平面
將三維物體表面剖分為一系列的三角形面，物體的光照亮度處理就轉(zhuǎn)化為對這些平面三角形的照明處理，從而可簡單地通過平面三角形的法向量與光的入射方向的夾角，來確定各種入射光對平面上每一點所貢獻的亮度值。
對于三角形3個頂點p0，p1，p2構(gòu)成的平面，選取一個垂直于該平面的法向量n。假設(shè)點p為平面上的任意一點，由于n . (p - p0) = 0，從而n . p
- n . p0 = 0，由此展開向量的點積運算可知，平面方程具有如下的一般形式：ax+by+cz+d=0。如果要具體求出平面的方程，可取n=(p1-p0) x
(p2 - p0)計算出來。
因為平面的方程式由4個系數(shù)a, b, c,
d就可確定，因此，DirectX提供了如下的D3DXPLANE結(jié)構(gòu)體保存方程式的4個系數(shù)，當然結(jié)構(gòu)體還提供了基于方程式系數(shù)的加減乘除等運算。
typedef struct D3DXPLANE
{
#ifdef __cplusplus
public:
D3DXPLANE() {}
D3DXPLANE( CONST FLOAT* );
D3DXPLANE( CONST D3DXFLOAT16* );
D3DXPLANE( FLOAT a, FLOAT b, FLOAT c, FLOAT d );
// casting
operator FLOAT* ();
operator CONST FLOAT* () const;
// assignment operators
D3DXPLANE& operator *= ( FLOAT );
D3DXPLANE& operator /= ( FLOAT );
// unary operators
D3DXPLANE operator + () const;
D3DXPLANE operator - () const;
// binary operators
D3DXPLANE operator * ( FLOAT ) const;
D3DXPLANE operator / ( FLOAT ) const;
friend D3DXPLANE operator * ( FLOAT, CONST D3DXPLANE& );
BOOL operator == ( CONST D3DXPLANE& ) const;
BOOL operator != ( CONST D3DXPLANE& ) const;
#endif //__cplusplus
FLOAT a, b, c, d;
} D3DXPLANE, *LPD3DXPLANE;
DirectX還提供了基于三點的向量求平面的
D3DXPlaneFromPoints函數(shù)以及基于一點和一個法向量求平面的D3DXPlaneFromPointNormal函數(shù)。
Constructs a
plane from three points.
D3DXPLANE * D3DXPlaneFromPoints(
D3DXPLANE * pOut,
CONST D3DXVECTOR3 * pV1,
CONST D3DXVECTOR3 * pV2,
CONST D3DXVECTOR3 * pV3
);
Parameters
- pOut
- [in, out] Pointer to the D3DXPLANE structure that is the result of the
operation.
- pV1
- [in] Pointer to a D3DXVECTOR3 structure, defining one of the points used
to construct the plane.
- pV2
- [in] Pointer to a D3DXVECTOR3 structure, defining one of the
points used to construct the plane.
- pV3
- [in] Pointer to a D3DXVECTOR3 structure, defining one of the
points used to construct the plane.
Return Values
Pointer to the D3DXPLANE structure constructed from the given points.
Remarks
The return value for this function is the same value returned in the pOut
parameter. In this way, the D3DXPlaneFromPoints function can be used as a
parameter for another function.
Constructs a plane from a point and a normal.
D3DXPLANE * D3DXPlaneFromPointNormal(
D3DXPLANE * pOut,
CONST D3DXVECTOR3 * pPoint,
CONST D3DXVECTOR3 * pNormal
);
Parameters
- pOut
- [in, out] Pointer to the D3DXPLANE structure that is the result of the
operation.
- pPoint
- [in] Pointer to a D3DXVECTOR3 structure, defining the point used to
construct the plane.
- pNormal
- [in] Pointer to a D3DXVECTOR3 structure, defining the normal used
to construct the plane.
Return Values
Pointer to the D3DXPLANE structure constructed from the point and the
normal.
