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OpenGL using own matrix in opengl

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Hi, I have been quite confused by matrix operations when developing games. I decided to learn to understand it but having a little confusion now. I created the following codes using opengl & glut: main.cpp
#pragma once
#include <stdio.h>
#include <stdlib.h>
#include <string.h>
#include <math.h>
#include <GL/glut.h>
#include <GL/glu.h>
#include <GL/gl.h>
#include <atlstr.h>
#include <time.h>

#include "Cube.h"
#include "Vector4.h"
#include "Matrix16.h"

// Set the size of the OpenGL Window
double winL = -300;
double winR = 300;
double winB = -300;
double winT = 300;

double start;
double last;
double now;

void UpdateScene(void);
void DrawScene(void);
void DrawAxis(void);
void renderBitmapString(float x, float y, void *font,char *string);
void Keyboard(int key, int x, int y);

CCube cube(5);

// This function is continuously called.
void Idle(void)
{
	DrawScene();
}


void renderBitmapString(float x, float y, void *font,char *string)
{
  
  char *c;
  glRasterPos2f(x, y);
  for (c=string; *c != '\0'; c++) {
    glutBitmapCharacter(font, *c);
  }
}  /* end renderBitmapString() */

void 
UpdateScene(void)
{
	last = clock();
	now = (last - start) / CLOCKS_PER_SEC;
	start = last;

	cube.Update(now);
}

void
DrawScene(void)
{
	UpdateScene();
	glLoadIdentity();
	gluLookAt(0, 0, 80, 0, 0, 0, 0, 1, 0);

	glClear(GL_COLOR_BUFFER_BIT | GL_DEPTH_BUFFER_BIT);

	// Your drawing code here

	DrawAxis();

	cube.Render();

	// End drawing code

	glutSwapBuffers();
}

void
DrawAxis(void)
{
	glBegin(GL_LINES);
	glColor3f(1, 0, 0);
	glVertex3f(-200, 0, 0);
	glVertex3f(200, 0, 0);

	glColor3f(0, 1, 0);
	glVertex3f(0, -200, 0);
	glVertex3f(0, 200, 0);

	glColor3f(0, 0, 1);
	glVertex3f(0, 0, -200);
	glVertex3f(0, 0, 200);
	glEnd();
}

void
Init(void)
{
	glShadeModel(GL_SMOOTH);
	glClearColor(0.0f, 0.0f, 0.0f, 0.0f);
	glClearDepth(1.0f);
	glEnable(GL_DEPTH_TEST);
	glDepthFunc(GL_LEQUAL);
	glHint(GL_PERSPECTIVE_CORRECTION_HINT, GL_NICEST);

	glMatrixMode(GL_PROJECTION);
	gluPerspective(45, (winR - winL) / (winT - winB), 1, 1000);
	glMatrixMode(GL_MODELVIEW);

	start = clock();

	glColorMaterial(GL_FRONT, GL_AMBIENT_AND_DIFFUSE);	
	glEnable ( GL_COLOR_MATERIAL ) ;

	glEnable(GL_LIGHTING);
	float pos[4] = {0, 100, 0, 1};
	glLightfv(GL_LIGHT0, GL_POSITION, pos);
	glEnable(GL_LIGHT0);
}

void
Reshape(int width, int height)
{
	glViewport(0, 0, width, height);
}

/* ARGSUSED1 */
void
Key(unsigned char key, int x, int y)
{
	switch (key) {
	case 27:
		exit(0);
		break;
	case ' ':
		glutIdleFunc(Idle);
		break;
	case 'q':
		cube.Rotate(-5, 1, 0, 0, 1);
		break;
	case 'w':
		cube.Rotate(5, 1, 0, 0, 1);
		break;
	case 'a':
		cube.Rotate(-5, 0, 1, 0, 1);
		break;
	case 's':
		cube.Rotate(5, 0, 1, 0, 1);
		break;
	case 'z':
		cube.Rotate(-5, 0, 0, 1, 1);
		break;
	case 'x':
		cube.Rotate(5, 0, 0, 1, 1);
		break;
	case 'r':
		cube.Translate(-5, 0, 0, 2);
		break;
	case 'f':
		cube.Translate(5, 0, 0, 2);
		break;
	case 'e':
		cube.Translate(-5, 0, 0, 1);
		break;
	case 'd':
		cube.Translate(5, 0, 0, 1);
		break;
	case 'l':
		cube.Scale(2, 2, 2);
		break;
	default:
		cube.Reset();
		break;
	}
}

void Keyboard(int key, int x, int y)
{
	switch(key)
	{
	case GLUT_KEY_UP:
		cube.Rotate(-5, 1, 0, 0, 2);
		break;

	case GLUT_KEY_DOWN:
		cube.Rotate(5, 1, 0, 0, 2);
		break;

	case GLUT_KEY_LEFT:
		cube.Rotate(-5, 0, 1, 0, 2);
		break;

	case GLUT_KEY_RIGHT:
		cube.Rotate(5, 0, 1, 0, 2);
		break;

	case GLUT_KEY_PAGE_UP:
		cube.Rotate(-5, 0, 0, 1, 2);
		break;

	case GLUT_KEY_PAGE_DOWN:
		cube.Rotate(5, 0, 0, 1, 2);
		break;
	}
}


int
main(int argc, char **argv)
{
	glutInitWindowSize((winR - winL), (winT - winB));
	glutInit(&argc, argv);
	glutInitDisplayMode(GLUT_DOUBLE);
	glutCreateWindow("Physics");

	Init();

	glutReshapeFunc(Reshape);
	glutKeyboardFunc(Key);
	glutSpecialFunc(Keyboard);
	glutDisplayFunc(DrawScene);
	glutIdleFunc(Idle);

	glutMainLoop();
	return 0;             /* ANSI C requires main to return int. */
}

Matrix16.h
#pragma once
#include <iostream>
using namespace std;

#include "Vector4.h"

static const float PI = 3.14159265359f;

inline float DEG2RAD(float a)
{
	return (PI/180*(a));
}

inline float RAD2DEG(float a)
{
	return (180/PI*(a));
}

class CMatrix16
{
public:
	CMatrix16(void);
	CMatrix16(float m11, float m12, float m13, float m14,
			float m21, float m22, float m23, float m24,
			float m31, float m32, float m33, float m34,
			float m41, float m42, float m43, float m44);
	~CMatrix16(void);

	// Modification
	static CMatrix16 & Add(const CMatrix16 & m1, const CMatrix16 & m2, CMatrix16 & result);
	static CMatrix16 & Subtract(const CMatrix16 & m1, const CMatrix16 & m2, CMatrix16 & result);
	static CMatrix16 & Multiply(const CMatrix16 & m1, const CMatrix16 & m2, CMatrix16 & result);
	static CMatrix16 & Multiply(const CMatrix16 & m1, float multiplyBy, CMatrix16 & result);
	static CVector4 & Multiply(const CMatrix16 & m1, const CVector4 & v, CVector4 & result);
	static CMatrix16 & Transpose(const CMatrix16 & m1, CMatrix16 & result);
	//static CMatrix16 & Invert(const CMatrix16 & m1, CMatrix16 & result);
	static CMatrix16 & SetIdentity(CMatrix16 & m1);
	static float Determinant(const CMatrix16 & m1);

	inline static CMatrix16 Translation(const CVector4 & v)
	{
		return Translation(v.X(), v.Y(), v.Z());
	}

