The point is to understand what a matrix means: a matrix is actually a transformation of coordinates

Understand the geometric functions of matrices:

  1. A matrix is a linear transformation (a line segment remains a line segment and the origin does not change)
  2. A matrix is a mapping, and a mapping can be one-to-one or one-to-many
  3. Matrix is a kind of spatial transformation, and each kind of matrix has its own geometric meaning, rather than a simple combination of numbers

Understand the formal meaning of matrices (matrices on the left, vectors on the right) :

  1. Square matrix full rank, is to carry out the coordinate transformation of the current space, dimension will not be raised or lowered
  2. M×N(M>N), this matrix is a kind of ascending dimension matrix, with geometric significance of 3×2. It is just like extending a plane in one direction and dimension (when rank is 2) and turning it into a three-dimensional space. Of course, the point (x,y) in the previous space is still a planar distribution in the three-dimensional space during the process of ascending dimension. It’s just an extra third dimension that’s all equal
  3. M×N(M
  4. Square matrix dissatisfied rank, in fact, is the same thing as before, dissatisfied rank means that this dimension is imaginary, it doesn’t matter, if the rank is one you’re squeezing into a line.

I. Translation matrix

1.1 Two-dimensional translation matrix

Form: What is the form of the translation matrix.

It has to be a full rank matrix, because we’re not going to transform the dimensions. The orientation and length of the axes are constant, because the essence of matrix operations is to change the X and Y of the reference frame. Find the matrix:

Now, what kind of matrix would allow a point to be shifted, because shifting doesn’t change the direction of the reference frame XY or the length of the unit vector, so in fact, in our current dimension, we can’t do that. Why? Because as we said before, the linear transformation of the matrix does not change the position of the origin, the matrix is the default origin is (0,0), and filling in the matrix with any values is not going to change this convention decision. So how do we do that? We operate with a higher dimensional matrix, a three dimensional matrix, in fact, in a higher dimension, to solve our origin shift problem.

In a geometric sense, the translation matrix is actually adding a non-orthogonal Z-axis (dx,dy,1) to redefine the coordinate axis and then transform the coordinate position.

The thing to notice here is that the point V= (x,y,1), is not a point in our two dimensional coordinates, but a point in the three dimensional space based on the column vectors of P, just to say that we change the two dimensional point to V= (x,y,1) and then apply the linear transformation of the corresponding matrix P to get the position of the shifted point.

The process may be difficult to understand, but it is also clever and precise. The matrix P constructed by us is formed by the column vectors of the basis in the new space, and the corresponding value of each coordinate value in the column vectors is actually determined by our original basis x=(1,0),y=(0,1). The matrix P, which is a linear transformation, maps the position of the point represented in its coordinate system to the space of the basis that we used to determine it.

This is essentially a projection operation, a projection of each of the coordinates of V.

And the inverse of P is just the operation that maps our coordinates to the coordinates of P.

1.2 THREE-DIMENSIONAL translation matrix

Once we understand the translation matrix in two dimensions, well, the same thing happens in three dimensions

Rotation matrix

The fundamental meaning of the rotation matrix is rotation about the origin.

2.1 Two-dimensional rotation matrix

Form: What is the form of the rotation matrix.

It must be full rank, because there’s no dimensional transformation and the length of the axes should be the same, because there’s no scaling, but the Angle should be changing. Find the matrix:

Let’s say we have some v is equal to (x,y), and the coordinate axis is the standard coordinate axis, and the Angle is theta, which is the Angle with the positive x axis.

First of all, we already know something, which is to find a new basis for the axis, and the basis is still of length 1. then

2.2 Three-dimensional rotation matrix

It’s pretty much the same for three dimensions, but the shape of the rotation matrix is different depending on the axis of rotation

So the transformation of the matrix is not V, it’s V1. Because the function of P is to transform a point in its own coordinates into a point in standard coordinates, so it’s not V in standard coordinates, it’s V1 in itself, because they have the same value, which is very misleading, but it doesn’t mean that their absolute positions in space are the same. They have the same value just because of the transformation of the reference frame.

Three. Scaling matrix

The scaling matrix is actually very simple, the scaling of the graph is actually the scaling of the basis of our own coordinate axes

Iv. Projection matrix

I project v onto A plane, A, in the direction of the normal vector n of B.

Form: What is the form of the orthogonal projection matrix?

It’s going to be a higher-dimensional matrix, because it involves translation and it’s going to be a lower-rank matrix, because our coordinates are all on the same plane because it’s an orthogonal projection, so what we’re going to compress is we’re going to find the matrix along the points on the B normal vector: Let’s say V= (vx,vy,vz),A has A point A = (Ax,ay,az), and the normal vector N= (Nx,Ny,Nz).

  1. First, the coordinate system on the plane is established, and the coordinate system (X*,Y*,Z*) is established through point A and normal vector N.
  2. Where Z is the direction of the normal vector N, it doesn’t matter what the direction is.
  3. If you’re done with that you’re not going to be able to translate it, you’re just going to be able to shift it in the direction of the coordinate system, you’re going to have to shift it in the position
  4. So we need the distance vector L from the origin to the plane
  5. To set up the final projection matrix note that all we need here, is to take the inverse, when we want the projection coordinate representation instead of the original coordinate representation