Matematika v Direct3DX

(English info)

V této kapitole je seznam matematických funkcí Direct3DX (což je jakýsi toolkit pro Direct3D (DirectXGraphics), jistá obdoba knihovny GLUT pro OpenGL), spolu s popisem matematiky, kterou tyto funkce provádí.

Upozorňuji, že toto není oficiální dokumentace k Direct3DX (ta se nalézá někde na MSDN).

Vektory

D3DXVecNAdd(Out, V1, V2)

Out = V1 + V2

D3DXVecNBaryCentric(Out, V1, V2, V3, f, g)

Out = V1 + f⋅(V2−V1) + g⋅(V3−V1)

D3DXVecNCatmullRom(Out, V1, V2, V3, V4, s)

/ 0 1 0 0 \ / V1 \

| | | |

| −0,5 0 0,5 0 | | V2 |

Out = (1,s,s²,s³) | | | |

| 1 −2,5 2 −0,5 | | V3 |

| | | |

\ −0,5 1,5 −1,5 0,5 / \ V4 /

D3DXVecNDot(U, V)

Result = U ⋅ V = ∑_iU_iV_i

D3DXVecNHermite(Out, V1, T1, V2, T2, s)

/ 1 0 0 0 \ / V1 \

| | | |

| 0 0 1 0 | | V2 |

Out = (1,s,s²s³) | | | |

| −3 3 −2 −1 | | T1 |

| | | |

\ 2 −2 1 1 / \ T2 /

D3DXVecNLength(V)

Result = |V| = √(∑_iV_i²)

D3DXVecNLengthSq(V)

Result = |V|² = ∑_iV_i²

D3DXVecNLerp(Out, V1, V2, s)

Out = V1 + s(V2−V1);

D3DXVecNMaximize(Out, U, V)

Out_i = max(U_i, V_i)

D3DXVecNMinimize(Out, U, V)

Out_i = min(U_i, V_i)

D3DXVecNNormalize(Out, V)

Out = ¹⁄_|V|⋅V

D3DXVecNScale(Out, V, s)

Out = sV

D3DXVecNSubtract(Out, V1, V2)

Out = V1 − V2

2D Vektory

D3DXVec2CCW(U, V)

Result = U_xV_y − U_yV_x

D3DXVec2Transform(Out, V, M)

a = (V_x, V_y, 0, 1)
b = (a×M)^T
Out = (b_x, b_y)

D3DXVec2TransformCoord(Out, V, M)

a = (V_x, V_y, 0, 1)
b = (a×M)^T
Out = ¹⁄_{b_w}(b_x, b_y)

D3DXVec2TransformNormal(Out, V, M)

a = (V_x, V_y, 0, 0)
b = (a×M)^T
Out = (b_x, b_y)

3D Vektory

D3DXVec3Cross(Out, V1, V2)

Out = V1 × V2

D3DXVec3Project(Out, V, Viewport, Proj, View, World)

a = (V_x, V_y, V_z, 1)
b = a×World×View×Proj
c = ^b⁄_{b_w}
Out_x = Viewport_X + Viewport_Width*(1+c_x)/2
Out_y = Viewport_Y + Viewport_Height*(1-c_y)/2
Out_z = Viewport_MinZ + c_z*(Viewport_MaxZ-Viewport_MinZ)

D3DXVec3Transform(Out, V, M)

a = (V_x, V_y, V_z, 1)
b = (a×M)^T
Out = (b_x, b_y, b_z)

D3DXVec3TransformCoord(Out, V, M)

a = (V_x, V_y, V_z, 1)
b = (a×M)^T
Out = ¹⁄_{b_w}(b_x, b_y, b_z)

D3DXVec3TransformNormal(Out, V, M)

a = (V_x, V_y, V_z, 0)
b = (a×M)^T
Out = (b_x, b_y, b_z)

D3DXVec3Unproject(Out, V, Viewport, Proj, View, World)

