Personal notes of graphics (1)

vector ：AB=B-A Indicates direction and length , The vector can move at will , After moving, it is still the same vector

Unit vector ： vector / The length of the vector （ It can mean direction ）

Vector addition ：A+B Quadrilateral rule

 

Triangle rule （ Put the ends together ）

  The default vector is column vector Transpose to row vector

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Vector point multiplication ：（ Give the vector a value , Give me two vectors, and I can use the formula to calculate the included angle or the cosine of the included angle ）

 （ If both vectors are unit vectors , Then their dot multiplication is directly the cosine of the included angle ）

  The nature of dot multiplication ：

  It is easier to express point multiplication in Cartesian coordinate system ：

  The use of dot multiplication ： Projection

  Ideas ： requirement b Vector in a The projection in the direction , Then the projection must be along a Vectorial , in other words a^（ A hat is expressed as a unit vector ） Multiply by a certain length , So find the length

Problem solving ：b Vector and b The projected vector forms a right triangle , therefore b The vector of the projection is equal to b The length of the vector times cos angle （ Angle cosine ）

Angle cosine =A and B Dot product of vector

Expansion of projection Calculate the vertical

  The use of dot multiplication ： Front and back

Dot multiplication is in the same direction Dot multiplication is in the negative opposite direction

   The use of dot multiplication ： Whether the direction is close （ The principle is the same as the above ）

The most important application ：

Specular reflection ：

Check the light vector of specular reflection Whether it is related to the vector of the angle of view Close to

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Vector cross product ：

Judge the result of cross multiplication ： Right-hand rule （ It is divided into spiral rule and three specified rules , This refers to the spiral rule ）

You can hold it with your right hand and have a try

（a Cross riding b The resulting vector is related to b Cross riding a The resulting vector is in the opposite direction - Yellow line ）

Some small laws of triangular coordinate system

  Some properties of cross product ：

 （0 It means the length is 0 Vector , Not numbers 0）

Matrix of vector cross multiplication ：

  The function of cross product ：

1. Decide left and right

  Sure b Cross riding a If the vector is z Negative then a stay b The right side of the Is regular a stay b The left side of the

2. Determine the inside and outside

 

  Judge p Whether it is inside the triangle

Anticlockwise or anticlockwise cross multiply （ such as ：ab*ap bc*bp ca*cp）

In the same direction

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The advantage of defining a rectangular coordinate system is that any vector can be decomposed into coordinate axes

You can put any vector （p vector ） It is decomposed into three axes （u v w）： With projection

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matrix ：

The product of matrices （ The premise of matrix multiplication is m*n n*p n==n）

For example, three rows and two columns and Two lines and three columns

skill ：

How to get The result of two rows and four columns Find the number in the second row in the first matrix 5 2

The second matrix finds the number in the fourth column 4 3

Then multiply these two groups of points （5*4+2*3=26）

  Matrix properties ：

  The distribution rate of associative law has

There is no exchange law

Matrix operations

A matrix is multiplied by a vector ：

  The transpose of the matrix ：

Unit matrix （ Diagonal matrix ）

  The inverse of the unique action matrix a and a Multiply the inverse matrix of Can get the identity matrix

Dot product of matrix ：

 

  Cross multiplication of matrices