Shader Shader. Shader 4.1

Transkript

1

2 d d 3 9 Shader Shader ( Shader ) ) ) ) ) 4.1 S 4.1

3 4.2 p p 4.2 Cartesian Coordinate System René Descartes 4.2 f t g V o X 41

4 x y 4.3 x y 30 x y OpenGL DirectX 4.4 f 1, C./ - 8 o ex f U. 50% ( U

5 4.2 3 basis vector 3 1 orthonormal basis 1 orthogonal basis 4.6 left-handed coordinate space right-handed coordinate space 43

6 4.2 x y OpenGL DirectX 180 y x x 4.6 +z +z x y handedness A B L. +x +y +z ) 3 forward 44

7 4.2 x y z z left-hand rule right-hand rule (( ~ ) 2 ~?

8 4.2 e ( 4.11 e X x 1 z z x 1 z 4 60 z z 5C Unity 4.6 Unity Scene 4.12 right up forward x y z 46

9 4.2 5C e O P Unity z z C w Unity d ) 1 3ds Max x y z 2 0, 0, 1 y +90 0, 0, 1 y Unity 0, 1, 10 Rotation (0, 0, 0) Scale 1, 1, 1 (0, 1, 0) 4.14 O P w O P z z 47

10 point n 2 3 P (P x, P y ) P (P x, P y, P z ) vector scalar n magnitude direction velocity 80km/h 80km/h distance v (x, y) v (x, y, z) v (x, y, z, w) a b x y z θ α a b u v A B S M R 4.15 head tail km/h 100km/h displacement ~

11 4.3 f o ( b ) c / kv (kv x, kv y, kv z ) v (x, y, z) 1 k k k (x, y, z) x k, y k, z,k 0 k 2(1, 2, 3) (2, 4, 6) 3.5(2, 0) ( 7, 0) (1,2,3) (0.5,1,1.5) 2 v k v k : 2 k <

12 4.3 a + b (a x + b x, a y + b y, a z + b z ) a b (a x - b x, a y - b y, a z - b z ) (1, 2, 3) + (4, 5, 6) (5, 7, 9) (5, 2, 7) - (3, 8, 4) (2, -6, 3) a b a b a b a b a+b triangle rule 4.18 ) f i c f a b a b b a b a 4.19 ( ( v v + v + v x y z (1,2,3) (3,4)

13 4.3 n

14 4.3 ( unit vector 1 normalized vector normalization v v ^v ^ v v,v v (3, 4) (3, 4) (3, 4) (3, 4) ( 4) 2 25 (3, 4) 3 5 5, 4 (0.6, 0.8) 5 0 v (0, 0, 0) ( dot product inner product cross product outer product ~ Shader Unity Shader dot(a, b) a b f s V 52

15 4.3 a b (a x, a y, a z ) (b x, b y, b z ) a x b x + a y b y + a z b z (1, 2, 3) (0.5, 4, 2.5) (-3, 4, 0) (5, -1, 7) a b b a projection ^a b b ^a ( ^a b b ^a ( ^a b ^a b ^a 4.22 ^a b V ^a a b b a a Shader. (ka) b a (kb) k(a b) :: a (b + c) a b + a c c c 53

16 4.3 v v v x v x + v y v y + v z v z v 2 a b a b cosθ ^a ^b 4.24 ^b 1 cosθ ^a ^b cosθ 4.24 ^a ^b cosθ a b ( a ^a ) ( b ^b ) a b ( ^a ^b ) a b cosθ 90 cosθ > 0 90 cosθ 0 90 cosθ < arcos ( θ arcos( ^a ^b ) ^a ^b cross product outer product a b a b (a x, a y, a z ) (b x, b y, b z ) (a y b z - a z b y, a z b x - a x b z, a x b y - a y b x )

