Simple mathematical knowledge related to 3D

Three dimensional coordinate system ：

 

 

Modules of vectors ： The distance from the start of the vector to the end

FLOAT D3DXVec3Length( // Returns the magnitude.

CONST D3DXVECTOR3* pV // The vector to compute the length of.

);

Vector normalization （ Normalization ） Scale the length of the vector to 1

D3DXVECTOR3 *D3DXVec3Normalize(

D3DXVECTOR3* pOut, // Result.

CONST D3DXVECTOR3* pV // The vector to normalize.

);

Vector addition and subtraction ：

D3DXVECTOR3 u(2.0f, 0.0f, 1.0f);

D3DXVECTOR3 v(0.0f, -1.0f, 5.0f);

D3DXVECTOR3 sum = u + (-)  v; // = (2.0f, -1.0f, 6.0f)

D3 Product of scalar and vector ：

 

D3DXVECTOR3 u(1.0f, 1.0f, -1.0f);

D3DXVECTOR3 scaledVec = u * 10.0f;

Dot product （ The result is scalar ）：

 

u · v =|u||v|cosθ

adopt u*v Judge the angle between two vectors ,0 vertical ,<0 Greater than 90 degree ,>0 Less than 90 degree

 

float dot = D3DXVec3Dot( &u, &v );

 

Cross product （ The result is another vector ）：

The length is the modulus of two vectors multiplied by the sine of the included angle , In any coordinate system, use the rule of any coordinate system to judge the direction of cross product .

Direction ： Point to the second vector according to the first vector

 

D3DXVec3Cross

 

Matrix multiplication ：

 

 

Inverse matrix ：

Only a square matrix has an inverse matrix , And not necessarily

p’= pR  And suppose we know p’ and R Sure seek p

D3DMatrixIverse

Transposition ：

 

D3DMatrixTranspose

 

Geometric transformation ：

Homogeneous coordinates （ use n+1 Dimensional coordinates to represent n Dimensional coordinates ）：

3*3 The transformation matrix of has certain limitations , Basic set transformations such as translation perspective projection cannot be completed , Therefore, it is necessary to expand the coordinates of three-dimensional points and vectors to 4 Homogeneous coordinates of dimensions

Homogeneous coordinates of points ：（x,y,z,1）

Homogeneous coordinates of vectors ：（x,y,z,0）w The value is 0 It can ensure that the vector is not affected by the translation transformation , So for points and vectors in the same graph , There is no need to distinguish them , Just use a transformation matrix to perform geometric transformation

Translation transformation matrix ：

D3DXMATRIX *D3DXMatrixTranslation(

D3DXMATRIX* pOut, //  Returns the translated matrix .

FLOAT x, // x Unit of axis movement

FLOAT y, // y Unit of axis movement

FLOAT z // z Unit of axis movement

);

 

The zoom ：

D3DXMATRIX *D3DXMatrixScaling(

D3DXMATRIX* pOut, //  Returns the scaled matrix

FLOAT sx, // x Scale of axis scaling

FLOAT sy, // y Scale of axis scaling

FLOAT sz // z Scale of axis scaling .

);

 

rotate ：

D3DXMATRIX *D3DXMatrixRotationX(

D3DXMATRIX* pOut, //  Returns the rotated matrix

FLOAT Angle // Angle Is the arc of rotation

);