M = (World×View×Proj)⁻¹
a_x = (V_x-Viewport_x)*2/Viewport_Width - 1
a_y = 1 - (V_y-Viewport_y)*2/Viewport_Height
a_z = (V_z-Viewport_MinZ)/(Viewport_MaxZ-Viewport_MinZ)
a_w = 1
b = (a×M)^T
Out = ¹⁄_{b_w}(b_x, b_y, b_z)

4D Vektory

D3DXVec4Cross(Out, U, V, W)

a = V_xW_y − V_yW_x
b = V_xW_z − V_zW_x
c = V_xW_w − V_wW_x
d = V_yW_z − V_zW_y
e = V_yW_w − V_wW_y
f = V_zW_w − V_wW_z
Out_x = fU_y − eU_z + dU_w
Out_y = fU_x + cU_z − bU_w
Out_z = eU_x − cU_y + aU_w
Out_w = dU_x + bU_y − aU_z

D3DXVec4Transform(Out, V, M)

Out = (V×M)^T

Matice (4×4)

D3DXMatrixAffineTransformation(Out, s, c, r, t)

/ s(2(r_y² + r_z²) − 1) 2s(r_xr_y + r_zr_w) 2s(r_xr_z − r_yr_w) 0 \

| |

| 2s(r_xr_y − r_zr_w) s(2(r_x² + r_z²) − 1) 2s(r_yr_z + r_xr_w) 0 |

Out = | |

| 2s(r_xr_z + r_yr_w) 2s(r_yr_z − r_xr_w) s(2(r_x² + r_y²) − 1) 0 |

| |

\ t_x+2(c_xr_y²+c_xr_z²−c_yr_xr_y+c_yr_zr_w−c_zr_xr_z−c_zr_yr_w) t_y+2(c_yr_x²+c_yr_z²−c_xr_xr_y−c_xr_zr_w−c_zr_yr_z+c_zr_xr_w) t_z+2(c_zr_x²+c_zr_y²−c_xr_xr_z+c_xr_yr_w−c_yr_yr_z−c_yr_xr_w) 1 /

D3DXMatrixfDeterminant(M)

Result = det M

D3DXMatrixIdentity(Out)

/ 1 0 0 0 \

| |

| 0 1 0 0 |

Out = E = | |

| 0 0 1 0 |

| |

\ 0 0 0 1 /

D3DXMatrixInverse(Out, D, M)

D = det M
Out = M⁻¹

D3DXMatrixIsIdentity(M)

Result = M≡E

D3DXMatrixLookAtRH(Out, Eye, At, Up)

D3DXMatrixLookAtLH(Out, Eye, At, Up)

D3DXMatrixMultiply(Out, M1, M2)

Out = M1×M2

D3DXMatrixOrthoRH(Out, w, h, n, f)

Q = (f−n)⁻¹

/ ²⁄_w 0 0 0 \

| |

| 0 ²⁄_h 0 0 |

Out = | |

| 0 0 −Q 0 |

| |

\ 0 0 −Qn 1 /

D3DXMatrixOrthoLH(Out, w, h, n, f)

Q = (f−n)⁻¹

/ ²⁄_w 0 0 0 \

| |

| 0 ²⁄_h 0 0 |

Out = | |

| 0 0 Q 0 |

| |

\ 0 0 −Qn 1 /

D3DXMatrixOrthoOffCenterRH(Out, l, r, t, b, n, f)

Q = (f−n)⁻¹
w = r−l
h = b−t

/ ²⁄_w 0 0 0 \

| |

| 0 ²⁄_h 0 0 |

Out = | |

| 0 0 −Q 0 |

| |

\ ^−(r+l)⁄₂ ^−(t+b)⁄₂ −Qn 1 /

D3DXMatrixOrthoOffCenterLH(Out, l, r, t, b, n, f)