Remarks
The return value for this function is the same value returned in the pOut
parameter. In this way, the D3DXPlaneFromPointNormal function can be used
as a parameter for another function.
要正確運行以下的示例程序，需要在工程中包含d3dx9.lib，或者在main函數(shù)前加入
#pragma comment(lib, "d3dx9.lib")
以指示編譯器鏈接d3dx9.lib。
代碼示例：
#include <stdio.h>
#include <D3DX9Math.h>
#pragma warning(disable : 4305)
int main()
{
D3DXVECTOR3 P0(8.0, -3.0, 6.0);
D3DXVECTOR3 P1(2.0, -3.0, -5.0);
D3DXVECTOR3 P2(-1.0, -3.0, 10.0);
D3DXPLANE plane;
D3DXPlaneFromPoints(&plane, &P0, &P1, &P2);
printf("Plane equation: %fx + %fy + %fz + %f = 0", plane.a, plane.b, plane.c, plane.d);
printf("\n\n\n");
return 0;
}
結(jié)果輸出：
Plane equation: 0.000000x + 1.000000y + 0.000000z + 3.000000 = 0
直線與平面的交點
Finds the intersection between a plane and a line.
D3DXVECTOR3 * D3DXPlaneIntersectLine(
D3DXVECTOR3 * pOut,
CONST D3DXPLANE * pP,
CONST D3DXVECTOR3 * pV1,
CONST D3DXVECTOR3 * pV2
);
Parameters
- pOut
- [in, out] Pointer to a D3DXVECTOR3 structure, identifying the
intersection between the specified plane and line.
- pP
- [in] Pointer to the source D3DXPLANE structure.
- pV1
- [in] Pointer to a source D3DXVECTOR3 structure, defining a line
starting point.
- pV2
- [in] Pointer to a source D3DXVECTOR3 structure, defining a line
ending point.
Return Values
Pointer to a D3DXVECTOR3 structure that is the intersection between
the specified plane and line.
Remarks
If the line is parallel to the plane, NULL is returned.
The return value for this function is the same value returned in the pOut
parameter. In this way, the D3DXPlaneIntersectLine function can be used
as a parameter for another function.
代碼示例
#include <stdio.h>
#include <D3DX9Math.h>
#pragma comment(lib, "d3dx9.lib")
#pragma warning(disable : 4305)
int main()
{
D3DXVECTOR3 A(0, 0, 0);
D3DXVECTOR3 B(5, 5, 5);
D3DXPLANE plane(0, 1, 0, -1); // equation: x = 1
D3DXVECTOR3 intersect_point;
D3DXPlaneIntersectLine(&intersect_point, &plane, &A, &B);
// intersect point must be: (1, 1, 1)
printf("Intersect point: (%f, %f, %f)\n\n", intersect_point.x, intersect_point.y, intersect_point.z);
return 0;
}
輸出：
Intersect point: (1.000000, 1.000000, 1.000000)
點和平面的位置關(guān)系
對于1個給定了法向量n和內(nèi)部一點p0的平面，任意1點p與該平面的位置關(guān)系可用兩向量的點積n . (p - p0) 進行判斷，分為如下的3種情形：
(1) 點p在法向量n的同一側(cè)，即n . (p - p0) > 0
(2) 點p在平面上，即n . (p - p0) = 0
(3) 點p在法向量n的反側(cè)，即n . (p - p0) < 0
對于以方程式ax+by+cz+d=0給出的平面，由系數(shù)a, b和c構(gòu)成的向量(a, b, c)必然是1個垂直于該平面的向量法向量。
（證明如下：設(shè)平面上任意兩個點(x1,y1,z1), (x2,y2,z2)，代入方程：
ax1+by1+cz1+d=0; ax2+by2+cz2+d=0 ;
====>
a(x1-x2)+b(y1-y2)+c(z1-z2)=0
====>
(a,b,c)(x1-x2, y1-y2, z1-z2)=0
即點積為0，因為兩點是任意的，所以具有一般性，即向量垂直與平面內(nèi)任意一條直線。）
稱向量(a, b, c)為平面ax + by + cz +d = 0的默認法向量。
判斷一點p(x0, y0, z0)與ax+by+z+d=0平面的位置關(guān)系，可以直接利用ax0+by0+cz0+d的符號來決定。
ax0 +by0+cz0+d > 0表示點p在默認法向量的一側(cè)，也可以說是位于平面的前方。
ax0+by0+cz0+d ＝ 0表示點p在平面上。
ax0+by0+cz0+d < 0表示點p在默認法向量的另一側(cè)，也可以說是位于平面的后方。
平面(a, b, c, d)與任意點(x0, y0, z0)的ax0 + by0 +
cz0值，可用DirectX提供的D3DXPlaneDotCoord函數(shù)計算出來，利用該函數(shù)返回值的符號可決定點在平面的那一側(cè)。
Computes the dot product of a plane and a 3D vector. The w parameter of the
vector is assumed to be 1.