17 4.3 U X e o n co f f c (1, 2, 3) ( 2, 1, 4) ((2)(4) (3)( 1), (3)( 2) (1)(4), (1)( 1) (2)( 2)) (8 ( 3), ( 6) 4, ( 1) ( 4)) (11, 10, 3) a b b a a b (b a) (a b) c a (b c). a b a b a b a b sinθ a b b h h a θ A b h b ( a sinθ) a b sinθ a b a b 0 a b ( 9 ) 55

18 4.3 a b a b a b a b 4.28 a b! ) d )! (2, 7, 3) 2 2.5(5, 4, 10) 3 (3,4) 2 4 (5, 12) 5 (1, 1, 1) 6 (7, 4) + (3, 5) 7 (9, 4, 13) (15, 3, 11) 3 (10, 13, 11) (2, 1, 1) 4 1 (4, 7) (3, 9) 2 (2, 5, 6) (3, 1, 2) 10 56

19 ( 3, 4) ( 2, 5) 4 (3, 1, 2) ( 5, 4, 1) 5 ( 5, 4, 1) (3, 1, 2) 5 a b a 4 b

20 4.4 1 a b 3 2 a b sin cos NPC p forward v 1 x NPC 2 1 p (4, 2), v ( 3, 4), x (10, 6) 3 NPC φ φ NPC 2 NPC x 4 3 NPC 7 3 p 1 p 2 p 3 p 1 p 2 p 3 xy z 0 z z matrix The Matrix matrix matrix The Matrix ( 1 ( a U U g V g 58

21 m n row column M m m m m21 m22 m23 m m m m ij M i j n 1 column matrix 1 n row matrix n v (3, 8, 6) [3 8 6] Shader c

22 4.4 m11 m12 m13 km11 km12 km13 kmm k k m m m km km km m31 m32 m 33 km31 km32 km 33 c r n A n c B AB r c A 4 3 B 3 6 AB 4 6 r n A n c B r c C AB C c ij A i B j c ij a i1 b 1j + a i2 b 2j + + a in b nj c ij A i B j c ij 4.30 A 4 2 B 2 4 C c 23 A 2 B 3 c 23 a 21 b 13 + a 22 b 23 Shader 4 4 AB BA (AB)C A(BC) ABCDE ((A(BC))D)E (AB)(CD)E ~ n k 1 ab ik kj 60

23 4.4 Shader. 1 square matrix diagonal elements m 11 m 22 m 33 ( 0 diagonal matrix identity matrix I n 3 3 I MI IM M 1 3 transposed matrix r c M M T c r i i j j M T ij M 6 7 T ji 61

24 4.4 z x [ x y z] T y T x y [ x y z] z (M T ) T M (AB) T B T A T 4 inverse matrix M M 1 M M -1 MM 1 M 1 M I 0 0 invertible nonsingular noninvertible singular determinant 0 Shader C++ Eigen Unity Unity Unity 4.8 M (M 1 ) 1 M I 1 I 62

25 4.4 (M T ) 1 (M 1 ) T (AB) 1 B 1 A 1 (ABCD) 1 D 1 C 1 B 1 A M v M 1 M 1 (Mv) (M 1 M)v Iv v orthogonal matrix M orthogonal M MM T M T M I. M M T M 1. MM T I 3 3 c1 T MM c c c c c3 c1 c1 c1 c2 c1 c3 c2 c1 c2 c2 c2 c3 c 3 c1 c3 c2 c3 c I

26 4.4 9 c 1 c 1 1 c 1 c 2 0 c 1 c 3 0 c 2 c 1 0 c 2 c 2 1 c 2 c 3 0 c 3 c 1 0 c 3 c 2 0 c 3 c 3 1 ~ c 1 c 2 c 3 1 c 1 c 2 c 3 0 ~ M M T orthogonal basis 1 1 orthonormal basis 1. v (x, y, z) v [x y z] v [x y z] T M m m m M m m m m m m M M M vm [ xm + ym + zm xm + ym + zm xm + ym + zm ] xm 11 + ym 12 + zm 13 Mv xm 21 + ym 22 + zm 23 xm 31 + ym 32 + zm 33 64