	inline static CMatrix16 Translation(float x, float y, float z)
	{
		return CMatrix16(1, 0, 0, 0,
						 0, 1, 0, 0,
						 0, 0, 1, 0,
						 x, y, z, 1);
	}

	inline static CMatrix16 Scale(float x, float y, float z)
	{
		return CMatrix16(x, 0, 0, 0,
						 0, y, 0, 0,
						 0, 0, z, 0,
						 0, 0, 0, 1);
	}

	inline static CMatrix16 Rotation(float angle, float x, float y, float z)
	{
		angle = angle - ((int) angle / 360);
		angle = DEG2RAD(angle);
		float c = cos(angle);
		float s = sin(angle);
		return CMatrix16(c + x * x * (1 - c), x * y * (1 - c) - s * z, x * z * (1 - c) + s * y, 0,
							x * y * (1 - c) + s * z, c + y * y * (1 - c), y * z * (1 - c) - s * x, 0,
							x * z * (1 - c) - s * y, y * z * (1 - c) + s * x, c + z * z * (1 - c), 0,
							0, 0, 0, 1);
	}

	inline CMatrix16 & Transpose() { CMatrix16 temp(*this); return CMatrix16::Transpose(temp, (*this)); }
	//inline CMatrix16 & Invert() { CMatrix16 temp((*this)); return CMatrix16::Invert(temp, (*this)); } 
	inline CMatrix16 & SetIdentity() { return CMatrix16::SetIdentity((*this)); }
	inline float Determinant() const { return CMatrix16::Determinant((*this)); }

	// Auxilliaries
	CMatrix16 & operator+=(const CMatrix16 & m1) { return CMatrix16::Add((*this), m1, (*this)); }
	CMatrix16 & operator-=(const CMatrix16 & m1) { return CMatrix16::Subtract((*this), m1, (*this)); }
	CMatrix16 & operator*=(const CMatrix16 & m1) { CMatrix16 temp((*this)); return CMatrix16::Multiply(temp, m1, (*this)); }
	CMatrix16 & operator*=(const float multiplyBy) { CMatrix16 temp((*this)); return CMatrix16::Multiply(temp, multiplyBy, (*this)); }

	// Conversion
	void ToArray(float * m) const 
	{
		m[0] = _m11;	m[4] = _m21;	m[8] = _m31;	m[12] = _m41;
		m[1] = _m12;	m[5] = _m22;	m[9] = _m32;	m[13] = _m42;
		m[2] = _m13;	m[6] = _m23;	m[10] = _m33;	m[14] = _m43;
		m[3] = _m14;	m[7] = _m24;	m[11] = _m34;	m[15] = _m44;
	}

	// Output and Input
	void Write(ostream & out) const
	{
		out << "[\t" << _m11 << ",\t" << _m12 << ",\t" << _m13 << ",\t" << _m14 << "\t]" << endl
			<< "[\t" << _m21 << ",\t" << _m22 << ",\t" << _m23 << ",\t" << _m24 << "\t]" << endl
			<< "[\t" << _m31 << ",\t" << _m32 << ",\t" << _m33 << ",\t" << _m34 << "\t]" << endl
			<< "[\t" << _m41 << ",\t" << _m42 << ",\t" << _m43 << ",\t" << _m44 << "\t]" << endl;
	}

	void Read(istream & in)
	{
		char ch;
		in >> _m11 >> ch >> _m12 >> ch >> _m13 >> ch >> _m14
			>> _m21 >> ch >> _m22 >> ch >> _m23 >> ch >> _m24
			>> _m31 >> ch >> _m32 >> ch >> _m33 >> ch >> _m34
			>> _m41 >> ch >> _m42 >> ch >> _m43 >> ch >> _m44;
	}

	float _m11, _m12, _m13, _m14,
		  _m21, _m22, _m23, _m24,
		  _m31, _m32, _m33, _m34,
		  _m41, _m42, _m43, _m44;
};

inline CMatrix16 operator+ (const CMatrix16 & m1, const CMatrix16 & m2) { CMatrix16 m; return CMatrix16::Add(m1, m2, m); }
inline CMatrix16 operator- (const CMatrix16 & m1, const CMatrix16 & m2) { CMatrix16 m; return CMatrix16::Subtract(m1, m2, m); }
inline CMatrix16 operator* (const CMatrix16 & m1, const CMatrix16 & m2) { CMatrix16 m; return CMatrix16::Multiply(m1, m2, m); }
inline CMatrix16 operator* (const CMatrix16 & m1, float multiplyBy) { CMatrix16 m; return CMatrix16::Multiply(m1, multiplyBy, m); }
inline CMatrix16 operator* (float multiplyBy, const CMatrix16 & m1) { CMatrix16 m; return CMatrix16::Multiply(m1, multiplyBy, m); }
inline CVector4 operator* (const CMatrix16 & m1, const CVector4 & v) { CVector4 temp; return CMatrix16::Multiply(m1, v, temp); }
inline CVector4 operator* (const CVector4 & v, const CMatrix16 & m1) { CVector4 temp; return CMatrix16::Multiply(m1, v, temp); }

inline ostream & operator<< (ostream & out, const CMatrix16 & mat) { mat.Write(out); return out; }
inline istream & operator>> (istream & in, CMatrix16 & mat) { mat.Read(in); return in; }

Matrix16.cpp
#include ".\matrix16.h"

CMatrix16::CMatrix16(void)
:	_m11(0), _m12(0), _m13(0), _m14(0),
	_m21(0), _m22(0), _m23(0), _m24(0),
	_m31(0), _m32(0), _m33(0), _m34(0),
	_m41(0), _m42(0), _m43(0), _m44(0)
{
}

CMatrix16::CMatrix16(float m11, float m12, float m13, float m14,
					float m21, float m22, float m23, float m24,
					float m31, float m32, float m33, float m34,
					float m41, float m42, float m43, float m44)
:	_m11(m11), _m12(m12), _m13(m13), _m14(m14),
	_m21(m21), _m22(m22), _m23(m23), _m24(m24),
	_m31(m31), _m32(m32), _m33(m33), _m34(m34),
	_m41(m41), _m42(m42), _m43(m43), _m44(m44)
{
}

CMatrix16::~CMatrix16(void)
{
}

CMatrix16 & CMatrix16::Add(const CMatrix16 & m1, const CMatrix16 & m2, CMatrix16 & result)
{
	result._m11 = m1._m11 + m2._m11;
	result._m12 = m1._m12 + m2._m12;
	result._m13 = m1._m13 + m2._m13;
	result._m14 = m1._m14 + m2._m14;

	result._m21 = m1._m21 + m2._m21;
	result._m22 = m1._m22 + m2._m22;
	result._m23 = m1._m23 + m2._m23;
	result._m24 = m1._m24 + m2._m24;

	result._m31 = m1._m31 + m2._m31;
	result._m32 = m1._m32 + m2._m32;
	result._m33 = m1._m33 + m2._m33;
	result._m34 = m1._m34 + m2._m34;

	result._m41 = m1._m41 + m2._m41;
	result._m42 = m1._m42 + m2._m42;
	result._m43 = m1._m43 + m2._m43;
	result._m44 = m1._m44 + m2._m44;

	return result;
}

CMatrix16 & CMatrix16::Subtract(const CMatrix16 & m1, const CMatrix16 & m2, CMatrix16 & result)
{
	result._m11 = m1._m11 - m2._m11;
	result._m12 = m1._m12 - m2._m12;
	result._m13 = m1._m13 - m2._m13;
	result._m14 = m1._m14 - m2._m14;

