## Plotting 4D Graphs With QOpenGLWidget

A 4D graph can be used for plotting a graph of a complex function, with complex numbers as its input and output. A complex number can be represented as a pair of two real numbers:

• A real part and an imaginary part.
• A magnitude and an argument.

An image has a width and a height, but you can add an illusion of depth using a linear transform and changes in hue. A common way to add the fourth dimension is to use colors to represent its value. If you use a color for the fourth dimension, you want to add a legend to tell the viewer how to estimate the value of the fourth coordinate.

A good example of one such graph is the graph of the Lambert function found in the Wikipedia.

The Lambert W function is defined as follows:

$x=ye^y\Rightarrow y=W_n(x)$

where $n \in \mathbb{Z}$ is the branch number.

In this post, I will discuss the use of Qt5 modules in creating my version of the graph:

The graph is not perfect and complete, but in this post I will explain how to create the components of the graph.

## Combining OpenGL and 2D Painting

In Qt5, the OpenGL widget is a painting device like any other widget. So, you can use a painter of class QPainter to paint 2D shapes on it. The painter is useful if you want to add text to an image. Remember to unbind every GL object (vertex array objects, buffers, textures) before painting with the painter.

Before painting with the GL functions, call the painter’s method beginNativePainting. After finishing the use of GL functions, call endNativePainting

If you want the text rotation, you can use SVG. A simple way to add SVG text is using the QSvgRenderer class, that can create an SVG image from a file or a string.

## Single Points

You can draw single points in OpenGL. To do so, you must enable GL_PROGRAM_POINT_SIZE using glEnable in your calling program, and define glPointSize in the vertex shader.

## Matrices

The main types of matrices used in my example are:

• Frustum – A perspective matrix defined using the minimum and maximum values of coordinates.
• Viewport – a matrix that transforms the coordinates used by GL into the coordinates of the input rectangle. That transform will be used for placing the axis ticks snd texts.

## The Program

The main window is a Qt Widget, so

### The QMake File: qmake.pro:

The following definition will be used for creating the make file.

TARGET=executable
SOURCES=main.cpp

DESTDIR=bin
QT=core
QT+=gui
QT+=svg

greaterThan(QT_MAJOR_VERSION, 4): QT+=widgets


The header files ‘graph.h’ contains some definition to be used by the main program and the widget extending the QT Widget:

#include <QtWidgets/QOpenGLWidget>
#include <QOpenGLBuffer>
#include <QOpenGLVertexArrayObject>
#include <QOpenGLTexture>
#include <QPainter>
#include <QtSvg>
#include <QStyleOptionGraphicsItem>

struct graph_point {
GLfloat coords[3];
GLfloat hue;
};

struct vertex2D {
GLfloat coords[2];
GLfloat hue;
};

struct axis_ticks {
GLfloat rect_coords[3];
GLfloat texture_coords[2];
};

class GraphWidget:public QOpenGLWidget{
private:
QOpenGLVertexArrayObject m_vao;
QOpenGLBuffer m_point_vbo;
QOpenGLBuffer m_ibo;

QPainter m_painter;
QFont m_font;
int fontSize;
int m_width, m_height;
float legend_left;
QMatrix4x4 transform;
QMatrix4x4 viewport;

void populatePointBuffer(void);
void paintGraph(void);
void draw3DLine(QVector4D fromVec, QVector4D toVec);

public:
GraphWidget(QWidget *parent=nullptr);
~GraphWidget();
void initializeGL();
void paintGL();
void resizeGL(int w, int h);
};


### HSV to RGB

The fourth coordinate of our graph represents arg(W(z)), an angle. In the graph we’ll use HSV (Hue, Saturation, Value) because the hue is given by the value of an angle. A shader-language function to covert from HSV to RGB is defined in file “shaders.h“:

#define TO_RGB_FUNC					\
"#version 140\n"					\
"vec4 to_rgb(float hue){ \n"				\
"float s=1.0;\n"					\
"float v=1.0;\n"					\
"float c=v*s;\n"					\
"float   x=c*(1-abs(mod(3*hue,2)-1));\n"		\
"float   m=v-c;\n"					\
"vec3  tempRGB=hue< -2./3?  vec3(0,x,c):\n"		\
"              hue< -1./3?  vec3(x,0,c): \n"		\
"              hue<     0?  vec3(c,0,x): \n"		\
"              hue<  1./3?  vec3(c,x,0): \n"		\
"              hue<  2./3?  vec3(x,c,0): \n"		\
"		              vec3(0,c,x); \n"		\
"\n"							\
"return vec4(tempRGB+vec3(m,m,m),1);\n"		\
"}\n"


