You can learn how to use these parameters by viewing the HTML source-code for this web-page.

Move point a left and right with the mouse and observe the motion of the point k.

green: y =  x2 4       f = (0,1)       red: y = -1

The point k is defined so that the distance between k and f is always equal to the shortest distance between k and the horizontal red line.

The green U-shaped curve is the set of all possible values for point k.  This curve is called a parabola.  The point f is called the focus of the parabola, and the horizontal red line is called the directrix of the parabola.

The curvature of a parabola depends on the distance between its focus and its directrix.  Move point c with the mouse to view all the parabolic curves.

green: y =  cy x2 cx2       f = (0,  cx2 4cy  )       red: y = -fy

All parabolas are similar to each other.  This means that any parabola can be transformed into any other parabola simply by increasing or decreasing its size, or by rotating it.  You can move c right and left to effectively "zoom-in" and "zoom-out" on the base of the parabola.

Move point a with the mouse in a circular pattern around the origin, and observe the motion of the point k.

green: y2 =  3 -  3x2 4       f = (1,0)       red: x = 4

The point k is defined so that the distance between k and f is exactly half of the distance between k and the vertical red line.

The green O-shaped curve is the set of all possible values for point k.  This curve is called an ellipse.  The point f is called a focus of the ellipse, and the vertical red line is called a directrix of the ellipse.

An ellipse actually has two focus points and two directrix lines.  The ellipse shown above has another focus at (-1,0), and another directrix at x = -4, which are mirror-symmetric to the ones displayed.

An ellipse does not have a "radius" like a circle does.  Observe that the points on the ellipse are located at different distances from the ellipse's center at the origin.  The maximum of these distances is called the semimajor axis, and the minimum of these distances is called the semiminor axis.  The ellipse shown above has a semimajor axis = 2, and a semiminor axis = V¯3.

For all ellipses, the distance between k and the focus is a constant fraction of the distance between k and the directrix.  This constant fraction is called the eccentricity of the ellipse.  (The ellipse shown above has an eccentricity of 0.5.)  All ellipses have an eccentricity between 0 and 1.

Move point d left and right between x=0 and x=1 to view the ellipse that has an eccentricity of d_x.

green: y2 = (1 - fx2)(1 - x2)       f = (dx,0)       red: x =  1 fx

Move point c to view all the ellipses that have their focus points at f and g.

green: y2 = (a2 - fx2)(1 -  x2 a2  )       a =  |c-f|+|c-g| 2

As you move c, explore the following properties:

1. The sum of the lengths of {c,f} and {c,g} is always equal to the length of the major axis.
    
2. The exterior line passing through c is called a tangent line, and it shows the direction of the ellipse's curvature at point c.  This tangent line creates two angles: one angle with {c,f}, and another angle with {c,g}.  These two opposing angles are always equal to each other.  This means that any ray of light that's emitted from one focus point, if it reflects off the ellipse, will always intersect the other focus point.

Move point a with the mouse in a circular pattern around the origin, and observe the motion of the point k.

green: y2 = 3x2 - 12       f = (4,0)       red: x = 1

The point k is defined so that the distance between k and f is exactly twice the distance between k and the vertical red line.

The two green curves are the set of all possible values for point k.  These curves (together) are called a hyperbola.  The point f is called a focus of the hyperbola, and the vertical red line is called a directrix of the hyperbola.

A hyperbola actually has two focus points and two directrix lines.  The hyperbola shown above has another focus at (-4,0), and another directrix at x = -1, which are mirror-symmetric to the ones displayed.

For all hyperbolas, the distance between k and the focus is a constant multiple of the distance between k and the directrix.  This constant multiple is called the eccentricity of the hyperbola.  (The hyperbola shown above has an eccentricity of 2.)  All hyperbolas have an eccentricity greater than 1.

Move point d left and right to view the hyperbola that has an eccentricity of d_x.

green: y2 = (1 - fx2)(1 - x2)       f = (dx,0)       red: x =  1 fx

The hyperbola with an eccentricity of V¯2 is called a rectangular hyperbola. The rectangular hyperbola is notable because it has several very simple mathematical equations.

The following graph shows some simple mathematical equations for the rectangular hyperbola.  Move point c around to explore the fact that these equations all yield similar rectangular hyperbolas.

green: x2 - y2 = 1 blue: y =  1 x red: (x, y) = (tan t, sec t) gray: (cosh t + i sinh t) c

Move c very close to the origin to see what a rectangular hyperbola looks like if you "zoom out" to view it from a long distance away.  The shape of the rectangular hyperbola approaches a 90° angle, which is the reason why it's called a rectangular hyperbola.

In general, all hyperbolas form an angle like this, which you could see more sharply if you were to "zoom out" far enough.  This angle is called the asymptotic angle, and it can be determined from the eccentricity (e), as = 2 arcsec e.

In the graph below, the gray curve is the parabola that has the simple equation y = x².

The green curve is a generalized parabola, and its size and position can be controlled with point a.

Move point a around with the mouse and observe how it controls the green parabola.

gray: y = x2       green: y = ax x2 + ay

Examine the green parabola's equation to understand exactly why the coordinates of a affect its curvature and position.

The green parabola can be used to generate all the ellipses and hyperbolas. This is accomplished by taking the square roots of each non-negative y value on the green parabola, and plotting them against x.

In the following graph, the green parabola is shown along with its square roots in blue (positive) and purple (negative).  Move point a to view all the ellipses and hyperbolas generated by these square roots.

green: y = ax x2 + ay       blue/purple: y2 = ax x2 + ay