	result._m21 = m1._m21 - m2._m21;
	result._m22 = m1._m22 - m2._m22;
	result._m23 = m1._m23 - m2._m23;
	result._m24 = m1._m24 - m2._m24;

	result._m31 = m1._m31 - m2._m31;
	result._m32 = m1._m32 - m2._m32;
	result._m33 = m1._m33 - m2._m33;
	result._m34 = m1._m34 - m2._m34;

	result._m41 = m1._m41 - m2._m41;
	result._m42 = m1._m42 - m2._m42;
	result._m43 = m1._m43 - m2._m43;
	result._m44 = m1._m44 - m2._m44;

	return result;
}

CMatrix16 & CMatrix16::Multiply(const CMatrix16 & m1, const CMatrix16 & m2, CMatrix16 & result)
{
	result._m11 = m1._m11 * m2._m11 + m1._m12 * m2._m21 + m1._m13 * m2._m31 + m1._m14 * m2._m41;
	result._m12 = m1._m11 * m2._m12 + m1._m12 * m2._m22 + m1._m13 * m2._m32 + m1._m14 * m2._m42;
	result._m13 = m1._m11 * m2._m13 + m1._m12 * m2._m23 + m1._m13 * m2._m33 + m1._m14 * m2._m43;
	result._m14 = m1._m11 * m2._m14 + m1._m12 * m2._m24 + m1._m13 * m2._m34 + m1._m14 * m2._m44;

	result._m21 = m1._m21 * m2._m11 + m1._m22 * m2._m21 + m1._m23 * m2._m31 + m1._m24 * m2._m41;
	result._m22 = m1._m21 * m2._m12 + m1._m22 * m2._m22 + m1._m23 * m2._m32 + m1._m24 * m2._m42;
	result._m23 = m1._m21 * m2._m13 + m1._m22 * m2._m23 + m1._m23 * m2._m33 + m1._m24 * m2._m43;
	result._m24 = m1._m21 * m2._m14 + m1._m22 * m2._m24 + m1._m23 * m2._m34 + m1._m24 * m2._m44;

	result._m31 = m1._m31 * m2._m11 + m1._m32 * m2._m21 + m1._m33 * m2._m31 + m1._m34 * m2._m41;
	result._m32 = m1._m31 * m2._m12 + m1._m32 * m2._m22 + m1._m33 * m2._m32 + m1._m34 * m2._m42;
	result._m33 = m1._m31 * m2._m13 + m1._m32 * m2._m23 + m1._m33 * m2._m33 + m1._m34 * m2._m43;
	result._m34 = m1._m31 * m2._m14 + m1._m32 * m2._m24 + m1._m33 * m2._m34 + m1._m34 * m2._m44;

	result._m41 = m1._m41 * m2._m11 + m1._m42 * m2._m21 + m1._m43 * m2._m31 + m1._m44 * m2._m41;
	result._m42 = m1._m41 * m2._m12 + m1._m42 * m2._m22 + m1._m43 * m2._m32 + m1._m44 * m2._m42;
	result._m43 = m1._m41 * m2._m13 + m1._m42 * m2._m23 + m1._m43 * m2._m33 + m1._m44 * m2._m43;
	result._m44 = m1._m41 * m2._m14 + m1._m42 * m2._m24 + m1._m43 * m2._m34 + m1._m44 * m2._m44;

	return result;
}

CMatrix16 & CMatrix16::Multiply(const CMatrix16 & m1, float multiplyBy, CMatrix16 & result)
{
	result._m11 = m1._m11 * multiplyBy;
	result._m12 = m1._m12 * multiplyBy;
	result._m13 = m1._m13 * multiplyBy;
	result._m14 = m1._m14 * multiplyBy;

	result._m21 = m1._m21 * multiplyBy;
	result._m22 = m1._m22 * multiplyBy;
	result._m23 = m1._m23 * multiplyBy;
	result._m24 = m1._m24 * multiplyBy;

	result._m31 = m1._m31 * multiplyBy;
	result._m32 = m1._m32 * multiplyBy;
	result._m33 = m1._m33 * multiplyBy;
	result._m34 = m1._m34 * multiplyBy;

	result._m41 = m1._m41 * multiplyBy;
	result._m42 = m1._m42 * multiplyBy;
	result._m43 = m1._m43 * multiplyBy;
	result._m44 = m1._m44 * multiplyBy;

	return result;
}

CVector4 & CMatrix16::Multiply(const CMatrix16 & m1, const CVector4 & v, CVector4 &result)
{
	result.X(m1._m11 * v.X() + m1._m21 * v.Y() + m1._m31 * v.Z() + m1._m41);
	result.Y(m1._m12 * v.X() + m1._m22 * v.Y() + m1._m32 * v.Z() + m1._m42);
	result.Z(m1._m13 * v.X() + m1._m23 * v.Y() + m1._m33 * v.Z() + m1._m43);

	return result;
}

CMatrix16 & CMatrix16::Transpose(const CMatrix16 & m1, CMatrix16 & result)
{
	result._m11 = m1._m11;
	result._m12 = m1._m21;
	result._m13 = m1._m31;
	result._m14 = m1._m41;

	result._m21 = m1._m12;
	result._m22 = m1._m22;
	result._m23 = m1._m32;
	result._m24 = m1._m42;

	result._m31 = m1._m13;
	result._m32 = m1._m23;
	result._m33 = m1._m33;
	result._m34 = m1._m43;

	result._m41 = m1._m14;
	result._m42 = m1._m24;
	result._m43 = m1._m34;
	result._m44 = m1._m44;

	return result;
}

/*CMatrix16 & CMatrix16::Invert(const CMatrix16 & m1, CMatrix16 & result)
{
	float temp = 1 / m1.Determinant();

	result._m11 = m1._m22 * m1._m33 * m1._m44 + m1._m23 * m1._m34 * m1._m42 + m1._m24 * m1._m32 * m1._m43
				- m1._m22 * m1._m34 * m1._m43 - m1._m23 * m1._m32 * m1._m44 - m1._m24 * m1._m33 * m1._m42;
	result._m21 = m1._m12 * m1._m34 * m1._m43 + m1._m13 * m1._m32 * m1._m44 + m1._m14 * m1._m33 * m1._m42
				- m1._m12 * m1._m33 * m1._m44 - m1._m13 * m1._m34 * m1._m42 - m1._m14 * m1._m32 * m1._m43;
	result._m31 = m1._m12 * m1._m23 * m1._m44 + m1._m13 * m1._m24 * m1._m42 + m1._m14 * m1._m22 * m1._m43
				- m1._m12 * m1._m24 * m1._m43 - m1._m13 * m1._m22 * m1._m44 - m1._m14 * m1._m23 * m1._m42;
	result._m41 = m1._m12 * m1._m24 * m1._m33 + m1._m13 * m1._m22 * m1._m34 + m1._m14 * m1._m23 * m1._m32
				- m1._m12 * m1._m23 * m1._m34 - m1._m13 * m1._m24 * m1._m32 - m1._m14 * m1._m22 * m1._m33;