The shader fragment for both the graph points and the legend is defined as follows:

#define FRAGMENT_SHADER				\
TO_RGB_FUNC					\
"varying float f_hue;\n"			\
"void main(void){\n"				\
"    gl_FragColor=to_rgb(f_hue);\n"		\
"}\n"


### Main Program Includes and Definitions

#include <stdio.h>

#include <graph.h>
#include <QApplication>
#include "svg.h"
#include <complex>
#include <iostream>

#define MIN_WIDTH 640
#define MIN_HEIGHT 480

using namespace std;



### Constructor and Destructor

GraphWidget::GraphWidget(QWidget *parent):
QOpenGLWidget(parent),
m_vao(),
m_point_vbo(),
m_ibo(QOpenGLBuffer::IndexBuffer),
m_font()
{
setGeometry(0,0,MIN_WIDTH,MIN_HEIGHT);
setMinimumWidth(MIN_WIDTH);
setMinimumHeight(MIN_HEIGHT);
}

GraphWidget::~GraphWidget(){
makeCurrent();
m_point_vbo.destroy();
m_ibo.destroy();
m_vao.destroy();
doneCurrent();

}

// NUM_ARGS - Args of complex numbers.
// NUM_MAGNS - Number of magnitudes.
#define NUM_ARGS 140
#define NUM_MAGNS 200


### Initializing, Populating the Point Buffer

Here we’ll define the variables that are initialize once. The following function, initializeGL, is called once upon the widget’s initization:

void GraphWidget::initializeGL(){
makeCurrent();
glEnable(GL_BLEND);
glEnable(GL_DEPTH_TEST);
glEnable( GL_PROGRAM_POINT_SIZE );
glBlendFunc(GL_SRC_ALPHA, GL_ONE_MINUS_SRC_ALPHA);

m_vao.create();

m_point_vbo.create();
m_point_vbo.bind();
m_point_vbo.allocate(NUM_MAGNS*NUM_ARGS*sizeof(graph_point));
populatePointBuffer();

GLshort indices[6]={0,1,2,2,3,0};
m_ibo.create();
m_ibo.bind();
m_ibo.allocate(indices,sizeof(indices));
m_ibo.release();
m_point_vbo.release();
}


A call to glEnable( GL_PROGRAM_POINT_SIZE ); enables drawing single points.

The function populatePointBuffer uses the method ‘write‘ of class QOpenGLBuffer to write points to the vertex buffer. I use the buffer this way so I can use functions of complex variables.

void GraphWidget::populatePointBuffer(void){
int vboOffset=0;
graph_point vec[NUM_MAGNS];
for (float magn=0;magn<1.4;magn+=1.4/NUM_MAGNS){
int pos=0;
for (float arg=-1;arg<1;arg+=2./NUM_ARGS){
// The arg of W(Z) divided by pi.
complex<float> iarg;
iarg=complex<float>(0,M_PI*arg);
complex<float> w=magn*exp(iarg);
complex<float> z=w*exp(w);
vec[pos].hue=arg;
vec[pos].coords[0]=imag(z);
vec[pos].coords[1]=magn;
vec[pos].coords[2]=real(z);
pos+=1;
}
m_point_vbo.write(vboOffset,vec,sizeof(vec));
vboOffset+=sizeof(vec);
}
}


### Resizing

The function resizeGL is called as a response to a widget resize event. In this function the values that change due to a resize event are set. The properties set here are:

• The font size for the painter
• The transform matrix
• The viewport matrix
void GraphWidget::resizeGL(int w, int h){
m_width=w;
m_height=h;
fontSize=12.* min((float)m_width/MIN_WIDTH,(float)m_height/MIN_HEIGHT);
m_font.setPixelSize(fontSize);
float z_far=0.05;
float z_near=0.01;
float a=2./(m_height-2*fontSize);
float b=1.8-a*m_height;
float y_floor=b-2.*a*fontSize;

