Editing Parabola (section)

== Inscribed angles and the 3-point form ==
[[File:Parabel-pws-s.svg|thumb|Inscribed angles of a parabola]]
A parabola with equation <math>y = ax^2 + bx + c,\ a \ne 0</math> is uniquely determined by three points <math>(x_1, y_1), (x_2, y_2), (x_3, y_3)</math> with different ''x'' coordinates. The usual procedure to determine the coefficients <math>a, b, c</math> is to insert the point coordinates into the equation. The result is a linear system of three equations, which can be solved by [[Gaussian elimination]] or [[Cramer's rule]], for example. An alternative way uses the ''inscribed angle theorem'' for parabolas.

In the following, the angle of two lines will be measured by the difference of the slopes of the line with respect to the directrix of the parabola. That is, for a parabola of equation <math>y = ax^2 + bx + c,</math> the angle between two lines of equations <math>y = m_1 x + d_1,\ y = m_2x + d_2</math> is measured by <math>m_1 - m_2.</math>

Analogous to the [[inscribed angle theorem]] for circles, one has the ''inscribed angle theorem for parabolas'':<ref>E. Hartmann, [http://www.mathematik.tu-darmstadt.de/~ehartmann/circlegeom.pdf Lecture Note ''Planar Circle Geometries'', an Introduction to Möbius-, Laguerre- and Minkowski Planes], p. 72.</ref><ref>W. Benz, ''Vorlesungen über Geomerie der Algebren'', [[Springer Science+Business Media|Springer]] (1973).</ref>
{{block indent | em = 1.5 | text = Four points <math>P_i = (x_i, y_i),\ i = 1, \ldots, 4,</math> with different {{mvar|x}} coordinates (see picture) are on a parabola with equation <math>y = ax^2 + bx + c</math> if and only if the angles at <math>P_3</math> and <math>P_4</math> have the same measure, as defined above. That is,
<math display="block">\frac{y_4 - y_1}{x_4 - x_1} - \frac{y_4 - y_2}{x_4 - x_2} = \frac{y_3 - y_1}{x_3 - x_1} - \frac{y_3 - y_2}{x_3 - x_2}.</math>}}

(Proof: straightforward calculation: If the points are on a parabola, one may translate the coordinates for having the equation <math>y = ax^2</math>, then one has <math>\frac{y_i - y_j}{x_i - x_j} = x_i + x_j</math> if the points are on the parabola.)

A consequence is that the equation (in <math>{\color{green}x}, {\color{red}y}</math>) of the parabola determined by 3 points <math>P_i = (x_i, y_i),\ i = 1, 2, 3,</math> with different {{mvar|x}} coordinates is (if two {{mvar|x}} coordinates are equal, there is no parabola with directrix parallel to the {{mvar|x}} axis, which passes through the points)
<math display="block">\frac{{\color{red}y} - y_1}{{\color{green}x} - x_1} - \frac{{\color{red}y} - y_2}{{\color{green}x} - x_2} = \frac{y_3 - y_1}{x_3 - x_1} - \frac{y_3 - y_2}{x_3 - x_2}.</math>
Multiplying by the denominators that depend on <math>{\color{green}x},</math> one obtains the more standard form
<math display="block">(x_1 - x_2){\color{red}y} = ({\color{green}x} - x_1)({\color{green}x} - x_2) \left(\frac{y_3 - y_1}{x_3 - x_1} - \frac{y_3 - y_2}{x_3 - x_2}\right) + (y_1 - y_2){\color{green}x} + x_1 y_2 - x_2 y_1.</math>

== Pole–polar relation ==
[[File:Parabel-pol-s.svg|thumb|Parabola: pole–polar relation]]
In a suitable coordinate system any parabola can be described by an equation <math>y = ax^2</math>. The equation of the tangent at a point <math>P_0 = (x_0, y_0),\ y_0 = ax^2_0</math> is
<math display="block">y = 2ax_0(x - x_0) + y_0 = 2ax_0x - ax^2_0 = 2ax_0x - y_0.</math>
One obtains the function
<math display="block">(x_0, y_0) \to y = 2ax_0x - y_0</math>
on the set of points of the parabola onto the set of tangents.

Obviously, this function can be extended onto the set of all points of <math>\R^2</math> to a bijection between the points of <math>\R^2</math> and the lines with equations <math>y = mx + d, \ m, d \in \R</math>. The inverse mapping is 
<math display="block">\text{line } y = mx + d ~~ \rightarrow ~~ \text{point } (\tfrac{m}{2a}, -d).</math>
This relation is called the ''[[Pole and polar|pole–polar relation]] of the parabola'', where the point is the ''pole'', and the corresponding line its ''polar''.

