How to draw such an openGL star?

Question

C++ / C#

You need to draw just such an asterisk:

As far as I understand, you must first calculate the location of the points on the generating circle in cycles (because there are 10 vertices, then n=10), and then somehow connect them.

float x = R * cos(2 * 3.14*i / n);
float y = R * sin(2 * 3.14*i / n);

But so far, the maximum that I have been able to draw is this: I

haven’t even tried to connect the dots yet, because in the original image two decagons are rotated relative to the center by a few degrees, but I haven’t figured out how to at least rotate the dots, unfortunately.
I would be grateful for advice, maybe I'm not digging there at all ...

void display(){
  glClear(GL_COLOR_BUFFER_BIT);
  glBegin(GL_POINTS); 

  float R = 100;
  for (int i = 0; i < 10; i++) {
    int x = R * cos(2 * 3.14*i / 10) + 320;
    int y = R * sin(2 * 3.14*i / 10)+ 240;
    glVertex2i(x, y);
  }
  R = 120;
  for (int i = 0; i < 10; i++) {
    int x = R * cos(2 * 3.14*i / 10) + 320;
    int y = R * sin(2 * 3.14*i / 10) + 240;
    glVertex2i(x, y);
  }
  R = 160;
  for (int i = 0; i < 10; i++) {
    int x = R * cos(2 * 3.14*i / 10) + 320;
    int y = R * sin(2 * 3.14*i / 10) + 240;
    glVertex2i(x, y);
  }

  glEnd();
  glFlush(); 
}

PS Please don't hit hard, I just got into openGL, maybe (most likely) I still don't understand a lot.

Answer 1

The figure consists of 10 vertices connected by 10 edges.
In the first step, you need to evenly distribute 10 vertices around the unit circle. This must be done in a specially designated array of vertices, because indexed primitives cannot be inferred in this old mode.
At the second stage, we switch the context to the output of a closed line (GL_LINE_LOOP) and output all 10 points from the vertex array, but with an offset of 3 points and modulo 10.
Pseudocode:

void display()
{
   glClear( GL_COLOR_BUFFER_BIT );
   glBegin( GL_LINE_LOOP );

   for( size_t index = 0; index < vertices.size(); ++index )
   {
      const Vertex& vertex = vertices[ ( index * 3 ) % vertices.size() ];
      glVertex2i( vertex.x, vertex.y );
   }

   glEnd();
   glFlush(); 
}

Answer 2

I'll post the full code, maybe someone will need it sometime (I sympathize in advance)

spoiler

#include "OpenGL\glut.h"
#include "OpenGL\gl.h"
#include "math.h"

#define PI 3.1415926535897931159979635

//Определение координаты x точки по формуле x = xc+R*cos(f+(2*pi*i/n)),
//где xc - координата х центра, а f - угловая координата первой вершины
float __x(float start_angle, int radius, int i, int n, int offset) {
  return radius * cos(start_angle + (2 * PI * i / n)) + offset;
}
//Определение координаты y точки по формуле y = xc+R*sin(f+(2*pi*i/n)),
//где xc - координата х центра, а f - угловая координата первой вершины
float __y(float start_angle, int radius, int i, int n, int offset) {
  return radius * sin(start_angle + (2 * PI * i / n)) + offset;
}

//Отрисовка звезды, в ф-цию можно передать радиус, кол-во вершин, смещение по оси х и у,
//а так же угловую координату начальной точки
void drawStar(int radius, int vertexes, int offset_x, int offset_y, float start_angle) {
  glClear(GL_COLOR_BUFFER_BIT);
  glBegin(GL_LINE_LOOP);

  float* o_x = new float[vertexes];
  float* o_y = new float[vertexes];

  for (int i = 0; i < vertexes; i++) {
    o_x[i] = __x(start_angle, radius, i, vertexes, offset_x);
    o_y[i] = __y(start_angle, radius, i, vertexes, offset_y);
  }
  for (int i = 0; i < vertexes; i++) {
    int ind = (i * 3) % vertexes;   //Точки выводим со смещением на 3
    glVertex2f(o_x[ind], o_y[ind]); //и по модулю 10ти
  }

  glEnd();
  glFlush();
}

void display(){
  //Такой вызов отрисовывает звезду аналогичную звезде из условия,
  //угловая координата выбрана "методом тыка"
  drawStar(200, 10, 320, 240, 6.6);
  //так же, кол-во вершин лучше ставить кратное 10ти, иначе фигура,
  //скорее всего, не отрисуется
}

int main(int argc, char** argv)
{
  glutInit(&argc, argv);
  glutInitDisplayMode(GLUT_SINGLE | GLUT_RGB);
  glutInitWindowSize(640, 480);
  glutInitWindowPosition(100, 150);

  glutCreateWindow("Star");

  glClearColor(1., 1., 1., 0.);
  glColor3ub(0, 0, 0); 		
  glLineWidth(4.);
  glPointSize(4.); 
  glMatrixMode(GL_PROJECTION); 
  glLoadIdentity(); 	
  gluOrtho2D(0., 640., 0., 480.);

  glutDisplayFunc(display); 
  glutMainLoop();
  return 0;
}