M applied to is the dot product of the transpose of M applied to v ith :

51 Recall Transposes and Dot-Products: If M is a matri, the dot product of v ith M applied to is the dot product of the transpose of M applied to v ith : t v, M v M

52 Recall Transposes and Dot-Products: If M is a matri, the dot product of v ith M applied to is the dot product of the transpose of M applied to v ith : t v, M v M v t M

53 Recall Transposes and Dot-Products: If M is a matri, the dot product of v ith M applied to is the dot product of the transpose of M applied to v ith : t v, M v M v t M t M v t

54 Recall Transposes and Dot-Products: If M is a matri, the dot product of v ith M applied to is the dot product of the transpose of M applied to v ith : t v, M v M v, M v t M t M v t t M v,

55 Appling a Transformation If e appl the transformation M to 3D space, ho does it act on normals?

56 Appling a Transformation If e appl the transformation M to 3D space, ho does it act on normals? A normal n is defined b being perpendicular to some vector(s) v. The transformed normal n should be perpendicular to M(v): n, v n, Mv

57 Appling a Transformation If e appl the transformation M to 3D space, ho does it act on normals? A normal n is defined b being perpendicular to some vector(s) v. The transformed normal n should be perpendicular to M(v): n, v n, Mv M t n, v

58 Appling a Transformation If e appl the transformation M to 3D space, ho does it act on normals? A normal n is defined b being perpendicular to some vector(s) v. The transformed normal n should be perpendicular to M(v): n, v n, Mv M t n, v t n M n

59 Appling a Transformation If e appl the transformation M to 3D space, ho does it act on normals? A normal n is defined b being perpendicular to some vector(s) v. The transformed normal n should be perpendicular to M(v): n, v n, Mv M t n, v t n M n n t M n

60 Quaternions Quaternions are etensions of comple numbers, ith 3 imaginar values instead of : a ib jc kd

61 Quaternions Quaternions are etensions of comple numbers, ith 3 imaginar values instead of : a ib jc kd Like the comple numbers, e can add quaternions together b summing the individual components: a ib jc kd + a ib jc kd a a i b b j c c k d d

62 Quaternions Quaternions are etensions of comple numbers, ith 3 imaginar values instead of : a ib jc kd Like the imaginar component of comple numbers, squaring the components gives: i j k

63 Quaternions Quaternions are etensions of comple numbers, ith 3 imaginar values instead of : a ib jc kd Like the imaginar component of comple numbers, squaring the components gives: i j k Hoever, the multiplication rules are more comple: ij k ik j jk i ji k ki j kj i

64 Quaternions Quaternions are etensions of comple numbers, ith 3 imaginar values instead of : a ib jc kd Note that multiplication of quaternions is Like the imaginar not commutative: component of comple numbers, The squaring result of the multiplication components depends gives: on the order i in hich j it as k done Hoever, the multiplication rules are more comple: ij k ik j jk i ji k ki j kj i

65 Quaternions More generall, the product of to quaternions is: kd jc ib a kd jc ib a c b b c d a a d k d b b d c a a c j d c d c b a a b i d d c c b b a a i kj j ki k ji i jk j ik k ij k j i

66 Quaternions As ith comple numbers, e define the conjugate of a quaternion q=a+ib+jc+kd as: q a ib jc kd

67 Quaternions As ith comple numbers, e define the conjugate of a quaternion q=a+ib+jc+kd as: q a ib jc kd As ith comple numbers, e define the norm of a quaternion q=a+ib+jc+kd as: q a b c d qq

68 Quaternions As ith comple numbers, e define the conjugate of a quaternion q=a+ib+jc+kd as: q a ib jc kd As ith comple numbers, e define the norm of a quaternion q=a+ib+jc+kd as: q a b c d qq As ith comple numbers, the reciprocal is defined b dividing the conjugate b the square norm: q q q

69 Quaternions One a to epress a quaternion is as a pair consisting of the real value and the 3D vector consisting of the imaginar components: q ith a, ( b, c,, d )

70 Quaternions One a to epress a quaternion is as a pair consisting of the real value and the 3D vector consisting of the imaginar components: The advantage of this representation is that it is easier to epress quaternion multiplication: ),, (,, d c b a q ith,,,, q q

71 Quaternions One a to epress a quaternion is as a pair consisting of the real value and the 3D vector consisting of the imaginar components: The advantage of this representation is that it is easier to epress quaternion multiplication: c b b c d a a d k d b b d c a a c j d c d c b a a b i d d c c b b a a q q ),, (,, d c b a q ith,,,, q q

72 Quaternions Given a 3D vector (,,z), e can think of the vector as an imaginar quaternion: v=i+j+kz

73 Quaternions Given a 3D vector (,,z), e can think of the vector as an imaginar quaternion: v=i+j+kz Then multipling v b a quaternion q on the left and its conjugate on the right gives: q v q,

74 Quaternions Given a 3D vector (,,z), e can think of the vector as an imaginar quaternion: v=i+j+kz Then multipling v b a quaternion q on the left and its conjugate This on means the that right unit quaternions gives: correspond to rotations in 3D. q v q, The mapping is linear in v, and preserves lengths hen q is a unit quaternion.

75 Quaternions and Rotations If q=a+ib+jc+kd is a unit quaternion ( q =), q corresponds to a rotation: R ( q ) c d bc ad bc ad b d bd ac cd ab bd ac cd ab b c Note that because all of the terms are quadratic, the rotation associated ith q is the same as the rotation associated ith -q.

76 Quaternions and Rotations If q=a+ib+jc+kd is a unit quaternion ( q =), q corresponds to a rotation: R ( q ) c d bc ad bd ac bc ad b d cd ab bd ac cd ab b c Since q is a unit quaternion, e can rite q as: q cos( / ),sin( / )

77 Quaternions and Rotations If q=a+ib+jc+kd is a unit quaternion ( q =), q corresponds to a rotation: R ( q ) c d bc ad bd ac bc ad b d cd ab Since q is a unit quaternion, e can rite q as: q cos( / ),sin( / ) It turns out that q corresponds to the rotation: Whose ais of rotation is, and Whose angle of rotation is. bd ac cd ab b c

78 R ( q ) Quaternions and Rotations c d bc ad bd ac bc ad b d cd ab bd ac cd ab b c Since R(q) preserves distances, shortest paths beteen rotations can be obtained b computing shortest paths beteen unit quaternions.

79 R ( q ) Quaternions and Rotations c d bc ad bd ac bc ad b d cd ab bd ac cd ab b c Since R(q) preserves distances, shortest paths beteen rotations can be obtained b computing shortest paths beteen unit quaternions. But shortest paths beteen to points on a sphere are just a great arcs (SLERP).