27 4.4 Unity CBAv ( C( B( Av ))) v A B C T T T T T T va B C ((( va ) B ) C ) ) 3 ( d cosθ sinθ 0 3 sinθ cosθ

28 4.5 as 3 (3, 2, 6) as ( ( ~ : 30? transform. linear transform ( f(x) + f(y) f(x + y) kf(x) f(kx) scale ( f(x) 2x 2 :. x f(x) 2x rotation ( ( 3 3 ( 66

29 4.5 as ( : shear mirroring reflection orthographic projection : ( f(x) x + (1, 2, 3) ( 既不 也 x (1, 1, 1) f(x) + f(x) (4, 6, 8) f(x + x) (3, 4, 5) 3 3 affine transform ( 4 4 homogeneous space N Y N Y Y N Y Y Y Y Y Y Y Y : : Y Y Y N Y Y Y N Y Y Y Y Y Y N N N N N N : ~ homogeneous coordinate w 1 w :. 4 4 : 67

30 4.5 as : 4 M 0 t M 3 3 : t [0 0 0] : tx x x+ tx t y y y+ ty t z z z+ t z x y z 3D (x, y, z) (t x, t y, t z ) tx x x t y y y t z z z t 3 1 M 3 3 I t x t y t z x y z : : 68

31 4.5 as kx x kxx 0 ky 0 0 y kyy 0 0 kz 0 z kzz : kx x kxx 0 ky 0 0 y kyy 0 0 kz 0 z kzz : k x k y k z : uniform scale nonuniform scale : : : : ( : 4.7 : : : kx k y kz : : : : : : ( x y z x θ cosθ sinθ 0 R x ( θ) 0 sinθ cosθ y 69

32 4.5 as cosθ 0 sinθ R y ( θ ) sinθ 0 cosθ z 70

33 4.5 as R z cosθ sinθ 0 0 sinθ cosθ 0 0 ( θ ) ) : (2, 2, 2) : y 30 z 4 p new M translation M rotation M scale p old : : : (0, 0, 5) (0, 0, 10): 2 : y : t x cosθ 0 sinθ 0 k x t M translation M rotation M scale y k y t z sinθ 0 cosθ k z k x cosθ 0 k z sinθ t x 0 k y 0 t y k x sinθ 0 k z cosθ t z : 71

34 4.6 cosθ 0 sinθ 0 k x t x M rotation M scale M translation 0 k y t y sinθ 0 cosθ k z t z k x cosθ 0 k z sinθ t x k x cosθ + t z k z sinθ 0 k y 0 t y k y k x sinθ 0 k z cosθ t x k x sinθ + t z k z cosθ x y z x y z (θ x, θ y, θ z ) Unity zxy API (θ x, θ y, θ z ) cosθ z sinθ z cosθ y 0 sinθ y 0 sinθ M rotatez M rotatex M rotatey z cosθ z 0 00 cosθ x sinθ x sinθ x cosθ x 0 sinθ y 0 cosθ y zxy (θ x, θ y, θ z ) E z θ z E y θ y E x θ x E z θ z E z θ z E' y θ y E' y θ y E'' x θ x zxy yxz Unity ) 4.6. : 2 ( ( 72

35 4.6.,. ( X Shader? Shader? ~ (. ( ( 3 73

36 4.6 parent P C 3 A c A p B p B c A p M c p A c B c M p c B p M c p M p c M c p (a, b, c) x a 3 y b 4 z c 4 C 3 P x c y c z c O c A c (a,b,c) 4 A n. O. x O + ax c O + ax + by c c c c c p O + ax + by + cz c c c c. M c p A O + ax + by + cz p c c c c 74

