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Parabola - Interesting Properties and Usages

1. From Cone to Parabola

Conic section is a curve obtained as the intersection of the surface of a cone with a plane. If the cutting plane is parallel to exactly one generating line of the cone, then the conic is unbounded and is called a parabola.

Look at the following figure.

1. Plane cross with the cone parallel to its generating line.

2. Pick a point on the intersecting line between plane and the cone (which is painted in brown)

3. Then there exist unique sphere such that it contacts with the plane and the cone.

Denote the point of contact between and as .

1. Now, define the intersection between and as . Since and only intersects (i.e. in contact) in , we get

1. Line where is the plane perpendicular to the axis of rotation of the cone which has point in it. Define to be the foot of perpendicular from to . Then we get a nice property

so that .

This means that the intersecting curve between the cone and plane is a parabola having focal point and directrix .

2. Examples and Usages

Example 1. In physics

Consider the following cylindrical container with liquid to be rotating at uniform angualr velocity .

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The width of the container (or the diameter) is . Consider an infinitesimal liquid element at height above the minimum of the parabola. The forces acting on it are

1. Gravitational force

2. Centripetal force .

Now the angle between the two forces will have the tangent value

This means that

Integrating both sides,

which is the parabola. By rotation, the surface of spinning water would be the paraboloid,

Example 2. Paraboloid mirror

Suppose a light is reflected by the parabolic mirror at point , where the initial path of light was . The Law of Reflection states that the angle between the incidence ray and mirror is equal to the angle between the reflected ray and mirror.

Angle between and :

Angle between and :

Using dot product between directional vectors,

(using ) .

Now we get

since all angles are accute. This means that every ray coming parallel with the axis of rotation will be concentrated to the focal point of parabola .

1. We can apply this fact in making parabolic antennas, an antenna that uses a parabolic reflector, a curved surface with the cross-sectional shape of a parabola, to direct the radio waves.

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1. Also in solar power generation, the solar dish in parabolic shape concentrates the solar energy to its focal point, which greatly increases the heat generation.

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3. Citation

[1] https://en.wikipedia.org/wiki/Conic_section (image is used)

[2] https://physics.stackexchange.com/questions/293106/shape-of-water-in-rotating-bucket (image is used)

[3] https://en.wikipedia.org/wiki/Parabolic_antenna (only image is used)

[4] https://www.researchgate.net/figure/Solar-dish-collector-with-conical-receiver-photographed-by-the-authors_fig2_269575959

All the other graphic images are made by myself using GeoGebra 3D plotter and Graphic Calculator.