1. Ellipses
Using OpenGL

2. Ellipse An ellipse is basically an oval.
The technical definition for an ellipse is a closed plane curve generated by a point moving in a way that the sums of its distances from two fixed points (called the foci) is a constant.

3. Ellipse Graphics
-Vertexes (V1 & V2 on the diagram) of the ellipse are defined as the intersections of the ellipse and a line passing through foci (F1 & F2 on the diagram). The distance between the vertexes are called major axis or focal axis. A line passing the center and perpendicular to the major axis is the minor axis.

4. Standard equation of an ellipse with center at (0, 0)
(x^2 / a^2) + (y^2 / b^2) = 1
Where a > b > 0
This equation is known as the Cartesian Equation. The drawing looks as follows.
The ellipse is horizontal

5. Equation of an ellipse cont.
(x^2 / b^2) + (y^2 / a^2) = 1
Where a > b > 0
The drawing looks as follows
The ellipse is vertical

6. Equation of an ellipse with center not at the origin
Ellipses don’t have to always have it’s center at the origin.
The equation of an ellipse with center not at the origin is …
where the center of the ellipse is at point (X0, Y0)

7. Equations of an ellipse cont.
Now that we have considered the standard equation of an ellipse now we can look at the general equation.
With this equation however, comes a few properties to consider …

8. Properties of the general equation
A is not equal to C
A is not equal to 0
C is not equal to 0
If A=C, then the graph will turn into a circle
If A=0, then the graph will turn into a parabola that opens horizontally (left of right).
If C=0, then the graph will turn into a parabola that opens vertically (up or down).

9. Area of an ellipse
The area of an ellipse with major axis radius, a, and minor axis radius, b, is given by …
The Area of an arbitrary ellipse given by the Quadratic Equation ( ) is

10. Drawing an ellipse in OpenGL
To draw an ellipse in OpenGL you need to first draw a circle and then scale the circle to look like an ellipse. You need to do this because there isn’t a built in command to draw an ellipse.
The procedure you need to do this is gluDisk( ).

11. The gluDisk function draws a circle (disk)
void gluDisk( GLUquadricObj *qobj, GLdouble innerRadius,
GLdouble outerRadius, GLint slices, GLint loops );
qobj – is the quadratic object
innerRadius – the inner radius of the disk ( it may be 0 )
outerRadius – the outer radius of the disk
slices – the number of subdivisions around the z-axis
loops – the number of concentric rings about the origin into which the disk is subdivided.

12. gluNewQuadric
Along with the gluDisk( ) function, you can’t draw an ellipse without using the function gluNewQuadric( ).
There are no parameters for this function and it is used to create and return a pointer to a new quadric object.

13. Code Segments for an ellipse program
void Display ()
{ glClear(GL_COLOR_BUFFER_BIT);
GLUquadricObj * quadricObj; //Creates a quadratic object
quadricObj = gluNewQuadric(); //Sets a pointer to a new quadratic object
gluQuadricDrawStyle(quadricObj, GLU_FILL);
glColor3f (0.0, 1.0, 0.0); //Sets ellipse color to green
glScalef(0.7, 1.5, 1.0); //Scales the ellipse
//glColor3f (1.0, 0.0, 0.0); //Red ellipse
//glScalef(1.4, 0.7, 1.0);
//glColor3f (0.0, 0.0, 1.0); //Blue ellipse
//glScalef(0.4, 1.6, 1.0);
gluDisk(quadricObj, 0.0, 3.0, 100, 100); //Sets the ellipse properties
glFlush(); }

14. Display ( ) function
Inside the Display function is where the most important commands to draw the ellipse occur. The essential lines of code that you need to have are …
GLUquadricObj * quadricObj; //Creates a quadratic object
quadricObj = gluNewQuadric(); //Sets a pointer to a new quadratic object
gluQuadricDrawStyle(quadricObj, GLU_FILL); //Sets the draw style
gluDisk(quadricObj, 0.0, 3.0, 100, 100); //Sets ellipse properties

15. Code Segments for an ellipse program cont.
void init() {
glClearColor(0.0, 0.0, 0.0, 0.0); //Sets background color to black
glMatrixMode( GL_PROJECTION);
glLoadIdentity();
glOrtho(-8.0, 8.0, -8.0, 8.0, -8.0, 8.0); }

16. Code Segments for an ellipse program cont.
void main( int argc, char **argv) {
glutInit ( &argc, argv);
glutInitDisplayMode(GLUT_SINGLE | GLUT_RGB);
glutInitWindowSize( 300, 300);
glutInitWindowPosition( 200, 200);
glutCreateWindow("Ellipses");
init();
glutDisplayFunc( Display );
glutMainLoop(); }

17. Ellipses
The three functions (display, init, and main) are all you need in order to draw a simple ellipse. As long as you have the essential few lines of code to implement the figure.
Demonstrate the simple ellipse.