// leftInPixels,RightInPixels - left and right boundaries
//                              of the graph.
float leftInPixels=12*fontSize+7;
float rightInPixels=m_width - 12 - 15*fontSize;
a=2/(rightInPixels-leftInPixels);
b=-1-a*leftInPixels;
float x_left=b;
float x_right=a*m_width+b;

transform.setToIdentity();

transform.frustum(x_left,x_right,y_floor,1.8,z_near,z_far);
transform.translate(0,0,-(z_far+z_near)/2);
transform.scale(1,1,(z_far-z_near)/2);
viewport.setToIdentity();
viewport.viewport(rect());
}


### Painting

The function paintGL is called as a response to paint events. Following is the code of paintGL:

void GraphWidget::paintGL(){
glClear(GL_COLOR_BUFFER_BIT | GL_DEPTH_BUFFER_BIT);
m_painter.begin(this);
m_painter.setPen(QColor(255,255,255,255));
m_painter.setFont(m_font);

paintGraph();

m_painter.end();
}


Let us look at the functions called by the paint event handler:

The function addLegend uses OpenGL function to create the HSV gradient and the painter to add ticks and text values:

void GraphWidget::addLegend(void){
QString str="arg(W(Z))";

QRect boundingRect=m_painter.boundingRect(rect(),0,str);

m_painter.beginNativePainting();

m_vao.bind();
QMatrix4x4 mat;
mat.ortho(rect());
GLfloat top=boundingRect.height()+20;
GLfloat bottom=top+0.75*m_height;
GLfloat left=m_width-boundingRect.width()-10;
GLfloat right=left+0.25*boundingRect.width();
legend_left = left; // Add it to the private members.
vertex2D rect_vertices[4]={{{left,top},1},{{right,top},1},
{{right,bottom},-1},{{left,bottom},-1}};
m_ibo.bind();
GLshort indices[6]={0,1,2,2,3,0};
m_ibo.write(0,indices,sizeof(indices));

m_program->bind();
m_program->setAttributeBuffer("legend_coords",
GL_FLOAT,
0,
2,
sizeof(vertex2D));
m_program->setAttributeBuffer("legend_hue",
GL_FLOAT,
sizeof(rect_vertices[0].coords),
1,
sizeof(vertex2D));
m_program->setUniformValue("ortho",mat);

m_vao.release();
m_vao.bind();
m_program->enableAttributeArray("legend_coords");
m_program->enableAttributeArray("legend_hue");
glDrawElements(GL_TRIANGLES,6,GL_UNSIGNED_SHORT,0);

m_vao.release();
m_program->disableAttributeArray("legend_coords");
m_program->disableAttributeArray("legend_hue");
m_program->release();
m_ibo.release();

m_painter.endNativePainting();
m_painter.drawText(m_width-boundingRect.width()-10,boundingRect.height()+10,str);

QString vals[7]={"pi","2pi/3","pi/3","0","-pi/3","-2pi/3","-pi"};
float curr_height=top;
float dist=(bottom-top)/6;
for (int i=0;i<7;i++){
m_painter.drawLine(right,curr_height,right+10,curr_height);
m_painter.drawText(right+15,curr_height+0.4*boundingRect.height(),vals[i]);
curr_height+=dist;
}
}


Following is the macro in shaders.h, which defines the vertex shader for the legend:

#define LEGEND_VERTEX_SHADER					\
"attribute vec2 legend_coords;\n"				\
"attribute float legend_hue;\n"				\
"uniform mat4 ortho;\n"					\
"varying float f_hue;\n"					\
"void main(void){\n"						\
"    gl_Position=ortho*vec4(legend_coords,0,1);\n"		\
"    f_hue=legend_hue;\n"					\
"}"


The function addAxes uses the painter and the viewport matrix to draw axes and add the ticks. You hove noticed that the matrices are of size 4×4 and the vectors are of length 4; when a matrix is multiplied by 4, you should divide coordinates x,y by coordinate w to find the vector’s location on the 2D window. Special treatment is added to the slanted z-axis, where SVG is used:

void GraphWidget::addAxes(void){

QVector4D startVec,endVec;

// Draw the x-axis
startVec=viewport*transform*QVector4D(-1,-0.2,1.1,1);
endVec=viewport*transform*QVector4D(1,-0.2,1.1,1);
draw3DLine(startVec,endVec);

// Draw the y-axis
startVec=viewport*transform*QVector4D(-1,-0.2,1.1,1);
endVec=viewport*transform*QVector4D(-1,1.4,1.1,1);
draw3DLine(startVec,endVec);