37 A p O c + ax c + by c + cz c (x Oc, y Oc, z Oc ) + a(x xc, y xc, z xc ) + b(x yc, y yc, z yc ) + c(x zc, y zc, z zc ) (x Oc, y Oc, z Oc ) + x xc x yc x zc y xc y yc y zc z xc z yc z zc a b c (x Oc, y Oc, z Oc ) + x c y c z c a b c. 3 3 A p (x Oc, y Oc, z Oc,1) + 0 x c y c z c a b c x Oc y Oc z Oc x c y c z c a b c 1 x Oc x c y c z c y Oc z Oc a b c 1 x c y c z c O c a b c 1 M c p c c c c c p x y z O M M c p M p c C

38 4.6 P P C M c p C P x c y c z c : 4 4 : x y z M c p C P C x (1, 0, 0, 0) M c p [ ] T M c p 3 3 Mc p xc yc zc Shader 3 3 ( M p c M c p M p c M p c M c p M c p 1 T M p c x p y p z p M c p M c p x c y c z c M c p x c y c z c x p y p z p M c p M A B A x B B x A B x y z A A B 76

39 4.6. P C C P A B. B x y z A x B y B z B A B M A B MA B xb yb z B M A B x B B x (1,0,0) B x (1, 0, 0) MA Bx B xb yb zb x xb xb xb 1 MA Bx B B B 0 B y xb y x z B zb xb 0. ( (., B 77

40 4.6 O P V O P V ( model space object space local space. forward back left right up down Unity +x +y +z x y z Unity Hierarchy 3? Unity (0, 2, 4) p (0, 2, 4, 1) 4.32 ( world space 78

41 4.6 Unity x y z Unity Transform Position Transform parent Transform 4.33 Transform Rotation Scale model transform Transform

42 4.6 5C 4 9CG D o 4 9CG D L0 G M 2DG DC W e O L,D MP T p 2DG DC n Transform (2, 2, 2) : (0, 150, 0) (5, 0, 25) : t x cosθ 0 sinθ 0 k x t M model y k y t z sinθ 0 cosθ k z P world M model P model (9, 4, ) 3 Unity 80

43 4.6 ( ( view space camera space ~ ~ Unity Unity +x +y +z ~ +z Unity OpenGL -z Unity Unity Camera.camera ToWorldMatrix Camera.worldToCameraMatrix projection view transform Transform 4.35 g O e P 5C e W e p n

44 4.6 Transform (30, 0, 0) (0, 10, 10) (0, 10, 10) ( 30, 0, 0) t x 0 cosθ sinθ 0 M view t y 0 sinθ cosθ t z z, M view M negate z M view P view M view P world (9, 8.84, 27.31) 82

45 4.6 () clip space clip matrix projection matrix view frustum clip planes orthographic projection perspective projection ( O P O ö入 门 精 要 4.36 P W e èé ìå 3D near øý 6 far clip plane ¼ òí clip plane 2D º HUD 4.37 ) w e w e w 83

46 4.6 homogeneous division. w x y z : 6. : w x y z w 1 w Unity Camera Game Camera Field of View FOV w Clipping Planes Near Far FOV nearclipplaneheight 2 Near tan 2 FOV farclipplaneheight 2 Far tan 2 ; Unity 84

47 4.6 Game Viewport Rect W H Unity Camera.aspect ~ Aspect nearclipplanewidth Aspect nearclipplaneheight Aspect farclipplanewidth farclipplaneheight Near Far FOV Aspect M frustum FOV cot Aspect FOV 0 cot Far + Near 2 Near Far 0 0 Far Near Far Near Unity z [-w, w] DirectX z [0, w] ~!!"#$!!"#$%#&!!"#$ cot!"# 2!"#$%& cot!"# !"# +!"#$ 2!"#$!"#!"#!"#$!!"!"#$ !"# cot! 2!"#$%&!cot!"# 2!"# +!"#$ 2!"#$!"#!!"#!"#$!"#!"#$! x y z : z : w 1 z, w x w!!! 1 85