	result._m12 = m1._m21 * m1._m34 * m1._m43 + m1._m23 * m1._m31 * m1._m44 + m1._m24 * m1._m33 * m1._m41
				- m1._m21 * m1._m33 * m1._m44 - m1._m23 * m1._m34 * m1._m41 - m1._m24 * m1._m31 * m1._m43;
	result._m22 = m1._m11 * m1._m33 * m1._m44 + m1._m13 * m1._m34 * m1._m41 + m1._m14 * m1._m31 * m1._m43
				- m1._m11 * m1._m34 * m1._m43 - m1._m13 * m1._m31 * m1._m44 - m1._m14 * m1._m33 * m1._m41;
	result._m32 = m1._m11 * m1._m24 * m1._m43 + m1._m13 * m1._m21 * m1._m44 + m1._m14 * m1._m23 * m1._m41
				- m1._m11 * m1._m23 * m1._m44 - m1._m13 * m1._m24 * m1._m41 - m1._m14 * m1._m21 * m1._m43;
	result._m42 = m1._m11 * m1._m23 * m1._m34 + m1._m13 * m1._m24 * m1._m31 + m1._m14 * m1._m21 * m1._m33
				- m1._m11 * m1._m24 * m1._m33 - m1._m13 * m1._m21 * m1._m34 - m1._m14 * m1._m23 * m1._m31;

	result._m13 = m1._m21 * m1._m32 * m1._m44 + m1._m22 * m1._m34 * m1._m41 + m1._m24 * m1._m31 * m1._m42
				- m1._m21 * m1._m34 * m1._m42 - m1._m22 * m1._m31 * m1._m44 - m1._m24 * m1._m32 * m1._m41;
	result._m23 = m1._m11 * m1._m34 * m1._m42 + m1._m12 * m1._m31 * m1._m44 + m1._m14 * m1._m32 * m1._m41
				- m1._m11 * m1._m32 * m1._m44 - m1._m12 * m1._m34 * m1._m41 - m1._m14 * m1._m31 * m1._m42;
	result._m33 = m1._m11 * m1._m22 * m1._m44 + m1._m12 * m1._m24 * m1._m41 + m1._m14 * m1._m21 * m1._m42
				- m1._m11 * m1._m24 * m1._m42 - m1._m12 * m1._m21 * m1._m44 - m1._m14 * m1._m22 * m1._m41;
	result._m43 = m1._m11 * m1._m24 * m1._m32 + m1._m12 * m1._m21 * m1._m34 + m1._m14 * m1._m22 * m1._m31
				- m1._m11 * m1._m22 * m1._m34 - m1._m12 * m1._m24 * m1._m31 - m1._m14 * m1._m21 * m1._m32;

	result._m14 = m1._m21 * m1._m33 * m1._m42 + m1._m22 * m1._m31 * m1._m43 + m1._m23 * m1._m32 * m1._m41
				- m1._m21 * m1._m32 * m1._m43 - m1._m22 * m1._m33 * m1._m41 - m1._m23 * m1._m31 * m1._m42;
	result._m24 = m1._m11 * m1._m32 * m1._m43 + m1._m12 * m1._m33 * m1._m41 + m1._m13 * m1._m31 * m1._m42
				- m1._m11 * m1._m22 * m1._m42 - m1._m12 * m1._m31 * m1._m43 - m1._m13 * m1._m32 * m1._m41;
	result._m34 = m1._m11 * m1._m23 * m1._m42 + m1._m12 * m1._m21 * m1._m43 + m1._m13 * m1._m22 * m1._m41
				- m1._m11 * m1._m22 * m1._m43 - m1._m12 * m1._m23 * m1._m41 - m1._m13 * m1._m21 * m1._m42;
	result._m44 = m1._m11 * m1._m22 * m1._m33 + m1._m12 * m1._m23 * m1._m31 + m1._m13 * m1._m21 * m1._m32
				- m1._m11 * m1._m23 * m1._m32 - m1._m12 * m1._m21 * m1._m33 - m1._m13 * m1._m22 * m1._m31;

	result = result * temp;

	return result;
}*/

CMatrix16 & CMatrix16::SetIdentity(CMatrix16 & m1)
{
	m1._m11 = 1;	m1._m12 = 0;	m1._m13 = 0;	m1._m14 = 0;
	m1._m21 = 0;	m1._m22 = 1;	m1._m23 = 0;	m1._m24 = 0;
	m1._m31 = 0;	m1._m32 = 0;	m1._m33 = 1;	m1._m34 = 0;
	m1._m41 = 0;	m1._m42 = 0;	m1._m43 = 0;	m1._m44 = 1;

	return m1;
}

float CMatrix16::Determinant(const CMatrix16 & m1)
{
	return m1._m11 * m1._m22 * m1._m33 * m1._m44 + m1._m11 * m1._m23 * m1._m34 * m1._m42 + m1._m11 * m1._m24 * m1._m32 * m1._m43 +
			m1._m12 * m1._m21 * m1._m34 * m1._m43 + m1._m12 * m1._m23 * m1._m31 * m1._m44 + m1._m12 * m1._m24 * m1._m33 * m1._m41 +
			m1._m13 * m1._m21 * m1._m32 * m1._m44 + m1._m13 * m1._m22 * m1._m34 * m1._m41 + m1._m13 * m1._m24 * m1._m31 * m1._m42 +
			m1._m14 * m1._m21 * m1._m33 * m1._m42 + m1._m14 * m1._m22 * m1._m31 * m1._m43 + m1._m14 * m1._m23 * m1._m32 * m1._m41 -
			m1._m11 * m1._m22 * m1._m34 * m1._m43 - m1._m11 * m1._m23 * m1._m32 * m1._m44 - m1._m11 * m1._m24 * m1._m33 * m1._m42 -
			m1._m12 * m1._m21 * m1._m33 * m1._m44 - m1._m12 * m1._m23 * m1._m34 * m1._m41 - m1._m12 * m1._m24 * m1._m31 * m1._m43 -
			m1._m13 * m1._m21 * m1._m34 * m1._m42 - m1._m13 * m1._m22 * m1._m31 * m1._m44 - m1._m13 * m1._m24 * m1._m32 * m1._m41 -
			m1._m14 * m1._m21 * m1._m32 * m1._m43 - m1._m14 * m1._m22 * m1._m33 * m1._m41 - m1._m14 * m1._m23 * m1._m31 * m1._m42;
}

Vector4.h
#pragma once
#include <iostream>
using namespace std;

#include <math.h>

class CVector4
{
public:
	// Constructors
	CVector4(void);
	CVector4(float x, float y, float z, float w = 1);
	~CVector4(void);

	// Selectors
	inline float X() const { return _x; }
	inline float Y() const { return _y; }
	inline float Z() const { return _z; }

	// Mutators
	inline void X(float x) { _x = x; }
	inline void Y(float y) { _y = y; }
	inline void Z(float z) { _z = z; }

	// Magnitude
	inline float Length() const { return sqrt(LengthSq()); }
	inline float LengthSq() const { return _x * _x + _y * _y + _z * _z; }
	inline void Zero() { X(0); Y(0); Z(0); }

	// Unit
	CVector4 & Unit();
	static void Unit(const CVector4 & v);

	// Modification
	static float Dot(const CVector4 & v1, const CVector4 & v2);
	static CVector4 & Cross(const CVector4 & v1, const CVector4 & v2, CVector4 & result);
	static CVector4 & Add(const CVector4 & v1, const CVector4 & v2, CVector4 & result);
	static CVector4 & Subtract(const CVector4 & v1, const CVector4 & v2, CVector4 & result);
	static CVector4 & Multiply(const CVector4 & v, float multiplyBy, CVector4 & result);
	static CVector4 & Invert(CVector4 & v);

