Unity and C# Study Notes - Computer Graphics Basics

09 December 2022 - 6 mins read time
Tags: Computer Graphics Unity C#

Vector

• In Unity, vector is defined as struct.
  • When you want to equal sign to copy the content (allocate new memory), it is more convenient to define the the type of object as struct. Otherwise, class will allow you to copy it as reference.
• Override ToString function (the print function will default call the object’s ToString function):

public override string ToString(){
    return string.Format("(0:F2),(1:F2),(2:F2)", x, y, z);
}

• In Unity, mesh.normals are the normals associated to the vertices: usually the shader calculates each pixel’s normal by interpolating among the normals. Flat surfaces have equal normals in their vertices (like if they were surface normals), but curved surfaces have different normals associated to each vertex. Notice that all data in a mesh is in local space, which doesn’t take the transform settings into account (rotation, scale, position). You should convert each vertex and its normal to world space in order to correctly display them

Matrix

using System;
using System.Text;
using System.Runtime.InteropServices;

[StructLayout(LayoutKind.Sequential)]
namespace My{
struct Matrix4x4{
    [MarshalAs(UnmanagedType.ByValArray, SizeConst = 4*4)]
    private float[,] m; // = null
    public Matrix4x4(float [,] m){
        this.m = m;
    }
    public float this[int row, int col]{
        get{
            if (row < 0 || row >= 4)
				throw new IndexOutOfRangeException();
			if (col < 0 || col >= 4)
				throw new IndexOutOfRangeException();
			
            return m[row, col];
        }
        set{
            if (row < 0 || row >= 4)
				throw new IndexOutOfRangeException();
			if (col < 0 || col >= 4)
				throw new IndexOutOfRangeException();

			m[row, col] = value;
        }
    }
    public Matrix4x4 transpose{
        get{
            Matrix4x4 mRet = new Matrix4x4(new float [4,4]);
            for(int i = 0; i < 4; i++){
                for(int j = 0; j < 4; j++){
                    mRet[i, j] = this[j, i];
                }
            }
            return mRet;
        }
    }
    public static Matrix4x4 operator *(Matrix4x4 a, Matrix4x4 b){
        Matrix4x4 c = new Matrix4x4(new float [4,4]);
        for(int i = 0; i < 4; i++){
            for(int j = 0; j < 4; j++){
                c[i, j] = a[i, 0] * b[0, j] +
                          a[i, 1] * b[1, j] +
                          a[i, 2] * b[2, j] +
                          a[i, 3] * b[3, j];
            }
        }
        return c;
    }
    public override string ToString(){
        StringBuilder sb = new StringBuilder();
        for(int i = 0; i < 4; i++){
            for(int j = 0; j < 4 ;j++){
                sb.AppendFormat("{0:F2} ", m[i, j]);
            }
            sb.Append("\n");
        }
        return sb.ToString();
    }
}
} // namespace

• In C#, struct does not allow to override the default constructor for a properties since it will automatically fill the content with null.

  If a struct declares any field initializers, it must explicitly declare a constructor. That constructor need not be parameterless. If a struct declares a field initializer but no constructors, the compiler reports an error. Any explicitly declared constructor (with parameters, or parameterless) executes all field initializers for that struct. All fields without a field initializer or an assignment in a constructor are set to the default value.

• To the implementer of a class, a property is one or two code blocks, representing a get accessor and/or a set accessor.
  • The code block for the get accessor is executed when the property is read.
  • The code block for the set accessor is executed when the property is assigned a value.
  • This is useful for Encapsulation, to make sure that “sensitive” data is hidden from users.
• throw (C# Reference) Signals the occurrence of an exception during program execution.
  • e.g. throw new IndexOutOfRangeException();
  • e.g. throw new NotImplementedException();

• Game Engine might optimize the matrix multiplication using SIMD.

• Unity 3D and OpenGL use column-major matrix, Direct3D use row-major matrix.
  • row-vector * row-major-matrix
  • col-major-matrix * col-vector

• When deploying Direct3D API in Unity, it will automatically convert the column-major matrix to a row-major matrix, along with the transformation of the coordinates’ axis directions.

• Transformation matrix:
  • M = M_translate * M_rotate * M_scale
  • M_translate.inverse = -M_translate
  • M_rotate.inverse = M_rotate.transpose
  • M_scale(x, y, z).inverse = M_scale(1/x, 1/y, 1/z)
• The inverse of the transformation matrix indicates the inverse transformation process.

Projection Matrix

• Direct3D transformation:

  p_screen = p_local * M_w * M_v * M_p * M_s
  M_w: local -> world
  M_v: world -> view 
  M_p: view -> projection
  M_s: projection -> screen
  

• OpenGL transformation:

  p_screen = M_s * M_p * M_v * M_w * p_local