By calculation, one checks the following properties of the pole–polar relation of the parabola:
* For a point (pole) ''on'' the parabola, the polar is the tangent at this point (see picture: <math>P_1,\ p_1</math>).
* For a pole <math>P</math> ''outside'' the parabola the intersection points of its polar with the parabola are the touching points of the two tangents passing <math>P</math> (see picture: <math>P_2,\ p_2</math>).
* For a point ''within'' the parabola the polar has no point with the parabola in common (see picture: <math>P_3,\ p_3</math> and <math>P_4,\ p_4</math>).
* The intersection point of two polar lines (see picture: <math>p_3, p_4</math>) is the pole of the connecting line of their poles (see picture: <math>P_3, P_4</math>).
* Focus and directrix of the parabola are a pole–polar pair.

''Remark:'' Pole–polar relations also exist for ellipses and hyperbolas.

== Tangent properties ==
=== Two tangent properties related to the latus rectum ===
Let the line of symmetry intersect the parabola at point Q, and denote the focus as point F and its distance from point Q as {{mvar|f}}. Let the perpendicular to the line of symmetry, through the focus, intersect the parabola at a point T. Then (1) the distance from F to T is {{math|2''f''}}, and (2) a tangent to the parabola at point T intersects the line of symmetry at a 45° angle.<ref>{{cite book|last=Downs |first=J. W. |title=Practical Conic Sections |publisher=Dover Publishing |date=2003}}{{ISBN missing}}</ref>{{rp|p=26}}
[[File:Parabel-orthop.svg|thumb|right|Perpendicular tangents intersect on the directrix]]

=== Orthoptic property ===
{{main|Orthoptic (geometry)}}
If two tangents to a parabola are perpendicular to each other, then they intersect on the directrix. Conversely, two tangents that intersect on the directrix are perpendicular. In other words, at any point on the directrix the whole parabola subtends a right angle.

=== Lambert's theorem ===
Let three tangents to a parabola form a triangle. Then '''[[Johann Heinrich Lambert|Lambert's]] theorem''' states that the focus of the parabola lies on the [[circumcircle]] of the triangle.<ref>{{cite journal|last=Sondow |first=Jonathan |arxiv=1210.2279 |title=The parbelos, a parabolic analog of the arbelos |journal=[[American Mathematical Monthly]] |volume=120 |date=2013 |issue=10 |pages=929–935 |doi=10.4169/amer.math.monthly.120.10.929|s2cid=33402874 }}</ref><ref name=ET/>{{rp|at=Corollary 20}}

Tsukerman's converse to Lambert's theorem states that, given three lines that bound a triangle, if two of the lines are tangent to a parabola whose focus lies on the circumcircle of the triangle, then the third line is also tangent to the parabola.<ref name=ET2>{{cite journal|last=Tsukerman |first=Emmanuel |title=Solution of Sondow's problem: a synthetic proof of the tangency property of the parbelos |journal=[[American Mathematical Monthly]] |volume=121 |date=2014 |issue=5 |pages=438–443 |arxiv=1210.5580 |doi=10.4169/amer.math.monthly.121.05.438|s2cid=21141837 }}</ref>

== Facts related to chords and arcs {{anchor|Chords|Arcs}} ==

=== Focal length calculated from parameters of a chord ===
Suppose a [[Chord (geometry)|chord]] crosses a parabola perpendicular to its axis of symmetry. Let the length of the chord between the points where it intersects the parabola be {{mvar|c}} and the distance from the vertex of the parabola to the chord, measured along the axis of symmetry, be {{mvar|d}}. The focal length, {{mvar|f}}, of the parabola is given by
<math display="block">f = \frac{c^2}{16d}.</math>

{{math proof | proof = Suppose a system of Cartesian coordinates is used such that the vertex of the parabola is at the origin, and the axis of symmetry is the {{mvar|y}} axis. The parabola opens upward. It is shown elsewhere in this article that the equation of the parabola is {{math|1=4''fy'' = ''x''<sup>2</sup>}}, where {{mvar|f}} is the focal length. At the positive {{mvar|x}} end of the chord, {{math|1=''x'' = {{sfrac|''c''|2}}}} and {{math|1=''y'' = ''d''}}. Since this point is on the parabola, these coordinates must satisfy the equation above. Therefore, by substitution, <math>4fd = \left(\tfrac{c}{2}\right)^2</math>. From this, <math>f = \tfrac{c^2}{16d}</math>.}}

=== Area enclosed between a parabola and a chord ===
[[File:Area between a parabola and a chord.svg|thumb|Parabola (magenta) and line (lower light blue) including a chord (blue). The area enclosed between them is in pink. The chord itself ends at the points where the line intersects the parabola.]]