	// Auxilliaries
	CVector4 & operator+=(const CVector4 & v) { return CVector4::Add((*this), v, (*this)); }
	CVector4 & operator-=(const CVector4 & v) { return CVector4::Subtract((*this), v, (*this)); }
	CVector4 & operator*=(const CVector4 & v) { CVector4 temp((*this)); return CVector4::Cross(temp, v, (*this)); }
	CVector4 & operator*=(float multiplyBy) { CVector4 temp((*this)); return CVector4::Multiply(temp, multiplyBy, (*this)); }
	CVector4 & operator/=(float divideBy) { CVector4 temp((*this)); return CVector4::Multiply(temp, (1 / divideBy), (*this)); }

	// Input and output
	void Write(ostream & out) const { out << "[" << _x << "," << _y << "," << _z <<  "]"; }
	void Read(istream & in) { char ch; in >> ch >> _x >> ch >> _y >> ch >> _z >> ch; }

private:
	float _x, _y, _z, _w;
};

inline CVector4 operator+(const CVector4 & v1, const CVector4 & v2) { CVector4 v; return CVector4::Add(v1, v2, v); }
inline CVector4 operator-(const CVector4 & v1, const CVector4 & v2) { CVector4 v; return CVector4::Subtract(v1, v2, v); }
inline CVector4 operator*(const CVector4 & v1, const CVector4 & v2) { CVector4 v; return CVector4::Cross(v1, v2, v); }
inline CVector4 operator*(const CVector4 & v1, float multiplyBy) { CVector4 v; return CVector4::Multiply(v1, multiplyBy, v); }
inline CVector4 operator/(const CVector4 & v1, float divideBy) { CVector4 v; float m = 1 / divideBy; return CVector4::Multiply(v1, m, v); }
inline float operator|(const CVector4 & v1, const CVector4 & v2) { return CVector4::Dot(v1, v2); }

inline ostream & operator<<(ostream & out, const CVector4 & v) { v.Write(out); return out; }
inline istream & operator>>(istream & in, CVector4 & v) { v.Read(in); return in; }

Vector4.cpp
#include ".\vector4.h"

CVector4::CVector4(void) 
: _x(0), _y(0), _z(0), _w(1)
{
}

CVector4::CVector4(float x, float y, float z, float w) 
: _x(x), _y(y), _z(z), _w(w)
{
}

CVector4::~CVector4(void)
{
}

float CVector4::Dot(const CVector4 & v1, const CVector4 & v2)
{
	return v1.X() * v2.X() + v1.Y() * v2.Y() + v1.Z() * v2.Z();
}

CVector4 & CVector4::Cross(const CVector4 & v1, const CVector4 & v2, CVector4 & result)
{
	result.X(v1.Y() * v2.Z() - v1.Z() * v2.Y());
	result.Y(v1.Z() * v2.X() - v1.X() * v2.Z());
	result.Z(v1.X() * v2.Y() - v1.Y() * v2.X());

	return result;
}

CVector4 & CVector4::Add(const CVector4 & v1, const CVector4 & v2, CVector4 & result)
{
	result.X(v1.X() + v2.X());
	result.Y(v1.Y() + v2.Y());
	result.Z(v1.Z() + v2.Z());

	return result;
}

CVector4 & CVector4::Subtract(const CVector4 & v1, const CVector4 & v2, CVector4 & result)
{
	result.X(v1.X() - v2.X());
	result.Y(v1.Y() - v2.Y());
	result.Z(v1.Z() - v2.Z());

	return result;
}

CVector4 & CVector4::Multiply(const CVector4 & v, float multiplyBy, CVector4 & result)
{
	result.X(v.X() * multiplyBy);
	result.Y(v.Y() * multiplyBy);
	result.Z(v.Z() * multiplyBy);

	return result;
}

CVector4 & CVector4::Invert(CVector4 & v)
{
	v.X(-v.X());
	v.Y(-v.Y());
	v.Z(-v.Z());

	return v;
}

cube.h
#pragma once
#include "Matrix16.h"
#include "Vector4.h"

class CCube
{
public:
	CCube(void);
	CCube(float size);
	~CCube(void);

	float GetSize() const { return _size; }

	void Update(float dt);
	void Render() const;

	void DrawAxis() const;

	void Rotate(float angle, float x, float y, float z, int Order)
	{
		switch(Order)
		{
		case 1:
			_r = CMatrix16::Rotation(angle, x, y, z) * _r;
			break;
		case 2:
			_r = _r * CMatrix16::Rotation(angle, x, y, z);
			break;
		}

		MarkDirty();
	}

	void Translate(float x, float y, float z, int Order)
	{
		switch(Order)
		{
		case 1:
			_t += (CVector4(x, y, z) * _r);
			break;
		case 2:
			_t += CVector4(x, y, z);
			break;
		}

		MarkDirty();
	}

	void Scale(float x, float y, float z)
	{
		_s._m11 *= x;
		_s._m22 *= y;
		_s._m33 *= z;

		MarkDirty();
	}

	const CMatrix16 & GetTransform() const
	{
		if(_dirty)
		{
			_trans.SetIdentity();
			_trans = _r * _s;
			
			_trans._m41 = _t.X();
			_trans._m42 = _t.Y();
			_trans._m43 = _t.Z();
			_trans._m44 = 1;
			
			_dirty = false;
		}

		return _trans;
	}

	void MarkDirty() { _dirty = true; }

	void Reset()
	{
		_s.SetIdentity();
		_r.SetIdentity();
		_t.Zero();

		MarkDirty();
		GetTransform();
	}

private:
	float _size;
	mutable bool _dirty;

	CMatrix16 _s;
	CMatrix16 _r;
	CVector4 _t;

	mutable CMatrix16 _trans;
};

cube.cpp
#include "Cube.h"
#include <GL\glut.h>

CCube::CCube(void)
:	_size(1), _dirty(true), _s(), _r(), _t(), _trans()
{
	_s.SetIdentity();
	_r.SetIdentity();

	GetTransform();
}

CCube::CCube(float size)
:	_size(size), _dirty(true), _s(), _r(), _t(), _trans()
{
	_s.SetIdentity();
	_r.SetIdentity();

	GetTransform();
}

CCube::~CCube(void)
{
}

void CCube::Update(float dt)
{
}

void CCube::Render() const
{
	glPushMatrix();

	float m[16];
	GetTransform().ToArray(m);
	glMultMatrixf(m);
	
	DrawAxis();

	glColor3f(1, 0, 0);
	glutSolidCube(_size);
	glPopMatrix();
}

void CCube::DrawAxis(void) const
{
	glBegin(GL_LINES);	
	glColor3f(1, 1, 0);
	glVertex3f(-30, 0, 0);
	glVertex3f(30, 0, 0);

	glColor3f(1, 0, 1);
	glVertex3f(0, -30, 0);
	glVertex3f(0, 30, 0);

	glColor3f(0, 1, 1);
	glVertex3f(0, 0, -30);
	glVertex3f(0, 0, 30);
	glEnd();
}

Keys: Q & W - Rotate clockwise-anticlockwise in along local X A & S - Rotate clockwise-anticlockwise in along local Y Z & X - Rotate clockwise-anticlockwise in along local Z E & D - Translate in local X axis A & S - Translate in world X axis Up & Down Arrow - Rotate clockwise-anticlockwise in along world X Left & Right Arrow - Rotate clockwise-anticlockwise in along world Y Pg Up & Pg Down - Rotate clockwise-anticlockwise in along world Z I am not sure whether the translation part is wrong or the rotation. When I translate, say (20, 0, 0), then rotate 90 in world space Y, the object still rotate on its own space. I would expect it to orbit around the world space Y. Please anyone could give me some pointers?