// Draw the z-axis
startVec=viewport*transform*QVector4D(1,-0.2,-1,1);
endVec=viewport*transform*QVector4D(1,-0.2,1.1,1);
draw3DLine(startVec,endVec);
}

void GraphWidget::draw3DLine(QVector4D fromVec, QVector4D toVec){
m_painter.drawLine(fromVec.x()/fromVec.w(),
m_height-fromVec.y()/fromVec.w(),
toVec.x()/toVec.w(),
m_height-toVec.y()/toVec.w());
}

QVector4D start_point,end_point;
QString vals[]={"-1","-0.5","0","0.5","1"};
int i=0;
for (float loc=-1;loc<=1;loc+=0.5){
start_point=viewport*transform*QVector4D(loc,-0.2,1.1,1);
start_point.setX(start_point.x()/start_point.w());
start_point.setY(m_height-start_point.y()/start_point.w());
end_point=start_point;
end_point.setY(end_point.y()+5);
m_painter.drawLine(start_point.x(),start_point.y(), end_point.x(),end_point.y());
m_painter.drawText(start_point.x(),end_point.y()+fontSize+1,vals[i++]);
}
const char *text="Im(Z)";
QVector4D start_vector=viewport*transform*QVector4D(-1,-0.2,1.1,1);
QVector4D end_vector=viewport*transform*QVector4D(1,-0.2,1.1,1);
float text_start=(start_vector.x()+end_vector.x())/(2.*end_vector.w())-2*fontSize;
float bottom=m_height-end_vector.y()/end_vector.w()+2*fontSize+5;
m_painter.drawText(text_start,bottom,text);
}

QVector4D start_point,end_point;
QString vals[]={"0.0","0.2","0.4","0.6","0.8","1.0","1.2","1.4"};
int i=0;
for (float loc=0;loc<=1.41;loc+=0.2){
start_point=viewport*transform*QVector4D(-1,loc,1.1,1);
start_point.setX(start_point.x()/start_point.w());
start_point.setY(m_height-start_point.y()/start_point.w());
end_point=start_point;
end_point.setX(end_point.x()-5);
m_painter.drawLine(start_point.x(),start_point.y(), end_point.x(),end_point.y());
m_painter.drawText(end_point.x()-2*fontSize,end_point.y()+0.5*fontSize,vals[i++]);

}
const char *text="|W(z)|";
QVector4D start_vector=viewport*transform*QVector4D(-1,-0.2,1.1,1);
QVector4D end_vector=viewport*transform*QVector4D(1,1,1.1,1);
float text_y_position=(start_vector.y()+end_vector.y())/(2.*end_vector.w())-2*fontSize;
float text_rightmost=start_vector.x()/start_vector.w()-6*fontSize;
m_painter.drawText(text_rightmost,text_y_position,text);
}

QVector4D start_point,end_point;
QString vals[]={"-1","-0.5","0","0.5","1"};
int i=0;
for (float loc=-1;loc<=1;loc+=0.5){
start_point=viewport*transform*QVector4D(1,-0.2,loc,1);
start_point.setX(start_point.x()/start_point.w());
start_point.setY(m_height-start_point.y()/start_point.w());
end_point=start_point;
end_point.setY(end_point.y()-5);
m_painter.drawLine(start_point.x(),start_point.y(), end_point.x(),end_point.y());
if (i!=1){
m_painter.drawText(end_point.x()-fontSize/2.,end_point.y()-1,vals[i]);
}
i+=1;
}
}

char *svg_str=(char *)calloc(strlen(SVG_TEXT_STR)+1,sizeof(char));
const char *text="Re(Z)";
QVector4D start_point,end_point;
start_point=viewport*transform*QVector4D(1,-0.2,1,1);
start_point.setX(start_point.x()/start_point.w());
start_point.setY(m_height-start_point.y()/start_point.w());
end_point=viewport*transform*QVector4D(1,-0.2,-1,1);
end_point.setX(end_point.x()/end_point.w());
end_point.setY(m_height-end_point.y()/end_point.w());
float centerX=(start_point.x()+end_point.x())/2.;
float centerY=(start_point.y()+end_point.y())/2.;
float lenOfText=(strlen(text)-1.5)*fontSize;
float textX=centerX-lenOfText;
float textY=centerY+fontSize;

sprintf(svg_str,SVG_TEXT_STR,textX,textY,fontSize,inDegrees,centerX,centerY,text);
cout<<svg_str<<endl;
cout<<"Length of svg_str"<<strlen(svg_str)<<endl;
cout<<"Lenght of SVG_TEXT_STR"<<strlen(SVG_TEXT_STR)<<endl;