The area enclosed between a parabola and a chord (see diagram) is two-thirds of the area of a parallelogram that surrounds it. One side of the parallelogram is the chord, and the opposite side is a tangent to the parabola.<ref>{{cite web |url=http://www.mathwarehouse.com/geometry/parabola/area-of-parabola.php |title=Sovrn Container |publisher=Mathwarehouse.com |access-date=2016-09-30}}</ref><ref>{{cite web |url=http://mysite.du.edu/~jcalvert/math/parabola.htm |title=Parabola |publisher=Mysite.du.edu |access-date=2016-09-30}}</ref> The slope of the other parallel sides is irrelevant to the area. Often, as here, they are drawn parallel with the parabola's axis of symmetry, but this is arbitrary.

A theorem equivalent to this one, but different in details, was derived by [[Archimedes]] in the 3rd century BCE. He used the areas of triangles, rather than that of the parallelogram.{{efn|Archimedes proved that the area of the enclosed parabolic segment was 4/3 as large as that of a triangle that he inscribed within the enclosed segment. It can easily be shown that the parallelogram has twice the area of the triangle, so Archimedes' proof also proves the theorem with the parallelogram.}} See [[The Quadrature of the Parabola]].

If the chord has length {{mvar|b}} and is perpendicular to the parabola's axis of symmetry, and if the perpendicular distance from the parabola's vertex to the chord is {{mvar|h}}, the parallelogram is a rectangle, with sides of {{mvar|b}} and {{mvar|h}}. The area {{mvar|A}} of the parabolic segment enclosed by the parabola and the chord is therefore
<math display="block">A = \frac{2}{3} bh.</math>

This formula can be compared with the area of a triangle: {{math|{{sfrac|1|2}}''bh''}}.

In general, the enclosed area can be calculated as follows. First, locate the point on the parabola where its slope equals that of the chord. This can be done with calculus, or by using a line that is parallel to the axis of symmetry of the parabola and passes through the midpoint of the chord. The required point is where this line intersects the parabola.{{efn|This method can be easily proved correct by calculus. It was also known and used by Archimedes, although he lived nearly 2000 years before calculus was invented.}} Then, using the formula given in [[Distance from a point to a line]], calculate the perpendicular distance from this point to the chord. Multiply this by the length of the chord to get the area of the parallelogram, then by 2/3 to get the required enclosed area.

=== Corollary concerning midpoints and endpoints of chords ===
[[File:Parabel-psehnen-s.svg|thumb|Midpoints of parallel chords]]

A corollary of the above discussion is that if a parabola has several parallel chords, their midpoints all lie on a line parallel to the axis of symmetry. If tangents to the parabola are drawn through the endpoints of any of these chords, the two tangents intersect on this same line parallel to the axis of symmetry (see [[parabola#Axis-direction of a parabola|Axis-direction of a parabola]]).{{efn|A proof of this sentence can be inferred from the proof of the [[#Orthoptic property|orthoptic property]], above. It is shown there that the tangents to the parabola {{math|1=''y'' = ''x''<sup>2</sup>}} at {{math|(''p'', ''p''<sup>2</sup>)}} and {{math|(''q'', ''q''<sup>2</sup>)}} intersect at a point whose {{mvar|x}} coordinate is the mean of {{mvar|p}} and {{mvar|q}}. Thus if there is a chord between these two points, the intersection point of the tangents has the same {{mvar|x}} coordinate as the midpoint of the chord.}}

=== Arc length ===
If a point X is located on a parabola with focal length {{mvar|f}}, and if {{mvar|p}} is the [[Distance from a point to a line|perpendicular distance]] from X to the axis of symmetry of the parabola, then the lengths of [[Arc (geometry)|arcs]] of the parabola that terminate at X can be calculated from {{mvar|f}} and {{mvar|p}} as follows, assuming they are all expressed in the same units.{{efn|In this calculation, the [[square root]] {{mvar|q}} must be positive. The quantity {{math|ln ''a''}} is the [[natural logarithm]] of&nbsp;{{mvar|a}}.}}
<math display="block">\begin{align}
 h &= \frac{p}{2}, \\
 q &= \sqrt{f^2 + h^2}, \\
 s &= \frac{hq}{f} + f \ln\frac{h + q}{f}.
\end{align}</math>

This quantity {{mvar|s}} is the length of the arc between X and the vertex of the parabola.

The length of the arc between X and the symmetrically opposite point on the other side of the parabola is {{math|2''s''}}.

The perpendicular distance {{mvar|p}} can be given a positive or negative sign to indicate on which side of the axis of symmetry X is situated. Reversing the sign of {{mvar|p}} reverses the signs of {{mvar|h}} and {{mvar|s}} without changing their absolute values. If these quantities are signed, ''the length of the arc between ''any'' two points on the parabola is always shown by the difference between their values of {{mvar|s}}''. The calculation can be simplified by using the properties of logarithms:
<math display="block">s_1 - s_2 = \frac{h_1 q_1 - h_2 q_2}{f} + f \ln\frac{h_1 + q_1}{h_2 + q_2}.</math>

This can be useful, for example, in calculating the size of the material needed to make a [[parabolic reflector]] or [[parabolic trough]].

This calculation can be used for a parabola in any orientation. It is not restricted to the situation where the axis of symmetry is parallel to the ''y'' axis.

== A geometrical construction to find a sector area ==
[[File: Sector Area Prop 30.png|400px|Sector area proposition 30]]

S is the focus, and V is the principal vertex of the parabola VG. Draw VX perpendicular to SV.

Take any point B on VG and drop a perpendicular BQ from B to VX. Draw perpendicular ST intersecting BQ, extended if necessary, at T. At B draw the perpendicular BJ, intersecting VX at J.

For the parabola, the segment VBV, the area enclosed by the chord VB and the arc VB, is equal to ∆VBQ / 3, also <math>BQ = \frac{VQ^2}{4SV}</math>.

The area of the parabolic sector <math>SVB = \triangle SVB + \frac{\triangle VBQ}{3} = \frac{SV \cdot VQ}{2} + \frac{VQ \cdot BQ}{6}</math>.

Since triangles TSB and QBJ are similar,
<math display="block">VJ = VQ - JQ = VQ - \frac{BQ \cdot TB}{ST} = VQ - \frac{BQ \cdot (SV - BQ)}{VQ} = \frac{3VQ}{4} + \frac{VQ \cdot BQ}{4SV}.</math>

Therefore, the area of the parabolic sector <math>SVB = \frac{2SV \cdot VJ}{3}</math> and can be found from the length of VJ, as found above.

A circle through S, V and B also passes through J.

Conversely, if a point, B on the parabola VG is to be found so that the area of the sector SVB is equal to a specified value, determine the point J on VX and construct a circle through S, V and J. Since SJ is the diameter, the center of the circle is at its midpoint, and it lies on the perpendicular bisector of SV, a distance of one half VJ from SV. The required point B is where this circle intersects the parabola.

If a body traces the path of the parabola due to an inverse square force directed towards S, the area SVB increases at a constant rate as point B moves forward. It follows that J moves at constant speed along VX as B moves along the parabola.

If the speed of the body at the vertex where it is moving perpendicularly to SV is ''v'', then the speed of J is equal to {{math|3''v''/4}}.

The construction can be extended simply to include the case where neither radius coincides with the axis SV as follows. Let A be a fixed point on VG between V and B, and point H be the intersection on VX with the perpendicular to SA at A. From the above, the area of the parabolic sector <math>SAB = \frac{2SV \cdot (VJ - VH)}{3} = \frac{2SV \cdot HJ}{3}</math>.

Conversely, if it is required to find the point B for a particular area SAB, find point J from HJ and point B as before. By Book 1, Proposition 16, Corollary 6 of Newton's [[Philosophiæ Naturalis Principia Mathematica|''Principia'']], the speed of a body moving along a parabola with a force directed towards the focus is inversely proportional to the square root of the radius. If the speed at A is ''v'', then at the vertex V it is <math>\sqrt{\frac{SA}{SV}} v</math>, and point J moves at a constant speed of <math>\frac{3v}{4} \sqrt{\frac{SA}{SV}}</math>.

The above construction was devised by Isaac Newton and can be found in Book 1 of [[Philosophiæ Naturalis Principia Mathematica]] as Proposition 30.