Parabola

A Parabola is a smooth planar curve that is equidistant from the curve directrix (a given line) and the curve focus (a given plane point).

• Context:
  • It can be represented by a Parabola Standard Equation (in Cartesian coordinates) as a Univariate Second-Order Polynomial Equation, [math]\displaystyle{ y=ax^2+bx+c }[/math]
  • It can be represented by a Parabola Vertex Equation (in Cartesian coordinates) as [math]\displaystyle{ y=a(x-h)^2+k }[/math] where the [math]\displaystyle{ (h,k) }[/math] are the coordinates of the parabola's vertex point.
  • It can be represented by a Parabola General Equation (in Cartesian coordinates) as [math]\displaystyle{ Ax^2+Bxy+Cy^2+Dx+Ey+F=0 }[/math] where [math]\displaystyle{ B^2=4AC }[/math]. In this case, the parabola is defined as a curve in the Cartesian plane defined by an irreducible equation of the general conic form.
  • Parabola standard in polar coordinates is [math]\displaystyle{ r(1+cos\theta)=l }[/math] where [math]\displaystyle{ l }[/math] is the distance from the focus to the parabola itself, measured along a line perpendicular to the axis of symmetry
  • It can also be defined as a conic section with an eccentricity of 1.
  • It can also be defined as an ellipse that has one focus at infinity.
  • It can also be defined as the inverse transform of a cardioid.
• Example(s):
  • [math]\displaystyle{ y=x^2+2x-8 }[/math] is a parabola that opens upwards with vertex point at (-1,-9)
  • [math]\displaystyle{ y=-3x^2+3 }[/math] is a parabola that opens downwards with vertex point at (0,3)
  • [math]\displaystyle{ x=y^2+2y-8 }[/math] is a parabola that opens to the right with vertex point at (-9,-1)
  • [math]\displaystyle{ x=-3y^2+3 }[/math] is a parabola that opens to the left with vertex point at (0,3)
  • 
• Counter-Example(s):
  • a Circle, such as [math]\displaystyle{ x^2+y^2=9 }[/math] of radius 3 centred at (0,0)
  • an Ellipse, such as [math]\displaystyle{ (x^2/5)+(y^2/4)=1 }[/math] with eccentricity 0.2 and focus at (0,1)
  • a Hyperbola, such as [math]\displaystyle{ x^2-y^2=1 }[/math] of length 2 and focal parameter 1.
  • a Cubic Function, such as [math]\displaystyle{ y=x^3+x^2+x+1= }[/math].
• See: Conic Section, Paraboloid, Reflection Symmetry, Curve, Plane (Geometry), Point (Geometry), Focus (Geometry), Line (Geometry), Directrix (Conic Section), Locus (Mathematics).

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References

• http://mathworld.wolfram.com/Parabola.html

2015

• (Wikipedia, 2015) ⇒ http://en.wikipedia.org/wiki/parabola Retrieved:2015-11-7.
  • A parabola (plural parabolas or parabolae, adjective parabolic, from ) is a two-dimensional, mirror-symmetrical curve, which is approximately U-shaped when oriented as shown in the diagram below, 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 lie on the directrix. The parabola is the locus of points in that plane that are equidistant from both the directrix and the focus. Another description of a parabola is as a conic section, created from the intersection of a right circular conical surface and a plane which is parallel to another plane which is tangential to the conical surface.A third description is algebraic. A parabola is a graph of a quadratic function, [math]\displaystyle{ y=x^2 }[/math] , for example. The line perpendicular to the directrix and passing through the focus (that is, the line that splits the parabola through the middle) is called the “axis of symmetry”. The point on the parabola that intersects the axis of symmetry is called the “vertex", and is the point where the parabola is most sharply curved. The distance between the vertex and the focus, measured along the axis of symmetry, is the "focal length". The "latus rectum" 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.

    Parabolas have the property that, if they are made of material that reflects light, then light which travels parallel to the axis of symmetry of a parabola and strikes its concave side 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 into a parallel ("collimated") 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.

    The parabola has many important applications, from a parabolic antenna or parabolic microphone to automobile headlight reflectors to the design of ballistic missiles. They are frequently used in physics, engineering, and many other areas.

    Strictly, the adjective parabolic should be applied only to things that are shaped as a parabola, which is a two-dimensional shape. However, as shown in the last paragraph, the same adjective is commonly used for three-dimensional objects, such as parabolic reflectors, which are really paraboloids. Sometimes, the noun parabola is also used to refer to these objects. Though not perfectly correct, this usage is generally understood.

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