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Applications and ProblemSolving
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

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.

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.

trilateration
The determination of the location of a point based on its distance from three other points.
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
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 .
Hyperbolas and Sundials
Hyperbolas as declination lines on a sundial.
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
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.
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).
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Key Term Reference
 circle
 Appears in these related concepts: Circles, Applications and ProblemSolving, and Circles
 distance
 Appears in these related concepts: Inequalities with Absolute Value, Symmetry, and The Distance Formula and Midpoints of Segments
 e
 Appears in these related concepts: Derivatives of Exponential Functions, Natural Logarithms, and e
 point
 Appears in these related concepts: The Intermediate Value Theorem, Relative Minima and Maxima, and Graphing Equations
 set
 Appears in these related concepts: Sequences, Introduction to Sequences, and Expressions and Sets of Numbers
 vertex
 Appears in these related concepts: Quadratic Functions of the Form f(x) = ax^2 + bx + c, Where a is not Equal to 0, Octahedral Complexes, and Quadratic Equations and Quadratic Functions
 zero
 Appears in these related concepts: The Quadratic Formula, The Discriminant, and Reducing Equations to a Quadratic
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Source: Boundless. “Applications and ProblemSolving.” Boundless Algebra. Boundless, 21 Jul. 2015. Retrieved 23 Jul. 2015 from https://www.boundless.com/algebra/textbooks/boundlessalgebratextbook/conicsections7/thehyperbola51/applicationsandproblemsolving22011100/