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Applications of Hyperbolas
A hyperbola is an open curve with two branches and a cut through both halves of a double cone, which is not necessarily parallel to the cone's axis.
Learning Objective

Apply the hyperbola to real world problems
Key Points
 Hyperbolas have applications to a number of different systems and problems including sundials and trilateration.
 Hyperbolas may be seen in many sundials. On any given 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.
 A hyperbola is the basis for solving trilateration problems, the task of locating a point from the differences in its distances to given points — or, equivalently, the difference in arrival times of synchronized signals between the point and the given points.
Terms

trilateration
The determination of the location of a point based on its distance from three other points.

hyperbola
A conic section formed by the intersection of a cone with a plane that intersects the base of the cone and is not tangent to the cone.

conic section
Any of the four distinct shapes that are the intersections of a cone with a plane, namely the circle, ellipse, parabola and hyperbola.
Example
 A hyperbola is the basis for solving trilateration problems, the task of locating a point from the differences in its distances to given points — or, equivalently, the difference in arrival times of synchronized signals between the point and the given points. Such problems are important in navigation, particularly on water; a ship can locate its position from the difference in arrival times of signals from GPS transmitters.
Full Text
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.
Hyperbola
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
Here are some examples of hyperbolas in the real world.
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. The angle between the ground plane and the sunlight cone depends on where you are on the Earth, and the axial tilt of Earth, which changes seasonally. At most populated latitudes and at most times of the year, this conic section is a hyperbola.
Sundials work by casting the shadow of a vertical marker, sometimes called a gnomon, over a clock face on the horizontal surface. The angle between the sunlight and the ground will be the same as the angle formed by the line connecting the tip of the gnomon with the end of its shadow.
If we mark where the end of the shadow falls over the course of the day, the line traced out by the shadow forms a hyperbola on the ground (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 angle of the cone of the sun's rays relative to the horizon.
The parameters of the traced hyperbola, such as its asymptotes and its eccentricity, are related to the specific physical conditions that produced it, namely the angle between the sunlight and the ground, and the latitude at which the sundial exists.
Hyperbolas and Sundials
Hyperbolas as declination lines on a sundial.
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 the case where a ship, or other object to be located, only knows the difference in distances between itself and two known points, the curve of possible locations is a hyperbola. One way of defining a hyperbola is as precisely this: the curve of points such that the absolute value of the difference between the distances to two focal points remains constant.
So if we call this difference in distances 2a, the hyperbola will have vertices separated by the same distance 2a, and the foci of the hyperbola will be 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, as long as gravity is the only force causing the orbital trajectory. Depending on the orbital properties, including size and shape (eccentricity), this orbit can be any of the four conic sections. In particular, if the eccentricity e of the orbit is greater than 1, the path of such a particle is a hyperbola. In the figure, the blue line shows the hyperbolic Kepler orbit. In the common case of a gravitational orbit, the massive object is one of the foci of the hyperbola (or other conic section).
Kepler Orbits
A diagram of the various forms of the Kepler Orbit and their eccentricities. Blue is a hyperbolic trajectory (e > 1). Green is a parabolic trajectory (e = 1). Red is an elliptical orbit (e < 1). Grey is a circular orbit (e = 0).
Physically, another way to understand hyperbolic orbits is in terms of the energy of the orbiting particle. Orbits which are circular or elliptical are bound orbits, which is to say the object never escapes its closed path around one of the focal points. This is associated with the particle's total energy E being less than the minimum energy required to escape, and so E is said to be negative in these cases.
A parabolic trajectory does have the particle escaping the system. However, this is the very special case when the total energy E is exactly the minimum escape energy, so E in this case is considered to be zero.
If there is any additional energy on top of the minimum (zero) value, the trajectory will become hyperbolic, and so E is positive in the hyperbolic orbit case.
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Key Term Reference
 absolute value
 Appears in these related concepts: Piecewise Functions, Equations with Absolute Value, and Absolute Value
 asymptote
 Appears in these related concepts: What Are Conic Sections?, Graphs of Exponential Functions, Base e, and Asymptotes
 circle
 Appears in these related concepts: Introduction to Circles, Types of Conic Sections, and Applications of Circles and Ellipses
 constant
 Appears in these related concepts: Graphing Quadratic Equations in Vertex Form, Inverse Variation, and Direct Variation
 difference
 Appears in these related concepts: Factoring a Difference of Squares, The Order of Operations, and Basic Operations
 distance
 Appears in these related concepts: Inequalities with Absolute Value, The Distance Formula and Midpoints of Segments, and Linear Mathematical Models
 e
 Appears in these related concepts: Natural Logarithms, Business Stakeholders: Internal and External, and The Number e
 factor
 Appears in these related concepts: Rational Algebraic Expressions, Factors, and Finding Factors of Polynomials
 minimum
 Appears in these related concepts: Parallel and Perpendicular Lines, Relative Minima and Maxima, and Combined Variation
 parameters
 Appears in these related concepts: Limited Growth, Applications of the Parabola, and Eccentricity
 point
 Appears in these related concepts: The Intermediate Value Theorem, Graphing Equations, and Polynomial and Rational Functions as Models
 term
 Appears in these related concepts: Basics of Graphing Polynomial Functions, The 22nd Amendment, and Introduction to Variables
 vertex
 Appears in these related concepts: Parabolas As Conic Sections, Completing the Square, and What is a Quadratic Function?
 zero
 Appears in these related concepts: Rational Inequalities, Other Equations in Quadratic Form, and Historical Traditions of Numerical Systems
 zeros
 Appears in these related concepts: Parts of a Parabola, A Graphical Interpretation of Quadratic Solutions, and Introduction to Complex Numbers
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Cite This Source
Source: Boundless. “Applications of Hyperbolas.” Boundless Algebra. Boundless, 08 Aug. 2016. Retrieved 25 Sep. 2016 from https://www.boundless.com/algebra/textbooks/boundlessalgebratextbook/conicsections341/thehyperbola51/applicationsofhyperbolas22011100/