QSvgRenderer renderer((QByteArray(svg_str)));
renderer.render(&m_painter,rect());

}



The function add_z_text renders the SVG text using an object of call QSvgRendered. It accepts a byte array made from a string. It can also take a file name. The macro SVG_TEXT_STR is defined in file “svg.h” as follows:/code

#define SVG_TEXT_STR							\
"<svg>"								\
"    <text x='%.0f'" 							\
"          y='%.0f'"							\
"          font-size='%dpx'"						\
"          stroke='white'"						\
"          fill='white'"						\
"          transform='rotate(%.0f ,  %.0f ,  %.0f)'>%s</text>"	\
"</svg>"



#### Painting the Graph

Following is the function that paints the graph:

void GraphWidget::paintGraph(){
m_painter.beginNativePainting();
m_vao.bind();
m_point_vbo.bind();
m_program->bind();
m_program->setAttributeBuffer("graph_coords",
GL_FLOAT,
0,
3,
sizeof(graph_point));
m_program->setAttributeBuffer("hue",
GL_FLOAT,
3*sizeof(GLfloat),
1,
sizeof(graph_point));
m_program->setUniformValue("transform",transform);
m_vao.release();
m_vao.bind();
m_program->enableAttributeArray("graph_coords");
m_program->enableAttributeArray("hue");
glDrawArrays(GL_POINTS,0,NUM_ARGS*NUM_MAGNS);
m_program->release();
m_point_vbo.release();
m_vao.release();
m_painter.endNativePainting();
}


### The Main Function

Following is the code of the main function:

int main(int argc, char *argv[]){
QApplication app(argc, argv);
GraphWidget w;
w.show();

return app.exec();
}


## OpenGL Made Simple with Qt5

Drawing and painting a 3D image with GL – the Graphic Library – is not as trivial as doing the same on a 2D canvas: no 3D versions of the 2D canvas drawing methods are defined in GL classes. Instead, programs written with GL send data to the GPU (Graphic Processing Unit), and directions for organizing the data in data structures and for the rendition of data to the screen. The data is stored in buffers, and processed by shader programs.

Once you know how to use the OpenGL API, drawing the following image is simple:

The way Qt simplifies the use of GL are as follows:

• it offers classes instead of integers to wrap GL resources: buffers, textures, vertex array and more.
• the classes include overloaded functions.
• the developer can create shaders without specifying the GL version.
• Qt manages the main loop.

The following sections will describe the process of painting a 3D surface.

## Initializing

To start working with GL, create an object of class QOpenGLWidget. Better create a subclass of QOpenGLWidget, if you work with Qt 5.4 or a later version. Don’t perform any GL operations before the method initializeGL is called. If you extend OpenGLWidget, any GL operation should be performed by one of the following methods: initializeGL, paintGL and resizeGL.

In initializeGL place operation to be performed once only at the initialization time. Some examples of initializeGL can be found in

Qt Assistnat->Qt Widgets->C++ Classes->QOpenGLWidget

Following is the initialization stage of my program to paint the triangle:

## The Class

Following is a definition of the ExtendedOpenGLWidget class:

#include <QtWidgets/QOpenGLWidget>
#include <QOpenGLVertexArrayObject>
#include <QOpenGLBuffer>
#include <QOpenGLContext>

class ExtendedOpenGLWidget:public QOpenGLWidget {
private:
QOpenGLVertexArrayObject m_vao;
QOpenGLBuffer m_vbo;
public:
ExtendedOpenGLWidget(QWidget *parent=nullptr);
~ExtendedOpenGLWidget();
void initializeGL();
void paintGL();
};


The class is defined in file “ui_opengl.h”.

#include <ui_opengl.h>
#include <iostream>

using namespace std;

The file “ui_opengl” is the header file generated by th Qt Designer. The rest can be used for debug printing.

### The vertices

Vertices coordinates and colors are defined in my program as follows:

struct vertex {
float coords[3];
float colors[4];
};

struct vertex {
float coords[3];
float colors[4];
};
vertex vertices[]={
{{-0.9, -0.9, 0.0}, {0.0, 0.0, 1.0, 1.0}},
{{ 0.9, -0.9, 0.0}, {0.0, 1.0, 0.0, 1.0}},
{{ 0.0,  0.9, 0.0}, {1.0, 0.0, 0.0, 1.0}}
};

The above is a definition of a global variable. If you put it in a class, you will have to specify array sizes.

