original in en Miguel A Sepulveda

This is the first article in a series on OpenGL, an industry standard for 2D/3D graphics (see also What is OpenGL). We will assume that the reader is familiar with his/hers C development platform, and has some knowledge of the GLUT library (otherwise just follow the "GLUT programming" series of articles in this magazine). Under Linux we recommend the usage of the Mesa-library which is a wonderful freeware implementation of OpenGL. Now there is even hardware support for Mesa (see 3Dfx graphics card).

Every presentation of new OpenGL commands will be accompanied by an example that tries to exploit its functionality, at least we will try !. By the end of our series we will present you with the source code of a game simulation completely written in OpenGL.

Before getting started I would like to mention that since I am a scientist, most of my experience with OpenGL is as a tool to write simulations of real quantum and classical systems. So my examples are a bit bias ;-). I hope readers find these examples accessible or at least amusing. If you would like to see other kinds of examples just let me know.

OpenGL is often associated with 3D graphics, fancy special effects, complex models with realistic light modeling etc.. However it is also a 2D graphics rendering machine. This is important because there are many things you can learn to do in 2D before starting to learn about the complexities of 3D perspectives, model rendering, lights, camera position, etc.. A large number of engineering and science applications can be render in 2D. So let us first learn how to do some simple 2D animations.

OpenGL has only a few geometric primitives: points, lines,
polygons. All of them are described in terms of their
respective vertices. A vertex is characterized by 2 or 3
floating points, the *Cartesian coordinates* of the
vertex, (x,y) in 2D and (x, y, z) in 3D. While Cartesian
coordinates are the most common, in computer graphics there is
also the *homogeneous coordinate system* in which every
point is described by 4 floating points (x, y, z, w). We will
come back to them after covering some elementary notions of 3D
rendering.

Since in OpenGL all geometric objects are eventually described as an ordered set of vertices, there is a family of routines to declare a vertex. Its syntax is:

`void glVertex{234}{sifd}[v](TYPE
coords);`

Get familiar with this notation. The curly brackets indicate
part of the name of the routine. The routines can take 2, 3 or
4 parameters in either short, long, float or double type.
Optionally these parameters can be supplied in a vector form,
in which case we would use the `v`-type routines. Here
are some examples:

`void glVertex2s(1, 3);`

`void glVertex2i(23L, 43L);`

`void glVertex3f(1.0F, 1.0F, 5.87220972F);`

`float vector[3];`

`void glVertex3fv(vector);`

To simplify all these routines are refered to as
** glVertex***.

OpenGL interprets any sequence of vertices according to its
context. The context is declared by the pair of routines
** glBegin(GLenum mode)** and

draws 5 points in 2D with the coordinates specified.
GL_POINTS is one of the labels defined in the OpenGL header
file `<GL/gl.h>.` There are many other modes
available but we will review then as necessary.

Every point is drawn with the color currently stored in the
OpenGL state variable associated with the color buffer. To
change the current color, use the family of routines
** glColor***; there is a lot to say about selecting
and manipulating colors (there will be another article only on
this subject). For the moment, we will be using three floating
point numbers from 0.0 to 1.0 - this is the RGB
(Red-Green-Blue) encoding;

This is already enough material to write our first two
examples of code. The first example is a simple OpenGL program
that draws a number of orbits in a chaotic map (The standard
map). If the reader is not familiar with mappings and the
*standard map* in particular, it does not matter. To put
it simply, a map takes a point and generates a new one using a
well defined formula:

`y _{n+1} = y_{n} + K
sin(x_{n})`

In the case of the standard map it represents a model for the trace left by a charged particle that circles around the tori of an accelerator of particles and crosses a section plane of the accelerator. Studying the properties of this and other maps is important in physics because it helps us understand the stability of the charged particle confined in the cyclotron. The standard map is very cool because for some values of its parameter, K clearly shows a mixture of chaotic and trapped motion. Finally even those who are not really interested in physics but still want to develop some nice graphics code should pay some attention to maps and their properties, many of the algorithms for generating textures, fire flares, trees, terrain, etc.. are based on fractal maps.

Here is the code example1.c:

