Hyperbolas Notes

Horizontal Hyperbola

  • Consists of x and y variables on one side.

  • Has a constant “a” and a constant “b”.

  • All is set equal to “1”.

  • Most Basic Form

  • Can be moved if “h” and “k” are added

  • Equation: (x−h)2a2−(y−k)2b2=1\frac{(x-h)^2}{a^2} - \frac{(y-k)^2}{b^2} = 1

Vertical Hyperbola

  • Same properties as a horizontal one

  • The “x” and “y” are switched

  • Hyperbola opens up

  • Can also be done by setting it equal to a negative

  • Equations:

    • y2a2−x2b2=1\frac{y^2}{a^2} - \frac{x^2}{b^2} = 1

    • x2a2−y2b2=−1\frac{x^2}{a^2} - \frac{y^2}{b^2} = -1

Axis & Asymptotes

  • Hyperbolas have a transverse and conjugate axis.

  • The transverse axis connects the two vertices and determines the way the hyperbola opens (d=2a)(d=2a).

  • The conjugate axis splits through the two parabolas (d=2b)(d=2b).

  • The slope of the rectangle formed by the two is the asymptotes. Equation for Asymptotes: y=±ba(x−h)y = \pm \frac{b}{a}(x-h)

Focal Point

  • Each parabola in the hyperbola has a focal point.

  • The focal points are the points equidistant from the center of the hyperbola.

  • They can be found using the Pythagorean theorem: a2+b2=c2a^2 + b^2 = c^2

Visuals/Real World Examples

  • Cooling towers

    • Structure of cooling towers is shaped like a hyperboloid

    • Shape gives the tower a narrowing middle and flaring base which creates a shape that looks like an hourglass

  • Dulles airport

    • saddle shaped surface

    • 3D surface with curves that are hyperbolas in one direction and parabolas in the other

    • Curved roof is shaped like a hyperbolic paraboloid and looks like it saddles between columns

  • Dorton arena

    • Roof is saddle shaped structure formed from a hyperbolic paraboloid

    • Roof is suspended between two parabolic arches

    • Roof cable system curves in a way that creates a surface with hyperbolic curvature

Hyperbolic Navigation

  • A hyperbola is the set of points where the difference in distance to two fixed points is constant.

  • In Tracking, the two fixed points are sensors or receivers.

  • The object being tracked releases a signal (radio, sound, etc.).

  • The signal reaches the two receivers at slightly different times because the object is closer to one than the other.

  • This time difference corresponds to a difference in distance.

  • The set of all possible positions the source could be in forms a hyperbola

Relationship to Acoustics

  • Hyperbolas find the source of a sound using the Time Difference of Arrival technique.

  • When sound is released it travels at a known speed.

  • If two microphones pick up the sound at different times, that time difference can be converted into a distance difference.

  • The set of all points that have a constant difference in distance to the two microphones forms a parabola.

  • The sound source lies somewhere on a hyperbola defined by that distance difference.

  • Add a third microphone and you can form another hyperbola.

  • Intersection of the two hyperbolas gives exact location of the sound source, which is called hyperbolic multilateration.