## Applications and Problem Solving

As we should know by now, a hyperbola is an open curve with two branches, the intersection of a plane with both halves of a double cone. The plane may or may not be parallel to the axis of the cone

### Sundials

Hyperbolas may be seen in many sundials.
Every day, the sun revolves in a circle on the celestial sphere, and its rays striking the point on a sundial traces out a cone of light.
The intersection of this cone with the horizontal plane of the ground forms a conic section.
At most populated latitudes and at most times of the year, this conic section is a hyperbola.
This conic section can be shown in .
The shadow of the tip of a pole traces out a hyperbola on the ground over the course of a day (this path is called the *declination line*).
The shape of this hyperbola varies with the geographical latitude and with the time of the year, since those factors affect the cone of the sun's rays relative to the horizon .

### Trilateration

Trilateration is the a method of pinpointing an exact location, using its distances to a given points. The can also be characterized as the difference in arrival times of synchronized signals between the desired point and known points.These types of problems arise in navigation, mainly nautical. A ship can locate its position using the arrival times of signals from GPS transmitters. Alternatively, a homing beacon can be located by comparing the arrival times of its signals at two separate receiving stations. This can be used to track people, cell phones, internet signals and many other things.In particular, the set of possible positions of a point that has a distance variation of 2a from two known points is a hyperbola of vertex separation 2a, and whose foci are the two known points.

### The Kepler Orbit of Particles

The Kepler orbit is the path followed by any orbiting body .
This can be applied to a particle of any size, a planet or even hydrogen atoms.
depending on the particles properties, including size and shape (eccentricity), this orbit can be one of six conic sections. In particular, if the total energy *E* of the particle is greater than zero (i.e., if the particle is unbound), the path of such a particle is a hyperbola.
In the figure, the blue line shows the hyperbolic Kepler orbit.