Note: this article aims to record the author’s learning process, only represents the author’s personal understanding, if there are inaccurate expressions, welcome to correct! Because the concept involved comes from the network, so if there is infringement, also hope to inform!
preface
In the development process we involve the transformation of graphics, will involve matrix, vector calculation, so this article will simply understand the concept of vector and matrix.
The body of the
What is a vector
Geometrically, a vector is a directed line segment with magnitude and direction.In 3D Cartesian coordinates, basically a vertex is a position in the XYZ coordinate space. Given in spaceA position is precisely defined by a single XYZ, and such XYZ is a vector.
The dot product of vectors
The dot product “.
Definition: We have two non-zero vectorsa,b,OA=a,OB=b, so Angle AOB is called the vectoraAnd the vectorbtheThe Angle, asTheta.0 or less and regulationsTheta.PI or less.The dot product of two vectorsPhi is a scalar (without direction), let’s call it phiA. b..ifa,bNot collinear, then
It’s the Angle between two vectors.
OpenGL uses: The Math3D library provides an API for dot products
M3DVector3f v);
The cross product of the vector
The same vector has another operation that is very valuable in development, called the “cross product.”
Definition: The geometric representation of the cross product of two vectors A and b is a vector, called a×b. If a and b are not collinear, a * b mode is: ∣ a * b ∣ = | a |, | | b, sin < a, b >; The directions of A × B are: perpendicular to A and B, and A, B, and a× B form the right-handed system in this order. If a and b, vertical ∣ a * b ∣ = | | | | * b (unlike the dot product here, please note), if a * b = 0, a and b is parallel. A cross product is a set of normal vectors in the plane of two noncollinear nonzero vectors.
Two vector
The cross product will give you another vector, and the new vector will be perpendicular to the plane defined by the original two vectors. Simultaneous fork
Times, it doesn’t have to be a unit vector.
OpenGL uses: The Math3D library provides an API for cross products
M3DVector3f v);
What is a matrix
ijThe number table of m rows and n columns is called the matrix of M rows and n columns, or m by N matrix for short.
The number AIj is located in the ith row and j column of matrix A and is called the (I,j) element of matrix A. The matrix with the number AIj as the (I,j) element can be denoted as (AIj) or (aij)m × N. The m× N matrix A is also denoted as Amn.
So how do we define matrices in OpenGL? In other programming standards, many matrix libraries use two-dimensional arrays when defining a matrix. OpenGL’s conventions tend to favor one-dimensional arrays, and the reason for this is that OpenGL uses the column-major matrix sorting convention.
Matrix calculation
It makes sense for a matrix to have only one row or one column, and just one row or one column of numbers can be called a vector or a matrix.
As we know, in OpenGL we render graphics to the screen, and we go through a lot of matrix transformations, among which the most commonly involved changes are model matrix transformation, view matrix transformation, projection matrix transformation. For example, the following formula:
