A parabola

A parabola (from the Greek: παραβολή) is a conic section generated by the intersection of a cone, and a plane tangent to the cone or parallel to some plane tangent to the cone. If the plane is itself tangent to the cone, one would obtain a degenerate parabola, a line. A parabola can also be defined as locus of points which are equidistant from a given point (the focus) and a given line (the directrix).

Definitions and overview

A graph showing the reflective property and the equidistant focus (blue) and directrix (green)

In Cartesian coordinates, a parabola with an axis parallel to the y axis with vertex (h, k), focus (h, k + p), and directrix y = k - p has the equation

A parabola may also be characterized as a conic section with an eccentricity of 1. As a consequence of this, all parabolas are similar. A parabola can also be obtained as the limit of a sequence of ellipses where one focus is kept fixed as the other is allowed to move arbitrarily far away in one direction.

A parabola has a single axis of reflective symmetry, which passes through its focus and is perpendicular to its directrix. The point of intersection of this axis and the parabola is called the vertex. A parabola spun about this axis in three dimensions traces out a shape known as a paraboloid of revolution. See also parabolic reflector.

A particle or body in motion under the influence of a uniform gravitational field (for instance, a baseball flying through the air, neglecting air friction) follows a parabolic trajectory.

In real-world cases, the trajectory of a body is more likely to be elliptical (if it lacks escape velocity) or hyperbolic (if it has sufficient energy to escape). For the trajectories of small bodies close to a massive body, the gravitational field is considered close enough to uniform to approximate a parabolic trajectory.

The parabola is an inverse transform of a cardioid.

Equations

Cartesian

Vertical axis of symmetry:

Horizontal axis of symmetry:

Quadratic (vertical axis of symmetry):

Quadratic (horizontal axis of symmetry):

x = ay² + by + c

a, b, and c are the same as above.

Parametric

Semi-latus rectum and polar coordinates

In polar coordinates, a parabola with the focus at the origin and the top on the negative x-axis, is given by the equation

where l is the semi-latus rectum: the distance from the focus to the parabola itself, measured along a line perpendicular to the axis. Note that this is twice the distance to the top.

Gauss-mapped form

A Gauss-mapped form: (tan²φ,2tanφ) has normal (cosφ,sinφ).

See also

• Paraboloid
• Ellipse
• Hyperbola
• conic section

Derivation of the focus

Given a parabola parallel to the y-axis with vertex (0,0) and with equation

then there is a point (0,f) — the focus — such that any point P on the parabola will be equidistant from both the focus and a line perpendicular to the axis of symmetry of the parabola (the linea directrix), in this case parallel to the x axis. Since the vertex is one of the possible points P, it follows that the linea directrix passes through the point (0,-f). So for any point P=(x,y), it will be equidistant from (0,f) and (x,-f). It is desired to find the value of f which has this property.

Let F denote the focus, and let Q denote the point at (x,-f). Line FP has the same length as line QP.

Square both sides,

Cancel out terms from both sides,

Cancel out the x² from both sides (x is generally not zero),

Now let p=f and the equation for the parabola becomes

Q.E.D.

Reflective property of the tangent

The tangent of the parabola described by equation (1) has slope

This line intersects the y-axis at the point (0,-y) = (0, - a x²), and the x-axis at the point (x/2,0). Let this point be called G. Point G is also the midpoint of points F and Q:

Since G is the midpoint of line FQ, this means that

and it is already known that P is equidistant from both F and Q:

and, thirdly, line GP is equal to itself, therefore:

It follows that .

Line QP can be extended beyond P to some point T, and line GP can be extended beyond P to some point R. Then and are vertical, so they are equal (congruent). But is equal to . Therefore is equal to .

The line RG is tangent to the parabola at P, so any light beam bouncing off point P will behave as if line RG were a mirror and it were bouncing off that mirror.

Let a light beam travel down the vertical line TP and bounce off from P. The beam's angle of inclination from the mirror is , so when it bounces off, its angle of inclination must be equal to . But has been shown to be equal to . Therefore the beam bounces off along the line FP: directly towards the focus.

Conclusion: Any light beam moving vertically downwards in the concavity of the parabola (parallel to the axis of symmetry) will bounce off the parabola moving directly towards the focus. (See parabolic reflector.)

Constructing a parabola

A parabola can be constructed geometrically as follows: draw focus F, vertex, linea directrix q, and linea verticis r (through the vertex, parallel to linea directrix). Choose a point Q₁ on linea directrix. Draw line FQ₁ which intersects linea verticis at R₁. A line (through R₁ and perpendicular to FQ₁ ) will intersect another line (through Q₁ and perpendicular to linea directrix) at point P₁. Point P₁ is on the parabola, and line R₁P₁ is tangential to the parabola. Choose another point Q₂ on linea directrix and repeat the steps of the paradigm above to obtain P₂. Continue with points Q₃,P₃,Q₄,P₄,Q₅,P₅, et cetera. If the points Q₁,Q₂,Q₃,... were drawn in a sequence, then the points P₁,P₂,P₃... can be connected sequentially to draw the parabola.

By paper folding

Draw a straight line on a piece of paper, and a point somewhere not on the line. Then fold the paper over so that the point touches the line and crease the fold. Do this several times. The envelope formed by the creases will make a nice parabola.

You can make an ellipse or hyperbola similarly by using a circle and a point.

External links

• MathWorld: Parabola (http://mathworld.wolfram.com/Parabola.html)
• Archimedes Triangle and Squaring of Parabola (http://www.cut-the-knot.org/Curriculum/Geometry/ArchimedesTriangle.shtml)
• Two Tangents to Parabola (http://www.cut-the-knot.org/Curriculum/Geometry/ParabolaLambert.shtml)
• Parabola As Envelope of Straight Lines (http://www.cut-the-knot.org/Curriculum/Geometry/ParabolaEnvelope.shtml)
• Parabolic Mirror (http://www.cut-the-knot.org/Curriculum/Geometry/ParabolaMirror.shtml)
• Three Parabola Tangents (http://www.cut-the-knot.org/Curriculum/Geometry/ThreeParabolaTangents.shtml)
• Two Tangents to Parabola (http://www.cut-the-knot.org/Curriculum/Geometry/ParabolaLambert.shtml)
• Focal Properties of Parabola (http://www.cut-the-knot.org/Curriculum/Geometry/ParabolaFocal.shtml)
• Parabola As Envelope II (http://www.cut-the-knot.org/Curriculum/Geometry/ParabolaMesh.shtml)