Please read the Programming GLUT article to understand the glut* routines, most of this code came from there. The graphics window is opened in single buffer and RGB mode. Then a callback function named display() draws the map: first we select the color black for the background; glClear(GL_COLOR_BUFFER_BIT) resets the color buffer to the current color (black), next after selecting white color with glColor, we run the NonlinearMap() a number of times and plot the points with#include <GL/glut.h> #include <math.h> const double pi2 = 6.28318530718; voidNonlinearMap(double *x, double *y){ static double K = 1.04295; *y += K * sin(*x); *x += *y; *x = fmod(*x, pi2); if (*x < 0.0) *x += pi2; }; void winInit(){ /* Set system of coordinates */ gluOrtho2D(0.0, pi2, 0.0, pi2); }; void display(void){ const int NumberSteps = 1000; const int NumberOrbits = 100; const double Delta_x = pi2/(NumberOrbits-1); int step, orbit; glColor3f(0.0, 0.0, 0.0); glClear(GL_COLOR_BUFFER_BIT); glColor3f(1.0, 1.0, 1.0); for (orbit = 0; orbit < NumberOrbits; orbit++){ double x, y; y = 3.1415; x = Delta_x * orbit; glBegin(GL_POINTS); for (step = 0; step < NumberSteps; step++){ NonlinearMap(&x, &y); glVertex2f(x, y); }; glEnd(); }; for (orbit = 0; orbit < NumberOrbits; orbit++){ double x, y; x = 3.1415; y = Delta_x * orbit; glBegin(GL_POINTS); for (step = 0; step < NumberSteps; step++){ NonlinearMap(&x, &y); glVertex2f(x, y); }; glEnd(); }; }; int main(int argc, char **argv) { glutInit(&argc, argv); glutInitDisplayMode(GLUT_SINGLE | GLUT_RGBA); glutInitWindowPosition(5,5); glutInitWindowSize(300,300); glutCreateWindow("Standard Map"); winInit(); glutDisplayFunc(display); glutMainLoop(); return 0; }

Notice that in the window initialization routine winInit() there is a single statement
from the OpenGL Utility toolkit, **gluOrtho2D()**. This
routine sets a 2D orthogonal system of coordinates. The
parameters passed are "minimum x, maximum x, minimum y, maximum
y".

I have chosen a single mode window and a large number of points
so that you have a chance to see the image as it is being
drawn. This is common of single mode with large and time
consuming images, things appear on your screen as they are
invoked with OpenGL routines.

After running ../../common/January1998/example1 you should see this image:

Let's go now to the second program, example2.c:

This program is based on ../../common/January1998/../../common/January1998/example1.c, the main difference is that the window is opened in a double buffer mode, and the map parameter K is a variable that changes during the life of the program. There is a new callback function idle() registered to the GLUT event processor by#include <GL/glut.h> #include <math.h> const double pi2 = 6.28318530718; const double K_max = 3.5; const double K_min = 0.1; static double Delta_K = 0.01; static double K = 0.1; void NonlinearMap(double *x, double *y){ /* Standard Map */ *y += K * sin(*x); *x += *y; /* Angle x is module 2Pi */ *x = fmod(*x, pi2); if (*x < 0.0) *x += pi2; }; /* Callback function: What to do in absence of use input */ voididle(void){ /* Increase the stochastic parameter */ K += Delta_K; if(K > K_max) K = K_min; /* Redraw the display */ glutPostRedisplay(); }; /* Initialization for the graphics window */ void winInit(void){ gluOrtho2D(0.0, pi2, 0.0, pi2); }; /* Callback function: What to do when the display needs redrawing */ void display(void){ const int NumberSteps = 1000; const int NumberOrbits = 50; const double Delta_x = pi2/(NumberOrbits-1); int step, orbit; glColor3f(0.0, 0.0, 0.0); glClear(GL_COLOR_BUFFER_BIT); glColor3f(1.0, 1.0, 1.0); for (orbit = 0; orbit < NumberOrbits; orbit++){ double x, y; y = 3.1415; x = Delta_x * orbit; glBegin(GL_POINTS); for (step = 0; step < NumberSteps; step++){ NonlinearMap(&x, &y); glVertex2f(x, y); }; glEnd(); }; for (orbit = 0; orbit < NumberOrbits; orbit++){ double x, y; x = 3.1415; y = Delta_x * orbit; glBegin(GL_POINTS); for (step = 0; step < NumberSteps; step++){ NonlinearMap(&x, &y); glVertex2f(x, y); }; glEnd(); }; glutSwapBuffers(); }; int main(int argc, char **argv) { /* GLUT Initializations */ glutInit(&argc, argv); glutInitDisplayMode(GLUT_DOUBLE | GLUT_RGBA); glutInitWindowPosition(5,5); glutInitWindowSize(300,300); /* Open Window */ glutCreateWindow("Order to Chaos"); /* Window initializations */ winInit(); /* Register callback functions */ glutDisplayFunc(display); glutIdleFunc(idle); /* Launch event processing */ glutMainLoop(); return 0; }

Another difference worth noticing is the use of
**glutSwapBuffers()** at the end of **display()**. The
window was initialized in double buffer mode, therefore all the
rendering directives are applied to the hidden buffer; the user
cannot see the image being draw in this case. After the whole
image (frame) has been finished then it is made visible by
switching hidden and visible buffers with
**glutSwapBuffers()**. Without this technique the animation
will not run smoothly.