### Constructor

The only thing the constructor does is call the parent constructor:

ExtendedOpenGLWidget::ExtendedOpenGLWidget(QWidget *parent):QOpenGLWidget(parent){
}

### The initializeGL method

This function creates the vertex array object(VAO). If the VAO cannot be created, it cannot be bound to the context. Your program can draw with or without it. In my example, I bind the VAO.

What actually does the drawing is the shader program. For it to draw you should link it at run time with the shaders. If you use Qt 5, the class QOpenGLShaderProgram exists with a method addShaderFromSourceCode, so you can create program in separate text files without the need to recompile your GL program when the shader code changes.

The example in Qt Assistant->Qt GUI->C++ Classes->QOpenGLShaderProgram does not complete the example in Qt Assistnat->Qt Widgets->C++ Classes->QOpenGLWidget; Don’t use it if you bind a buffer, because the coordinates and colors defined by setAttributeArray and setUniformValue will not set the correct locations of the colors of coordinates for the drawing method. In addition, only use coordinates specified in pixel if the GL Widget is of fixed size.

Following is an initializeGL method for coordintes whose values are between -1 and 1, and the color RGBA values are floats between 0 and 1:

void ExtendedOpenGLWidget::initializeGL(){
glEnable(GL_BLEND);
glEnable(GL_DEPTH_TEST);
glBlendFunc(GL_SRC_ALPHA, GL_ONE_MINUS_SRC_ALPHA);
m_vao.create();
if (m_vao.isCreated()){
cout<<"Created"<<endl;
m_vao.bind();
}
m_vbo.create();
m_vbo.bind();

// Allocating buffer memory and populating it
m_vbo.allocate(vertices, sizeof(vertices));

m_program->bind();

// Arguments of setAttributeBuffer in this example:
//    1: A string, the corresponding attribute in the shaders.
//    2: the type of each array element.
//    3: the start point relative to the beginning of the current buffer.
//    4: the number of elements in each vector.
//    5: stride: the difference in bytes between two successive vectors
m_program->setAttributeBuffer("vertex", GL_FLOAT,0,3,sizeof(vertex));
m_program->setAttributeBuffer("color", GL_FLOAT,sizeof(vertices[0].coords),4,sizeof(vertex));
m_program->enableAttributeArray("vertex");
m_program->enableAttributeArray("color");
m_vao.release();
}


• a vertex shader – outputs the coordinates of which point
• a fragment shader – outputs the color of which point.
• Following are the files:

 attribute highp vec3 vertex;
attribute highp vec4 color;
varying highp vec4 f_color;
void main(void){
gl_Position=vec4(vertex,1.0);
f_color=color;
}

 varying mediump vec4 f_color;
void main(void){
gl_FragColor = f_color;
}

## Drawing

The method paintGL draws the image. In this example, it uses the function glDrawArrays with the mode GL_TRIANGLES. In 3D the only 3D polygon type a graphical program always knows how to fill because the vertices are on the same plan.

Following is the paintGL method:

void ExtendedOpenGLWidget::paintGL(){
cout<<"paint"<<endl;
glClear(GL_COLOR_BUFFER_BIT | GL_DEPTH_BUFFER_BIT);
m_vao.bind();
// Arguments of setAttributeBuffer in this example:
//    1. mode - how the shaders will draw the array
//    2. start - first enabled array
//    3. count - the number of indices.
glDrawArrays(GL_TRIANGLES,0,3);
m_vao.release();
}

## The main function

An Open GL Widget may be a top-level window or a widget inside another container. In the following main function, the widget is inside a main window:

int main(int argc, char *argv[]){
QApplication app(argc, argv);
QMainWindow mainWindow;
Ui_MainWindow mainObject;
mainObject.setupUi(&mainWindow);
mainWindow.show();
return app.exec();
}

## A Destructor

If you want to avoid warnings at the end of the execution, you better define a destructor to clean up the allocated objects.

Following is the code of one:

ExtendedOpenGLWidget::~ExtendedOpenGLWidget(){
makeCurrent();
m_vao.destroy();
m_vbo.destroy();
}