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First of all, how come in the Translate function the second "order" case doesn't have _r on either side?
Now, the problem is obviously with the order of the matrix multiplication. If you want to rotate the world space and R is the rotation matrix and M is the current matrix then you do R*P. The last column of P should the the position (_t in your code). I think that the problem is that after you change _t with the transformation you don't change the last column of _r. So when you rotate _r you aren't taking the transformation into account.

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Quote:
Original post by daniel_i_l
First of all, how come in the Translate function the second "order" case doesn't have _r on either side?
Now, the problem is obviously with the order of the matrix multiplication. If you want to rotate the world space and R is the rotation matrix and M is the current matrix then you do R*P. The last column of P should the the position (_t in your code). I think that the problem is that after you change _t with the transformation you don't change the last column of _r. So when you rotate _r you aren't taking the transformation into account.


Hi daniel,

Second 'order' in the Translate function is to translate based on world space.
I am not quite sure if that's done correctly.
And I tried to modify only the Translate function to:


void Translate(float x, float y, float z, int Order)
{
switch(Order)
{
case 1:
_t += (CVector4(x, y, z) * _r);
break;
case 2:
_t += CVector4(x, y, z);
break;
}

_r._m41 = _t.X();
_r._m42 = _t.Y();
_r._m43 = _t.Z();

MarkDirty();
}



the rotation works as what I thought it should i.e. orbit around the world space Y.
However, the when I translate along world X after that, the object assumes the old position before the orbit around the world space Y and then moves.

Any idea about this?

Thanks.

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first of all opengl's matrices are stored in the following order


/*
00,04,08,12
01,05,09,13
02,06,10,14
03,07,11,15
*/

float mat[16];
glMultiMatrixf(mat);



12,13,14 is the translation vector

if you perform matrix transformations remember the the last matrix multiplied onto the matrix stack is the first one applied to your vertices.

e.g.:
rot(0,1,0,90)*trans(1,1,1) will translate your vertices by (1,1,1) and finally rotate the translated vertices around the y axis with the origin(0,0,0) as your rotation pivot.


if you want to rotate your object in object coordinates you need to first translate its center to the origin, rotate it, translate it back.
e.g.:
trans(1,1,1)*rot(0,1,0,90)*trans(-1,-1,-1)




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Quote:
Original post by Basiror
first of all opengl's matrices are stored in the following order

*** Source Snippet Removed ***

12,13,14 is the translation vector

if you perform matrix transformations remember the the last matrix multiplied onto the matrix stack is the first one applied to your vertices.

e.g.:
rot(0,1,0,90)*trans(1,1,1) will translate your vertices by (1,1,1) and finally rotate the translated vertices around the y axis with the origin(0,0,0) as your rotation pivot.


if you want to rotate your object in object coordinates you need to first translate its center to the origin, rotate it, translate it back.
e.g.:
trans(1,1,1)*rot(0,1,0,90)*trans(-1,-1,-1)


Hi Basiror,

Thanks for your reply.
I was told about pre and post-multiply to achieve world and local transformation.
Would you advise doing pre or post-multiply for performing world and local matrix?

Kindly advice from anyone is welcome too, please.
I have been struggling and reading a lot of articles on matrices.
I understand the basic but when it comes to applying transformation, I would still get confused.
Hope could get some helps.
Thanks.

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By convention opengl matrices are post multiplied

Think about it like this:

Example post multiply(OPENGL):
1) Modelview; glTranslate(...);
2) Modelview*trans(...); glRotate(....);
3) Modelview*trans(...)*rotate(...);

Post multiply == multiply from right
Pre multiply == multiply from left

Example pre multiply:
1) trans(...); glRotate(....);
2) trans(...)*rotate(...); premult(modelview);
3) Modelview*trans(...)*rotate(...);


thats all.

some notes:
your matrix is orthogonal:
-> the transpose of the matrix is its inverse
-> the column vectors are perpendicular to each other
e.g.: a*b == 0 -> a perpendicular to b

the determinant of a square diagonal matrix is its determinant

there is really a lot of information I could provide you, but I would suggest your to get a math script of a lecture at university, there tons of them on the net.

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Quote:
Original post by Basiror
By convention opengl matrices are post multiplied

Think about it like this:

Example post multiply(OPENGL):
1) Modelview; glTranslate(...);
2) Modelview*trans(...); glRotate(....);
3) Modelview*trans(...)*rotate(...);

Post multiply == multiply from right
Pre multiply == multiply from left

Example pre multiply:
1) trans(...); glRotate(....);
2) trans(...)*rotate(...); premult(modelview);
3) Modelview*trans(...)*rotate(...);


thats all.

some notes:
your matrix is orthogonal:
-> the transpose of the matrix is its inverse
-> the column vectors are perpendicular to each other
e.g.: a*b == 0 -> a perpendicular to b

the determinant of a square diagonal matrix is its determinant

there is really a lot of information I could provide you, but I would suggest your to get a math script of a lecture at university, there tons of them on the net.


I was actually looking at the implementation in the Ogre3d engine too.
In there the order of matrix multiplication is according to the transform space that is specified.
I wanted to re-create that example to understand on the matrix transformation.
I was thinking that doing that way will be much more flexible in terms of implementing user interaction.
Any advice on that?
Thanks.

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Quote:
Original post by _neutrin0_
Actually you can use any order for storing matrices. OpenGL does not enforce ordering of matrices.

The only thing I like to add is this link. Hope that helps.


Of course you can use whatever order you wish, but remember that you should take care about sequential access to the matrix elements when multiplying matrices with vectors

I would adopt my internal order in such a way that you can directly use OpenGL matrices for the sake of simplicity.

For my personal projects I multiply from the right by convention.
Its just more intuitive and most frameworks I have worked with do it the same way.

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Quote:
Original post by Basiror
For my personal projects I multiply from the right by convention.
Its just more intuitive and most frameworks I have worked with do it the same way.
I'm kind of nitpicking here, but for the benefit of the OP I'd like to suggest that neither left-multiplication nor right-multiplication is any more intuitive than the other. One or the other may be more intuitive to a particular person (depending on how they prefer to think about sequential transforms), but neither is objectively better than the other (or at least I haven't come across any convincing arguments to this effect).

Also, I personally find the terms 'right multiply' and 'left multiply' to be somewhat confusing; the terms 'row-vector notation' and 'column-vector notation' are, on the other hand, unambiguous.

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May I ask for more advices, please?

With OpenGL, we have glTranslatef(), glRotatef(), glScalef() which have to be called within glPushMatrix() and glPopMatrix().
I find that this might limit the possibility of my key interaction as I will need to fix the function calls before I render the object.
For example, the sample I created above, I want the object to be able to change its direction (rotate on it's own space), move forward according to direction, orbit a point, then move left and right.
Therefore i decided to keep my own matrix.

Any advices, please?

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Quote:
Original post by juxie
May I ask for more advices, please?

With OpenGL, we have glTranslatef(), glRotatef(), glScalef() which have to be called within glPushMatrix() and glPopMatrix().
I find that this might limit the possibility of my key interaction as I will need to fix the function calls before I render the object.
For example, the sample I created above, I want the object to be able to change its direction (rotate on it's own space), move forward according to direction, orbit a point, then move left and right.
Therefore i decided to keep my own matrix.

Any advices, please?
Whether or not you use your own math functions or OpenGL's has nothing to do with when, whether, or how an object transform can be modified. I tend to find using one's own math code to be more flexible and convenient than relying exclusively on OpenGL function calls, but there's no fundamental difference in functionality between the two methods (at least not of the sort you describe above).

