Hyperbola

Did you know that the orbit of a spacecraft can sometimes be a hyperbola?

A spacecraft can use the gravity of a planet to alter its path and propel it at high speed away from the planet and back out into space using a technique called "gravitational slingshot".

If this happens, then the path of the spacecraft is a hyperbola.

(See this happen in Gravity Freeplay)

Definition

A hyperbola is a curve where the distances of any point from:

  • a fixed point (the focus), and
  • a fixed straight line (the directrix) are always in the same ratio.

 

This ratio is called the eccentricity, and for a hyperbola it is always greater than 1.

The hyperbola is an open curve (has no ends).

But that isn't the full story! Because a hyperbola is actually two separate curves in mirror image like this:

On the diagram you can see:

  • a directrix and a focus (one on each side)
  • an axis of symmetry (that goes through each focus, at right angles to the directrix)
  • two vertices (where each curve makes its sharpest turn)

The "asymptotes" (shown on the diagram) are not part of the hyperbola, but show where the curve would go if continued indefinitely in each of the four directions.

And, strictly speaking, there is also another axis of symmetry that reflects the two separate curves of the hyperbola.

Conic Section

You can also get a hyperbola when you slice through a cone.

The slice must be steeper than that for a parabola, but does not
have to be parallel to the cone's axis for the hyperbola to be symmetrical.

So the hyperbola is a conic section (a section of a cone).

  conic section hyperbola

Equation

By placing a hyperbola on an x-y graph (centered over the x-axis and y-axis), the equation of the curve is:

x2/a2 − y2/b2 = 1

hyperbola on xy graph  

Also:

One vertex is at (a, 0), and the other is at (−a, 0)

The asymptotes are the straight lines:

  • y = (b/a)x
  • y = −(b/a)x

And the equation is also similar to the equation of the ellipse: x2/a2 + y2/b2 = 1, except for a "−" instead of a "+")

Eccentricity

We already mentioned the eccentricity (usually shown as the letter e), it shows how "uncurvy" (varying from being a circle) the hyperbola is.

On this diagram:

  • P is a point on the curve,
  • F is the focus and
  • N is the point on the directrix so that PN is perpendicular to the directrix.

The ratio PF/PN is the eccentricity of the hyperbola (for a hyperbola the eccentricity is always greater than 1).

It can also given by the formula:

e = hyperbola eccentricity

Using "a" and "b" from the diagram above.

Latus Rectum

The Latus Rectum is the line through the focus and parallel to the directrix.

The length of the Latus Rectum is 2b2/a.