FLOAT D3DXPlaneDotCoord(
CONST D3DXPLANE * pP,
CONST D3DXVECTOR3 * pV
);
Parameters
- pP
- [in] Pointer to a source D3DXPLANE structure.
- pV
- [in] Pointer to a source D3DXVECTOR3 structure.
Return Values
The dot product of the plane and 3D vector.
Remarks
Given a plane (a, b, c, d) and a 3D vector (x, y, z) the return value of this
function is a*x + b*y + c*z + d*1. The D3DXPlaneDotCoord function is
useful for determining the plane's relationship with a coordinate in 3D space.
如果要計算平面法線和另一法線的夾角，可使用D3DXPlaneDotNormal。
Computes the dot product of a plane and a 3D vector. The w parameter of the
vector is assumed to be 0.
FLOAT D3DXPlaneDotNormal(
CONST D3DXPLANE * pP,
CONST D3DXVECTOR3 * pV
);
Parameters
- pP
- [in] Pointer to a source D3DXPLANE structure.
- pV
- [in] Pointer to a source D3DXVECTOR3 structure.
Return Values
The dot product of the plane and 3D vector.
Remarks
Given a plane (a, b, c, d) and a 3D vector (x, y, z) the return value of this
function is a*x + b*y + c*z + d*0. The D3DXPlaneDotNormal function is
useful for calculating the angle between the normal of the plane, and another
normal.
平面單位化：
如果平面ax+by+cz+d=0的默認法向量n =(a,b,c)為單位向量，那么該平面稱為單位化平面。
Normalizes the
plane coefficients so that the plane normal has unit length.
D3DXPLANE * D3DXPlaneNormalize(
D3DXPLANE * pOut,
CONST D3DXPLANE * pP
);
Parameters
- pOut
- [in, out] Pointer to the D3DXPLANE structure that is the result of the
operation.
- pP
- [in] Pointer to the source D3DXPLANE structure.
Return Values
Pointer to a D3DXPLANE structure that represents the normal of the
plane.
Remarks
This function normalizes a plane so that |a,b,c| == 1.
The return value for this function is the same value returned in the pOut
parameter. In this way, this function can be used as a parameter for another
function.
代碼示例：
#include <stdio.h>
#include <D3DX9Math.h>
#pragma warning(disable : 4305)
int main()
{
D3DXPLANE plane(1, 0, 0, -1);
D3DXVECTOR3 p(5, 0, 0);
float dot = D3DXPlaneDotCoord(&plane, &p);
printf("dot = %f\n\n", dot);
dot = D3DXPlaneDotNormal(&plane, &p);
printf("dot = %f\n\n", dot);
D3DXPLANE plane2(3, 4, 0, 0);
D3DXPlaneNormalize(&plane2, &plane2);
printf("After normalize: (%f, %f, %f, %f)\n\n", plane2.a, plane2.b, plane2.c, plane2.d);
return 0;
}
輸出：
dot = 4.000000
dot = 5.000000
After normalize: (0.600000, 0.800000, 0.000000, 0.000000)