d3dx9math.h 수학적 함수들 수학적 설명

2012. 11. 2. 11:25

D3DXmath.htm

Matematika v Direct3DX

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)

v = Normalized(Eye−At)
l = Up×v
u = v×l

	/	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	/

D3DXMatrixLookAtLH(Out, Eye, At, Up)

v = Normalized(At−Eye)
r = Up×v
u = v×r

	/	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	/

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)

s = sin Angle
c = cos Angle
d = 1−c
(x,y,z) = V

	/	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	/

D3DXMatrixRotationQuaternion(Out, Q)

(x,y,z,w) = Q

	/	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	/

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)

(sa,sb,sc) = sin (Roll, Pitch, Yaw)
(ca,cb,cc) = cos (Roll, Pitch, Yaw)

	/	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	/

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

저작자표시 비영리 동일조건

'수학 (Mathematics) > 3D수학' 카테고리의 다른 글

회전행렬에서 회전축'회전각의 분리(angular displacement) (0)	2012.11.02
d3dx9math.h 함수가 하는 일 설명 (0)	2012.11.02
언프로젝션과 픽킹 (0)	2012.11.02
D3DXVec3TransformNormal & D3DXVec3TransformCoord (0)	2012.11.02
D3DXVec3TransformCoord & D3DXVec3Transform 둘의 차이는 w 로 나누었는가의 차이 (0)	2012.11.02

3DMP

d3dx9math.h 수학적 함수들 수학적 설명

http://mormegil.wz.cz/prog/3dref/D3DXmath.htm

Matematika v Direct3DX

Vektory

D3DXVecNAdd(Out, V1, V2)

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

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

D3DXVecNDot(U, V)

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

D3DXVecNLength(V)

D3DXVecNLengthSq(V)

D3DXVecNLerp(Out, V1, V2, s)

D3DXVecNMaximize(Out, U, V)

D3DXVecNMinimize(Out, U, V)

D3DXVecNNormalize(Out, V)

D3DXVecNScale(Out, V, s)

D3DXVecNSubtract(Out, V1, V2)

2D Vektory

D3DXVec2CCW(U, V)

D3DXVec2Transform(Out, V, M)

D3DXVec2TransformCoord(Out, V, M)

D3DXVec2TransformNormal(Out, V, M)

3D Vektory

D3DXVec3Cross(Out, V1, V2)

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

D3DXVec3Transform(Out, V, M)

D3DXVec3TransformCoord(Out, V, M)

D3DXVec3TransformNormal(Out, V, M)

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

4D Vektory

D3DXVec4Cross(Out, U, V, W)

D3DXVec4Transform(Out, V, M)

Matice (4×4)

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

D3DXMatrixfDeterminant(M)

D3DXMatrixIdentity(Out)

D3DXMatrixInverse(Out, D, M)

D3DXMatrixIsIdentity(M)

D3DXMatrixLookAtRH(Out, Eye, At, Up)

D3DXMatrixLookAtLH(Out, Eye, At, Up)

D3DXMatrixMultiply(Out, M1, M2)

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

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

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

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

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

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

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

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

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

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

D3DXMatrixReflect(Out, Plane)

D3DXMatrixRotationAxis(Out, V, Angle)

D3DXMatrixRotationQuaternion(Out, Q)

D3DXMatrixRotationX(Out, Angle)

D3DXMatrixRotationY(Out, Angle)

D3DXMatrixRotationYawPitchRoll(Out, Yaw, Pitch, Roll)

D3DXMatrixRotationZ(Out, Angle)

D3DXMatrixScaling(Out, x, y, z)

D3DXMatrixShadow(Out, Light, Plane)

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

D3DXMatrixTranslation(Out, x, y, z)

D3DXMatrixTranspose(Out, M)

Roviny

D3DXPlaneDot(P, V)

D3DXPlaneDotCoord(P, V)

D3DXPlaneDotNormal(P, V)

D3DXPlaneIntersectLine(Out, P, U, V)

D3DXPlaneFromPointNormal(Out, P, N)

D3DXPlaneNormalize(Out, P)

D3DXPlaneFromPoints(Out, A, B, C)

D3DXPlaneTransform(Out, P, M)

'수학 (Mathematics) > 3D수학' 카테고리의 다른 글

+ Recent posts

티스토리툴바