Q = (f−n)⁻¹
w = r−l
h = b−t

/ ²⁄_w 0 0 0 \

| |

| 0 ²⁄_h 0 0 |

Out = | |

| 0 0 Q 0 |

| |

\ ^−(r+l)⁄₂ ^−(t+b)⁄₂ −Qn 1 /

D3DXMatrixPerspectiveRH(Out, w, h, n, f)

Q = (f−n)⁻¹

/ ²⁄_w 0 0 0 \

| |

| 0 ²⁄_h 0 0 |

Out = | |

| 0 0 −Qf −1 |

| |

\ 0 0 −Qnf 0 /

D3DXMatrixPerspectiveLH(Out, w, h, n, f)

Q = (f−n)⁻¹

/ ²⁄_w 0 0 0 \

| |

| 0 ²⁄_h 0 0 |

Out = | |

| 0 0 Qf 1 |

| |

\ 0 0 −Qnf 0 /

D3DXMatrixPerspectiveFovRH(Out, Fovy, Aspect, n, f)

Q = (f−n)⁻¹
y = ^cotg(Fovy)⁄₂

/ ^y⁄_Aspect 0 0 0 \

| |

| 0 y 0 0 |

Out = | |

| 0 0 −Qf −1 |

| |

\ 0 0 −Qnf 0 /

D3DXMatrixPerspectiveFovLH(Out, Fovy, Aspect, n, f)

Q = (f−n)⁻¹
y = ^cotg(Fovy)⁄₂

/ ^y⁄_Aspect 0 0 0 \

| |

| 0 y 0 0 |

Out = | |

| 0 0 Qf 1 |

| |

\ 0 0 −Qnf 0 /

D3DXMatrixPerspectiveOffCenterRH(Out, l, r, t, b, n, f)

Q = (f−n)⁻¹
w = r−l
h = b−t

/ ²⁄_w 0 0 0 \

| |

| 0 ²⁄_h 0 0 |

Out = | |

| ^(r+l)⁄_w ^(t+b)⁄_h −Qf −1 |

| |

\ 0 0 −Qnf 0 /

D3DXMatrixPerspectiveOffCenterLH(Out, l, r, t, b, n, f)

Q = (f−n)⁻¹
w = r−l
h = b−t

/ ²⁄_w 0 0 0 \

| |

| 0 ²⁄_h 0 0 |

Out = | |

| ^−(r+l)⁄_w ^−(t+b)⁄_h Qf 1 |

| |

\ 0 0 −Qnf 0 /

D3DXMatrixReflect(Out, Plane)

(a,b,c,d) = ^{(Plane_a,Plane_b,Plane_c,Plane_d)}⁄_{√(Plane_a²+Plane_b²+Plane_c²)}

/ 1−2a² −2ba −2ca 0 \

| |

| −2ab 1−2b² −2cb 0 |

Out = | |

| −2ac −2bc 1−2c² 0 |

| |

\ −2ad −2bd −2cd 1 /

D3DXMatrixRotationAxis(Out, V, Angle)

D3DXMatrixRotationQuaternion(Out, Q)

D3DXMatrixRotationX(Out, Angle)

s = sin Angle
c = cos Angle

/ 1 0 0 0 \

| |

| 0 c s 0 |

Out = | |

| 0 −s c 0 |

| |

\ 0 0 0 1 /

D3DXMatrixRotationY(Out, Angle)

s = sin Angle
c = cos Angle

/ c 0 −s 0 \

| |

| 0 1 0 0 |

Out = | |

| s 0 c 0 |

| |

\ 0 0 0 1 /

D3DXMatrixRotationYawPitchRoll(Out, Yaw, Pitch, Roll)

D3DXMatrixRotationZ(Out, Angle)

s = sin Angle
c = cos Angle

/ c s 0 0 \

| |

| −s c 0 0 |

Out = | |

| 0 0 1 0 |

| |

\ 0 0 0 1 /

D3DXMatrixScaling(Out, x, y, z)

/ x 0 0 0 \

| |

| 0 y 0 0 |

Out = | |

| 0 0 z 0 |

| |

\ 0 0 0 1 /

D3DXMatrixShadow(Out, Light, Plane)