48 4.6 w y w w z w e e n i o 4.39 z 6 Unity Camera Game 4.40 Camera Size Clipping Planes Near Far w nearclipplaneheight 2 Size farclipplaneheight nearclipplaneheight ; Aspect nearclipplanewidth Aspect nearclipplaneheight farclipplanewidth nearclipplanewidth 86

49 4.6 Near Far Size Aspect Aspect Size M ortho Size 2 Far + Near 0 0 Far Near Far Near Unity P clip Aspect Size x y Size z 2 Far + Near Far Near Far Near x Aspect Size y Size 2z Far + Near Far Near Far Near 1 M ortho P view w 1 [ ] [ ] (9, 8.84, ) 3D Game

50 4.6 e e n i o FOV 60 Near 5 Far 40 Aspect 4/

51 4.6 cot FOV Aspect M frustum 0 cot FOV Far + Near 0 0 Far Near Near Far Far Near P clip M frustum P view (11.691, , , 27.31) Unity w x w w x y w z w (. screen space. 2D homogeneous division perspective division w x y z OpenGL Normalized Device Coordinates NDC. NDC.. OpenGL x y z [ 1, 1] DirectX API z [0, 1] Unity OpenGL

52 4.6 s w.. w 1 x y z 4.44 s w. x y Unity 0, 0 (pixelwidth, pixelheight) x y [ 1, 1] : 90

53 4.6 clipx pixelwidth pixelwidth screenx + 2 clip 2 screen y w clipy pixelheight pixelheight + 2 clip 2 w x y z z clip z clip w clip w. NDC Unity (11.691, , , 27.31), NDC clipx pixelwidth pixelwidth screenx + 2 clip 2 screen y clipy pixelheight pixelheight + 2 clip ( , ) ( w w Unity 4.46 Unity 4.46 Unity tangent space ( (

54 4.6 ( 5C 92

55 ( normal normal vector. ( ( ( M A B A B ( ( tangent tangent vector ( ( ( 4.47 ( ( 3 3 M A B ) h 3 3 ( ( ( T B M A B T A T A T B A B ( M A B ( ( 4.48 s X ( ( T A ( N A T A N A 0 M A B. T B M A B T A G ( N A ( 93

56 4.8 Unity Shader O P ( T B N B (M A B T A ) (GN A ) 0 T T T T T ( M T ) ( GN ) ( M T ) ( GN ) T M GN T ( M G) N 0 A B A A A B A A A A B A A A B A T A N A 0 M T A BG I T 1 1 T G ( M ) ( M ) ( A B A B M A B M 1 M T T 1 ( M ) M A B A B A B A B ( : : T : k M A B ( MA B) M A B k ( 4.8 Unity Shader O P 1 1 Unity Shader Unity UnityShaderVariables.cginc 4.2 Unity 5.2 float4x4 Unity Unity Unity x 4.2 Unity UNITY_MATRIX_MVP UNITY_MATRIX_MV UNITY_MATRIX_V UNITY_MATRIX_P UNITY_MATRIX_VP UNITY_MATRIX_T_MV UNITY_MATRIX_IT_MV _Object2World _World2Object / / / / / UNITY_MATRIX_MV UNITY_MATRIX_MV ( UNITY_MATRIX_MV / _Object2World / UNITY_MATRIX_T_MV 94

