Conic Sections 2.0

Figure 1 Cone sliced at an angle parallel with a side produces a parabola Directrix Focus Axis of symmetry (Not to scale) Figure 2 Right circular cone sliced at a greater angle to the base than that of a side produces a hyperbola

On our previous page we mentioned 2 of the 4 conic sections: circle and ellipse.

Now we will discuss the parabola and hyperbola. These may not be so familiar even though we see them every day.

Parabola:

The locus of all points in a plane such that every point on the line is an equal distance from a fixed point (focus) as it is from a fixed straight line (directrix).

For example, the familiar satellite dishes we use for television and Internet are parabolic curves. As are the dishes that are used by astronomers to collect light and other media (even radio and x-ray) from stars.

This is due to the fact that a ray of anything that is parallel to the axis of symmetry is reflected to a single point called the focus. We use this shape as the reflector of the light in flashlights and automobile headlights. Radar dishes are also parabolic.

When you kick a soccer ball, or throw a stone, or fire a missile, the path of each is a parabola.

General parametric equation for

where a = distance from origin to focus

Hyperbola

The set of all points where the absolute value of the difference in the distance from two fixed points (called the foci) is constant.

A parametric equation is:
x = a*sec(t) + h
y = b*tan(t) + k

where:
h,k are the center coordinates of the hyperbola
a is the semi-major axis length
b is the semi-minor axis length

The hyperbola is a two-branched curve. It is the path of some comets that travel too fast to produce an elliptical orbit around the sun, so never return to our solar system.

The trajectory of a space vehicle that uses the gravitational force of a planet to gain a gravity assist or 'sling shot' is a hyperbolic curve, which will be used to continue its path.