(a,b,c,d) = ^{(Plane_a,Plane_b,Plane_c,Plane_d)}⁄_{√(Plane_a²+Plane_b²+Plane_c²)}
(x,y,z,w) = Light
f = Light_x⋅Plane_a + Light_y⋅Plane_b + Light_z⋅Plane_c + Light_w⋅Plane_d

/ f−xa −ya −za −wa \

| |

| −xb f−yb −zb −wb |

Out = | |

| −xc −yc f−zc −wc |

| |

\ −xd −yd −zd f−wd /

D3DXMatrixTransformation(Out, Scenter, Srot, Scaling, Rotcenter, Rot, Trans)

D3DXMatrixTranslation(A, -Scenter_x, -Scenter_y, -Scenter_z)
D3DXMatrixScaling(B, Scaling_x, Scaling_y, Scaling_z)
D3DXMatrixRotationQuaternion(C, Srot)
u = Scenter − Rotcenter
D3DXMatrixTranslation(D, u_x, u_y, u_z)
D3DXMatrixRotationQuaternion(E, Rot)
v = Rotcenter + Trans
D3DXMatrixTranslation(F, v_x, v_y, v_z)
Out = A×C^T×B×C×D×E×F

D3DXMatrixTranslation(Out, x, y, z)

/ 1 0 0 0 \

| |

| 0 1 0 0 |

Out = | |

| 0 0 1 0 |

| |

\ x y z 1 /

D3DXMatrixTranspose(Out, M)

Out = M^T

Roviny

D3DXPlaneDot(P, V)

Result = (P_a, P_b, P_c, P_d)⋅V

D3DXPlaneDotCoord(P, V)

Result = (P_a, P_b, P_c)⋅V + P_d

D3DXPlaneDotNormal(P, V)

Result = (P_a, P_b, P_c)⋅V

D3DXPlaneIntersectLine(Out, P, U, V)

n = (Plane_a, Plane_b, Plane_c)
d = V − U
Out = U − d⋅^{(P_d + n⋅U)}⁄_(d⋅n) [iff d⋅n ≠ 0]

D3DXPlaneFromPointNormal(Out, P, N)

Plane_a = N_x
Plane_b = N_y
Plane_c = N_z
Plane_d = −N⋅P

D3DXPlaneNormalize(Out, P)

q = ¹⁄_{√(P_a² + P_b² + P_c²)}
Out_a = q⋅P_a
Out_b = q⋅P_b
Out_c = q⋅P_c
Out_d = q⋅P_d

D3DXPlaneFromPoints(Out, A, B, C)

v = (B − A) × (C − A)
n = ¹⁄_|v| v
Out_a = n_x
Out_b = n_y
Out_c = n_z
Out_d = −n⋅A

D3DXPlaneTransform(Out, P, M)

Q = ^P⁄_|P|
u = (Q_a, Q_b, Q_c, 0)
D = Q_d
A = (−Du_x, −Du_y, −Du_z, 1)
B = A×M
v = u×M
q = ¹⁄_|v|
Out_a = qv_x
Out_b = qv_y
Out_c = qv_z
Out_d = −qv⋅B

	/	0	1	0	0	\	/	V1	\
	\|					\|	\|		\|
	\|	−0,5	0	0,5	0	\|	\|	V2	\|
Out = (1,s,s²,s³)	\|					\|	\|		\|
	\|	1	−2,5	2	−0,5	\|	\|	V3	\|
	\|					\|	\|		\|
	\	−0,5	1,5	−1,5	0,5	/	\	V4	/

	/	1	0	0	0	\	/	V1	\
	\|					\|	\|		\|
	\|	0	0	1	0	\|	\|	V2	\|
Out = (1,s,s²s³)	\|					\|	\|		\|
	\|	−3	3	−2	−1	\|	\|	T1	\|
	\|					\|	\|		\|
	\	2	−2	1	1	/	\	T2	/