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Quote:
Original post by Brother Bob
Where did you get the idea from that you have to call matrix functions within a glPush/PopMatrix pair? You're free not to if you don't want to.


i mean if i don't call the matrix functions within glPush/PopMatrix, it will affect the view matrix, right?

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Quote:
Original post by jyk

Whether or not you use your own math functions or OpenGL's has nothing to do with when, whether, or how an object transform can be modified. I tend to find using one's own math code to be more flexible and convenient than relying exclusively on OpenGL function calls, but there's no fundamental difference in functionality between the two methods (at least not of the sort you describe above).


I am really sorry for my confusion.
Is it right to say that, OpenGL matrix is always according to local matrix?
Whereas DirectX is according to world matrix?

How can I flexibly combine world transform and local transform?

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Quote:
Original post by juxie
Is it right to say that, OpenGL matrix is always according to local matrix?
Whereas DirectX is according to world matrix?
I've heard people say that, but I'm not exactly sure what they mean by it, and in any case I would argue that it's not really accurate (OpenGL and DirectX deal with transforms in basically the same way).

I think what people are referring to here is the difference in notational convention between the two APIs (row-vector vs. column-vector notation) and the implications for multiplication order.

When using the DirectX math library this is directly evident, but in OpenGL everything happens 'under the hood'. It could probably be argued that OpenGL itself doesn't really assume a notational convention; rather, it is simply the case that transforms are applied in the opposite of the order in which the corresponding function calls appear in the code. (Most OpenGL references use column-vector notation, however, so this is how people tend to think of things when working with OpenGL transform functions.)

It's all a bit confusing, but I think the first thing you need to understand is that OpenGL and D3D/DirectX are fundamentally the same in terms of how they deal with transforms. I say 'fundamentally' because there are a number of superficial differences - for example, D3D maintains separate world and model matrices, while OpenGL combines them into one - but the concepts are essentially the same.

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Quote:
Original post by jyk
Quote:
Original post by Basiror
For my personal projects I multiply from the right by convention.
Its just more intuitive and most frameworks I have worked with do it the same way.
I'm kind of nitpicking here, but for the benefit of the OP I'd like to suggest that neither left-multiplication nor right-multiplication is any more intuitive than the other. One or the other may be more intuitive to a particular person (depending on how they prefer to think about sequential transforms), but neither is objectively better than the other (or at least I haven't come across any convincing arguments to this effect).

Also, I personally find the terms 'right multiply' and 'left multiply' to be somewhat confusing; the terms 'row-vector notation' and 'column-vector notation' are, on the other hand, unambiguous.


The western world writes from left to right.
So lets say you do multiply from left, you had to write down the entire matrix stack again if you forgot a transformation and there is no room left to insert it.

Just an example, but frankly in each math/computer graphic lecture I have attended so far we multiply from right.

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The western world writes from left to right.
So lets say you do multiply from left, you had to write down the entire matrix stack again if you forgot a transformation and there is no room left to insert it.

Just an example, but frankly in each math/computer graphic lecture I have attended so far we multiply from right.
Just to make sure I'm not misunderstanding, when you say 'multiply from the right' are you referring to column-vector notation, or row-vector notation?

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I've heard people say that, but I'm not exactly sure what they mean by it, and in any case I would argue that it's not really accurate (OpenGL and DirectX deal with transforms in basically the same way).

I think what people are referring to here is the difference in notational convention between the two APIs (row-vector vs. column-vector notation) and the implications for multiplication order.

When using the DirectX math library this is directly evident, but in OpenGL everything happens 'under the hood'. It could probably be argued that OpenGL itself doesn't really assume a notational convention; rather, it is simply the case that transforms are applied in the opposite of the order in which the corresponding function calls appear in the code. (Most OpenGL references use column-vector notation, however, so this is how people tend to think of things when working with OpenGL transform functions.)

It's all a bit confusing, but I think the first thing you need to understand is that OpenGL and D3D/DirectX are fundamentally the same in terms of how they deal with transforms. I say 'fundamentally' because there are a number of superficial differences - for example, D3D maintains separate world and model matrices, while OpenGL combines them into one - but the concepts are essentially the same.


I feel bad that I am still pretty much confused.
I tried to read up on quite a number of article, it seems to touch on simpler transformation.
I am not sure when I did the transformation correct or when I did it incorrectly.

Anywhere I can read up on this?

Thanks.

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The western world writes from left to right.
So lets say you do multiply from left, you had to write down the entire matrix stack again if you forgot a transformation and there is no room left to insert it.

Just an example, but frankly in each math/computer graphic lecture I have attended so far we multiply from right.
Just to make sure I'm not misunderstanding, when you say 'multiply from the right' are you referring to column-vector notation, or row-vector notation?


column

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Original post by jyk
I've heard people say that, but I'm not exactly sure what they mean by it, and in any case I would argue that it's not really accurate (OpenGL and DirectX deal with transforms in basically the same way).

I think what people are referring to here is the difference in notational convention between the two APIs (row-vector vs. column-vector notation) and the implications for multiplication order.

When using the DirectX math library this is directly evident, but in OpenGL everything happens 'under the hood'. It could probably be argued that OpenGL itself doesn't really assume a notational convention; rather, it is simply the case that transforms are applied in the opposite of the order in which the corresponding function calls appear in the code. (Most OpenGL references use column-vector notation, however, so this is how people tend to think of things when working with OpenGL transform functions.)

It's all a bit confusing, but I think the first thing you need to understand is that OpenGL and D3D/DirectX are fundamentally the same in terms of how they deal with transforms. I say 'fundamentally' because there are a number of superficial differences - for example, D3D maintains separate world and model matrices, while OpenGL combines them into one - but the concepts are essentially the same.


I feel bad that I am still pretty much confused.
I tried to read up on quite a number of article, it seems to touch on simpler transformation.
I am not sure when I did the transformation correct or when I did it incorrectly.

Anywhere I can read up on this?

Thanks.


Do you have a math program like octave installed?
I would suggest you to do some transformations manually and to examining the results to get an intuition how they behave.

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Do you have a math program like octave installed?
I would suggest you to do some transformations manually and to examining the results to get an intuition how they behave.


I have just downloaded octave.
I will give it a try and hopefully will understand matrices better.
Thanks everyone.

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Just to make sure I'm not misunderstanding, when you say 'multiply from the right' are you referring to column-vector notation, or row-vector notation?
column
Oh, ok. Well, I'm with you on column-vector notation, and it does seem that most academic references use this convention (as do, of course, most OpenGL references). However, since DirectX (which uses row-vector notation) is so prevalent, I would guess that row-vector and column-vector notation are used about equally overall.

As far as intuitiveness goes, row vector advocates point out that sequences of transforms read naturally from left to right when written using row-vector notation, while column vector advocates sometimes use the counter-argument that column vector notation closely mirrors function composition.

Again, I tend to use column vectors myself, but I'm still not sure if one or the other convention can be said to be more intuitive (or more prevalent overall) than the other.