57 4.8 Unity Shader O P UNITY_MATRIX_MV UNITY_MATRIX_T_MV UNITY_MATRIX_T_MV UNITY_MATRIX_MV 4.5 : UNITY_MATRIX_MV : : k UNITY_MATRIX_MV : 1 k : UNITY_MATRIX_MV 1 k UNITY_MATRIX_T_MV UNITY_MATRIX_T_MV : : UNITY_MATRIX_IT_MV 4.7. ( UNITY_MATRIX_IT_MV ( UNITY_MATRIX_MV // S transpose UNITY_MATRIX_IT_MV // UNITY_MATRIX_MV c // float4 modelpos mul(transpose(unity_matrix_it_mv), viewpos); // fsx transpose mul c // float4 modelpos mul(viewpos, UNITY_MATRIX_IT_MV); mul Unity Camera 4.3 Unity Unity _WorldSpaceCameraPos float3 _ProjectionParams _ScreenParams _ZBufferParams float4 float4 float4 x y Near z Far w /Far Near Far x width y height z /width w /height width height render target x 1 Far/Near y Far/Near z x/far w y/far ( Z

58 4.9 unity_orthoparams float4 x width y heigth z w 1.0 w 0.0 width height unity_cameraprojection float4x4 unity_camerainvprojection float4x4 unity_cameraworldclipplanes[6] float ) ( : w 1 3 3, Unity Shader Cg Cg float Cg float3x3 float4x4 float3 float4 1 x n n x 1 float4 a float4(1.0, 2.0, 3.0, 4.0); float4 b float4(1.0, 2.0, 3.0, 4.0); // y float result dot(a, b); Cg mul float4 v float4(1.0, 2.0, 3.0, 4.0); float4x4 M float4x4(1.0, 0.0, 0.0, 0.0, 0.0, 2.0, 0.0, 0.0, 0.0, 0.0, 3.0, 0.0, 0.0, 0.0, 0.0, 4.0); // v M c float4 column_mul_result mul(m, v); // v M c float4 row_mul_result mul(v, M); // Scolumn_mul_result X g row_mul_result S // mul(m,v) mul(v, tranpose(m)) // mul(v,m) mul(tranpose(m), v) 96

59 4.9 Unity UNITY_MATRIX_MVP Cg Cg float4x (1, 2, 3, 4, 5, 6, 7, 8, 9) 3 x Cg Cg // r M float3x3 M float3x3(1.0, 2.0, 3.0, 4.0, 5.0, 6.0, 7.0, 8.0, 9.0); // M (1.0, 2.0, 3.0) float3 row M[0]; // M float ele M[1][0]; Unity Shader Cg Cg Unity API Unity Matrix4x4 Unity Shader 5C S,D 3 C2DG Shader / VPOS WPOS 5.4 VPOS HLSL WPOS Cg Unity Shader HLSL/Cg / / VPOS WPOS 97

60 4.9 fixed4 frag(float4 sp : VPOS) : SV_Target { // o _ScreenParams.xy return fixed4(sp.xy/_screenparams.xy,0.0,1.0); } 4.49 VPOS/WPOS float4. xy 400 x 300 x [0.5, 400.5] y [0.5, 300.5] OpenGL DirectX zw Unity VPOS/WPOS z [0,1] z 0 z 1 w w 1 1, Near Far Near Far Camera w 1. w viewport space (0, 0) (1, 1) xy Unity ComputeScreenPos UnityCG.cginc ComputeScreenPos struct vertout { float4 pos : SV_POSITION; float4 scrpos : TEXCOORD0; }; vertout vert(appdata_base v) { vertout o; o.pos mul (UNITY_MATRIX_MVP, v.vertex); // S ComputeScreenPos scrpos o.scrpos ComputeScreenPos(o.pos); return o; } fixed4 frag(vertout i) : SV_Target { // f S scrpos.xy o scrpos.w float2 wcoord (i.scrpos.xy/i.scrpos.w); return fixed4(wcoord,0.0,1.0); }

