Ellipses
Using OpenGL
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.
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.
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
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
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)
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 …
Properties of the general equation
• A is not equal to C
If A=C, then the graph will
turn into a circle
• A is not equal to 0
If A=0, then the graph will
turn into a parabola that
opens horizontally (left of
right).
• C is not equal to 0
If C=0, then the graph will
turn into a parabola that
opens vertically (up or
down).
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
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( ).
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.
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.
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);
glFlush();
}
//Sets the ellipse properties
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
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);
}
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();
}
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.