Here are some of the frames displayed during the animation:

IMPORTANT: The display() callback function always gets invoked at least once, before idle(). Keep this in mind when writing your animations and deciding what goes to display() and what to idle().

example3.c

As previously mentioned **glBegin(GLenum mode)**
accepts various modes and the sequence of vertices
v

- GL_POINTS Draws a points at each of the n vertices.
- GL_LINES Draws a series of unconnected
lines. The segments are drawn between v
_{0}and v_{1}, v_{2}and v_{3},...etc. If n is odd v_{n-1}is ignored. - GL_POLYGON Draws a polygon using
v
_{0}, v_{1},..,v_{n-1}as vertices. n must be at least 3 or nothing is drawn, also the polygon can not intersect itself and must be convex (due to the hardware's algorithm limitations). - GL_TRIANGLES Draws a series of triangles
using vertices v
_{0}, v_{1}and v_{2}, then v_{3}, v_{4}and v_{5}etc. If n is not a multiple of 3 the remaining points are ignored. - GL_LINE_STRIP Draws a line from
v
_{0}to v_{1}, them from v_{1}to v_{2}and so on. Finally from v_{n-2}to v_{n-1}for a total of n-1 line segments. There are no restrictions on the vertices describing a line strip, lines can intersect arbitrarily. - GL_LINE_LOOP Same as GL_LINE_STRIP except
that a final line segment is drawn from v
_{n-1}to v_{0}, closing the loop. - GL_QUADS Draws a series of quadrilaterals
using vertices v
_{0}, v_{1}, v_{2}, v_{3}and v_{4}, v_{5}, v_{6}, v_{7}and so on. - GL_QUAD_STRIP Draws a series of
quadrilaterals using vertices v
_{0}, v_{1}, v_{3}, v_{2}then v_{2}, v_{3}, v_{5}, v_{4}and so on. - GL_TRIANGLE_STRIP Draws a series of
triangles using vertices in the following order
v
_{0}, v_{1}, v_{2}, then v_{2}, v_{1}, v_{3}, then v_{2}, v_{3}, v_{4}, etc. The ordering is to ensure that the triangles has the correct orientation and the strip can be used to form part of a surface. - GL_TRIANGLE_FAN Similar to GL_TRIANGLE_STRIP
except that the triangles are v
_{0}, v_{1}, v_{2}, then v_{0}, v_{2}, v_{3}, then v_{0}, v_{3}, v_{4}, and so on. All the triangles have v_{0}as a common vertix.

As before there is an **idle()** callback function whose
aim here is to keep the clock running (updating the variable
*time*). The **display()** draws two objects; the
pendulum cord and weight (in white and red respectively). The
motion of the pendulum coordinates is implicit in the formulas
for xcenter and ycenter:

void display(void){ static double radius = 0.05; const double delta_theta = pi2/20; double xcenter , ycenter; double x, y; double theta = 0.0; double current_angle = cos(omega * time); glColor3f(0.0, 0.0, 0.0); glClear(GL_COLOR_BUFFER_BIT); glColor3f(1.0, 1.0, 1.0); /* Draw pendulum cord */ glColor3f(1.0, 1.0, 1.0); glBegin(GL_LINES); glVertex2f(0.0, 0.0);xcenter = -cord_length * sin(current_angle); ycenter = -cord_length * cos(current_angle); glVertex2f(xcenter, ycenter); glEnd(); /* Draw pendulum dish */ glColor3f(1.0, 0.0, 0.0); glBegin(GL_POLYGON); while (theta <= pi2) { x = xcenter + radius * sin(theta); y = ycenter + radius * cos(theta); glVertex2f(x, y); theta += delta_theta; }; glEnd(); glutSwapBuffers(); };

Here are some suggestions for practicing what you have learned so far:

- In example1.c try other maps. Go to the library and pick any book on Chaos and Fractals, surely you will find many examples. Experiment changing parameters, system of coordinates, and applying several maps consecutively before drawing the points. Have fun with it.
- In example2.c
you could add colors to each of the points. For example, a
very interesting color coding would be based on assigning to
each dot a color according to the local stability of the
orbit
`(Physics Review Letters Vol 63, (1989) 1226)`, when the trajectory goes through a chaotic region it becomes more red. For instance, while near stable islands it could become more blue. If you code this effect, the fractal nature of our example map will become obvious. It is a bit advanced for those of you without course work on differential equations, but it is worthwhile learning about it if you want to take advantage of mappings and fractals in your computer graphics. - In example3.c , try changing the type of lines used for drawing the disc. Use GL_LINES, GL_TRIANGLES, etc.. See what happens. Try optimizing the disc generation, it is not necessary to evaluate sines and cosines so many times for drawing the same disc in each frame, you could save it into an array. Using polygon drawing try attaching boxes, diamonds, or whatever to the end of the pendulum. Write two pendulums per frame, moving independently or even colliding with each other.

This is all for now. There are still many things to discuss
about polygons. In the next issue (March 1998) we will continue
to explore polygons, modeling and cover in more detail some of
the commands your are already familiar with.

- Read Hardware Review: 3Dfx Graphics Card.
- Consult http://www.opengl.org/.
- Read other articles by the same author:What is OpenGL?, GLUT Programming: Windows and Animations.