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      Static variables (SHADER_VARIABLE_TYPE_STATIC) are variables that are expected to be set only once. They may not be changed once a resource is bound to the variable. Such variables are intended to hold global constants such as camera attributes or global light attributes constant buffers. Mutable variables (SHADER_VARIABLE_TYPE_MUTABLE) define resources that are expected to change on a per-material frequency. Examples may include diffuse textures, normal maps etc. Dynamic variables (SHADER_VARIABLE_TYPE_DYNAMIC) are expected to change frequently and randomly. Shader variable type must be specified during shader creation by populating an array of ShaderVariableDesc structures and initializing ShaderCreationAttribs::Desc::VariableDesc and ShaderCreationAttribs::Desc::NumVariables members (see example of shader creation above).
      Static variables cannot be changed once a resource is bound to the variable. They are bound directly to the shader object. For instance, a shadow map texture is not expected to change after it is created, so it can be bound directly to the shader:
      PixelShader->GetShaderVariable( "g_tex2DShadowMap" )->Set( pShadowMapSRV ); Mutable and dynamic variables are bound via a new Shader Resource Binding object (SRB) that is created by the pipeline state (IPipelineState::CreateShaderResourceBinding()):
      m_pPSO->CreateShaderResourceBinding(&m_pSRB); Note that an SRB is only compatible with the pipeline state it was created from. SRB object inherits all static bindings from shaders in the pipeline, but is not allowed to change them.
      Mutable resources can only be set once for every instance of a shader resource binding. Such resources are intended to define specific material properties. For instance, a diffuse texture for a specific material is not expected to change once the material is defined and can be set right after the SRB object has been created:
      m_pSRB->GetVariable(SHADER_TYPE_PIXEL, "tex2DDiffuse")->Set(pDiffuseTexSRV); In some cases it is necessary to bind a new resource to a variable every time a draw command is invoked. Such variables should be labeled as dynamic, which will allow setting them multiple times through the same SRB object:
      m_pSRB->GetVariable(SHADER_TYPE_VERTEX, "cbRandomAttribs")->Set(pRandomAttrsCB); Under the hood, the engine pre-allocates descriptor tables for static and mutable resources when an SRB objcet is created. Space for dynamic resources is dynamically allocated at run time. Static and mutable resources are thus more efficient and should be used whenever possible.
      As you can see, Diligent Engine does not expose low-level details of how resources are bound to shader variables. One reason for this is that these details are very different for various APIs. The other reason is that using low-level binding methods is extremely error-prone: it is very easy to forget to bind some resource, or bind incorrect resource such as bind a buffer to the variable that is in fact a texture, especially during shader development when everything changes fast. Diligent Engine instead relies on shader reflection system to automatically query the list of all shader variables. Grouping variables based on three types mentioned above allows the engine to create optimized layout and take heavy lifting of matching resources to API-specific resource location, register or descriptor in the table.
      This post gives more details about the resource binding model in Diligent Engine.
      Setting the Pipeline State and Committing Shader Resources
      Before any draw or compute command can be invoked, the pipeline state needs to be bound to the context:
      m_pContext->SetPipelineState(m_pPSO); Under the hood, the engine sets the internal PSO object in the command list or calls all the required native API functions to properly configure all pipeline stages.
      The next step is to bind all required shader resources to the GPU pipeline, which is accomplished by IDeviceContext::CommitShaderResources() method:
      m_pContext->CommitShaderResources(m_pSRB, COMMIT_SHADER_RESOURCES_FLAG_TRANSITION_RESOURCES); The method takes a pointer to the shader resource binding object and makes all resources the object holds available for the shaders. In the case of D3D12, this only requires setting appropriate descriptor tables in the command list. For older APIs, this typically requires setting all resources individually.
      Next-generation APIs require the application to track the state of every resource and explicitly inform the system about all state transitions. For instance, if a texture was used as render target before, while the next draw command is going to use it as shader resource, a transition barrier needs to be executed. Diligent Engine does the heavy lifting of state tracking.  When CommitShaderResources() method is called with COMMIT_SHADER_RESOURCES_FLAG_TRANSITION_RESOURCES flag, the engine commits and transitions resources to correct states at the same time. Note that transitioning resources does introduce some overhead. The engine tracks state of every resource and it will not issue the barrier if the state is already correct. But checking resource state is an overhead that can sometimes be avoided. The engine provides IDeviceContext::TransitionShaderResources() method that only transitions resources:
      m_pContext->TransitionShaderResources(m_pPSO, m_pSRB); In some scenarios it is more efficient to transition resources once and then only commit them.
      Invoking Draw Command
      The final step is to set states that are not part of the PSO, such as render targets, vertex and index buffers. Diligent Engine uses Direct3D11-syle API that is translated to other native API calls under the hood:
      ITextureView *pRTVs[] = {m_pRTV}; m_pContext->SetRenderTargets(_countof( pRTVs ), pRTVs, m_pDSV); // Clear render target and depth buffer const float zero[4] = {0, 0, 0, 0}; m_pContext->ClearRenderTarget(nullptr, zero); m_pContext->ClearDepthStencil(nullptr, CLEAR_DEPTH_FLAG, 1.f); // Set vertex and index buffers IBuffer *buffer[] = {m_pVertexBuffer}; Uint32 offsets[] = {0}; Uint32 strides[] = {sizeof(MyVertex)}; m_pContext->SetVertexBuffers(0, 1, buffer, strides, offsets, SET_VERTEX_BUFFERS_FLAG_RESET); m_pContext->SetIndexBuffer(m_pIndexBuffer, 0); Different native APIs use various set of function to execute draw commands depending on command details (if the command is indexed, instanced or both, what offsets in the source buffers are used etc.). For instance, there are 5 draw commands in Direct3D11 and more than 9 commands in OpenGL with something like glDrawElementsInstancedBaseVertexBaseInstance not uncommon. Diligent Engine hides all details with single IDeviceContext::Draw() method that takes takes DrawAttribs structure as an argument. The structure members define all attributes required to perform the command (primitive topology, number of vertices or indices, if draw call is indexed or not, if draw call is instanced or not, if draw call is indirect or not, etc.). For example:
      DrawAttribs attrs; attrs.IsIndexed = true; attrs.IndexType = VT_UINT16; attrs.NumIndices = 36; attrs.Topology = PRIMITIVE_TOPOLOGY_TRIANGLE_LIST; pContext->Draw(attrs); For compute commands, there is IDeviceContext::DispatchCompute() method that takes DispatchComputeAttribs structure that defines compute grid dimension.
      Source Code
      Full engine source code is available on GitHub and is free to use. The repository contains two samples, asteroids performance benchmark and example Unity project that uses Diligent Engine in native plugin.
      AntTweakBar sample is Diligent Engine’s “Hello World” example.

       
      Atmospheric scattering sample is a more advanced example. It demonstrates how Diligent Engine can be used to implement various rendering tasks: loading textures from files, using complex shaders, rendering to multiple render targets, using compute shaders and unordered access views, etc.

      Asteroids performance benchmark is based on this demo developed by Intel. It renders 50,000 unique textured asteroids and allows comparing performance of Direct3D11 and Direct3D12 implementations. Every asteroid is a combination of one of 1000 unique meshes and one of 10 unique textures.

      Finally, there is an example project that shows how Diligent Engine can be integrated with Unity.

      Future Work
      The engine is under active development. It currently supports Windows desktop, Universal Windows and Android platforms. Direct3D11, Direct3D12, OpenGL/GLES backends are now feature complete. Vulkan backend is coming next, and support for more platforms is planned.
    • By michaeldodis
      I've started building a small library, that can render pie menu GUI in legacy opengl, planning to add some traditional elements of course.
      It's interface is similar to something you'd see in IMGUI. It's written in C.
      Early version of the library
      I'd really love to hear anyone's thoughts on this, any suggestions on what features you'd want to see in a library like this? 
      Thanks in advance!
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