# Mathematical Symphony of Curves and Reflections

Categories: Focus

A parabola is a two-dimensional, mirror-symmetrical curve, which is around U-shaped when oriented as shown in the diagram, but which can be in any orientation in its plane. It fits any of several superficially different mathematical descriptions which can all be proved to define curves of exactly the same shape.

One description of a parabola involves a point (the focus) and a line (the directrix). The focus does not push the directrix. The locus of points in that aircraft that are equidistant from both the directrix and the focus is the parabola.

Another description of a parabola is as a conic area, developed from the crossway of a right circular cone-shaped surface area and a planewhich is parallel to another plane which is digressive to the conical surface. [a] A 3rd description is algebraic. A parabola is a graph of a quadratic function, such as

The line perpendicular to the directrix and travelling through the focus (that is, the line that splits the parabola through the middle) is called the "axis of proportion".

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The point on the axis of proportion that converges the parabola is called the "vertex", and it is the point where the curvature is greatest. The range in between the vertex and the focus, measured along the axis of balance, is the "focal length". The "latus anus" is the chord of the parabola which is parallel to the directrix and passes through the focus. Parabolas can open up, down, left, right, or in some other arbitrary direction. Any parabola can be repositioned and rescaled to fit exactly on any other parabola — that is, all parabolas are geometrically similar.

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Parabolas have the property that, if they are made of material that reflects light, then light which enters a parabola travelling parallel to its axis of symmetry is reflected to its focus, regardless of where on the parabola the reflection occurs. Conversely, light that originates from a point source at the focus is reflected ("collimated") into a parallel beam, leaving the parabola parallel to the axis of symmetry. The same effects occur with sound and other forms of energy. This reflective property is the basis of many practical uses of parabolas.

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Updated: Dec 12, 2023