	/	s(2(r_y² + r_z²) − 1)	2s(r_xr_y + r_zr_w)	2s(r_xr_z − r_yr_w)	0	\
	\|					\|
	\|	2s(r_xr_y − r_zr_w)	s(2(r_x² + r_z²) − 1)	2s(r_yr_z + r_xr_w)	0	\|
Out =	\|					\|
	\|	2s(r_xr_z + r_yr_w)	2s(r_yr_z − r_xr_w)	s(2(r_x² + r_y²) − 1)	0	\|
	\|					\|
	\	t_x+2(c_xr_y²+c_xr_z²−c_yr_xr_y+c_yr_zr_w−c_zr_xr_z−c_zr_yr_w)	t_y+2(c_yr_x²+c_yr_z²−c_xr_xr_y−c_xr_zr_w−c_zr_yr_z+c_zr_xr_w)	t_z+2(c_zr_x²+c_zr_y²−c_xr_xr_z+c_xr_yr_w−c_yr_yr_z−c_yr_xr_w)	1	/

	/	l_x	u_x	v_x	0	\
	\|					\|
	\|	l_y	u_y	v_y	0	\|
Out =	\|					\|
	\|	l_z	u_z	v_z	0	\|
	\|					\|
	\	−l⋅Eye	−u⋅Eye	−v⋅Eye	1	/

	/	r_x	u_x	v_x	0	\
	\|					\|
	\|	r_y	u_y	v_y	0	\|
Out =	\|					\|
	\|	r_z	u_z	v_z	0	\|
	\|					\|
	\	−r⋅Eye	−u⋅Eye	−v⋅Eye	1	/

	/	²⁄_w	0	0	0	\
	\|					\|
	\|	0	²⁄_h	0	0	\|
Out =	\|					\|
	\|	0	0	−Q	0	\|
	\|					\|
	\	^−(r+l)⁄₂	^−(t+b)⁄₂	−Qn	1	/

	/	^y⁄_Aspect	0	0	0	\
	\|					\|
	\|	0	y	0	0	\|
Out =	\|					\|
	\|	0	0	−Qf	−1	\|
	\|					\|
	\	0	0	−Qnf	0	/

	/	1−2a²	−2ba	−2ca	0	\
	\|					\|
	\|	−2ab	1−2b²	−2cb	0	\|
Out =	\|					\|
	\|	−2ac	−2bc	1−2c²	0	\|
	\|					\|
	\	−2ad	−2bd	−2cd	1	/

	/	dx²+c	dxy+zs	dxz−ys	0	\
	\|					\|
	\|	dxy−zs	dy²+c	dyz+xs	0	\|
Out =	\|					\|
	\|	dxy+ys	dyz−xs	dz²+c	0	\|
	\|					\|
	\	0	0	0	1	/

	/	1−2y²−2z²	2xy+2zw	2xz−2yw	0	\
	\|					\|
	\|	2xy−2zw	1−2x²−2z²	2yz+2xw	0	\|
Out =	\|					\|
	\|	2xz+2yw	2yz−2xw	1−2x²−2y²	0	\|
	\|					\|
	\	0	0	0	1	/

	/	ca⋅cc+sa⋅sb⋅sc	−sa⋅cc+ca⋅sb⋅sc	cb⋅sc	0	\
	\|					\|
	\|	sa⋅cb	ca⋅cb	−sb	0	\|
Out =	\|					\|
	\|	−ca⋅sc+sa⋅sb⋅cc	sa⋅sc+ca⋅sb⋅cc	cb⋅cc	0	\|
	\|					\|
	\	0	0	0	1	/

	/	f−xa	−ya	−za	−wa	\
	\|					\|
	\|	−xb	f−yb	−zb	−wb	\|
Out =	\|					\|
	\|	−xc	−yc	f−zc	−wc	\|
	\|					\|
	\	−xd	−yd	−zd	f−wd	/