61 4.9 viewport viewport x y clipx clip 2 w clipy clip 2 [-1, 1] NDC [0, 1] ComputeScreenPos UnityCG.cginc ComputeScreenPos inline float4 ComputeScreenPos (float4 pos) { float4 o pos * 0.5f; #if defined(unity_half_texel_offset) o.xy float2(o.x, o.y*_projectionparams.x) + o.w * _ScreenParams.zw; #else o.xy float2(o.x, o.y*_projectionparams.x) + o.w; #endif } o.zw pos.zw; return o; ComputeScreenPos pos. MVP UNITY_HALF_TEXEL_OFFSET Unity DirectX #else _ProjectionParams.x 1 1 Output Output x Output Output y z w w clipx clip clipy clip clipz clip w xy w ComputeScreenPos w Unity ComputeScreenPos w Unity x y w x w y w w w x w y w ( (. xy [0, 1] zw zw zw 99

62 4.11 d z [-Near, Far] w [Near, Far] z [ 1, 1] w [1][2] [3 ] ( [4] [1] Fletcher Dunn, Ian Parberry. 3D Math Primer for Graphics and Game Development (2nd Edition). November 2, 2011 by A K Peters/CRC Press [2] Eric Lengyel. Mathematics for 3D game programming and computer graphics (3rd Edition) by Charles River Media. [3] David Eberly. Conversion of Left-Handed Coordinates to Right-Handed Coordinates [4] d (1, 0, 0) (1, 0, 0) 4.50 m g O P O P X u p e z (0, 0, 1) (0, 0, 1) y

63 4.11 d X Unity z z 10 z z a b (a x, a y, a z ) (b x, b y, b z ) (a y b z a z b y, a z b x a x b z, a x b y a y b x ) (12.5, 10, 25) 3 (1.5, 2) (10, 9) 7 ( 6, 1, 2)

64 4.11 d ( 9, 13, 7) 5 (9, 13, 7) x - p v x NPC (x p) v v x p cosθ θ x - p v 0 θ < 90 x NPC 0 θ > 90 x NPC 0 θ 90 x NPC 2 (x p) v ((10, 6) (4, 2)) ( 3, 4) (6, 4) ( 3, 4) < 0 x NPC φ φ φ 3 cosθ cos cos θ > cos θ < NPC φ φ cosθ < cos θ > NPC cosθ 2 2 ( x p) v cosθ x p v cos φ 2 4 p 7 u p 2 - p 1 v p 3 - p 1 xy u (u x, u y, 0), v (v x, v y, 0) u v (0, 0, u x v y u y v x ) u x v y u y v x

65 4.11 d (1)( 1) + (3)(0) (1)(5) + (3)(2) (2)( 1) + (4)(0) (2)(5) + (4)(2) (1)( 5) + ( 2)(4) + (3)(8) (5)( 5) (1)(4) (4)(8) (6)( 5) + (0)(4) + (3)(8) z θ (3)(1) + (2)(0) + (6)(0) 1 [3 2 6] (3)(0) (2)(1) (6)(0) + + [3 2 6] (3)(0) + (2)(0) + (6)(1) (1)(3) + (0)(2) + (0)(6) (0)(3) (1)(2) (0)(6) (3)(0)(3) + (0)(2) + (1)(6) 6 (3, 2, 6) 2 103

66 4.11 d (3)(1) + (2)(0) + (6)(0) [3 2 6] (3)(0) (2)(1) (6)(0) + + [3 2 18] (3)(2) + (2)( 3) + (6)(3) (1)(3) + (0)(2) + (2)(6) (0)(3) (1)(2) ( 3)(6) (0)(3) + (0)(2) + (3)(6) 18 T [3 2 6] [3 2 6] (3)(1) + (2)(0) + (6)(2) (3)(0) (2)(1) (6)( 3) + + [ ] (3)(0) + (2)(0) + (6)(3) (3)(2) + (2)( 1) + (6)(3) [3 2 6] (3)( 1) (2)(5) (6)( 3) + + [ ] (3)(3) + (2)( 3) + (6)(4) (2)(3)+(-1)(2)+(3)(6) ( 1)(3)+(5)(2)+( 3)(6) (3)(3)+( 3)(2)+(4)(6) 27 (22, 11, 27) symmetric matrix 104