Editing Polar coordinate system (section)

==Polar equation of a curve==
[[File:Cartesian to polar.gif|thumb|A curve on the Cartesian plane can be mapped into polar coordinates. In this animation, <math>y = \sin (6\!\cdot\!x) + 2</math> is mapped onto <math>r = \sin (6\!\cdot\!\theta) + 2</math>. Click on image for details.]]The equation defining a [[plane curve]] expressed in polar coordinates is known as a ''polar equation''. In many cases, such an equation can simply be specified by defining ''r'' as a [[function (mathematics)|function]] of ''φ''. The resulting curve then consists of points of the form (''r''(''φ''),&nbsp;''φ'') and can be regarded as the [[graph of a function|graph]] of the polar function ''r''. Note that, in contrast to Cartesian coordinates, the independent variable ''φ'' is the ''second'' entry in the ordered pair.

Different forms of [[symmetry]] can be deduced from the equation of a polar function ''r'':

* If {{math|1=''r''(−''φ'') = ''r''(''φ'')}} the curve will be symmetrical about the horizontal (0°/180°) ray;
* If {{math|1=''r''(''π'' − ''φ'') = ''r''(''φ'')}} it will be symmetric about the vertical (90°/270°) ray:
* If {{math|1=''r''(''φ'' − α) = ''r''(''φ'')}} it will be [[rotational symmetry|rotationally symmetric]] by α clockwise and counterclockwise about the pole.

Because of the circular nature of the polar coordinate system, many curves can be described by a rather simple polar equation, whereas their Cartesian form is much more intricate. Among the best known of these curves are the [[Rose (mathematics)|polar rose]], [[Archimedean spiral]], [[Lemniscate of Bernoulli|lemniscate]], [[limaçon]], and [[cardioid]].

For the circle, line, and polar rose below, it is understood that there are no restrictions on the domain and range of the curve.

===Circle===
[[Image:circle r=1.svg|thumb|right|A circle with equation {{math|1=''r''(''φ'') = 1}}]]
The general equation for a circle with a center at <math>(r_0, \gamma)</math> and radius ''a'' is
<math display="block">r^2 - 2 r r_0 \cos(\varphi - \gamma) + r_0^2 = a^2.</math>

This can be simplified in various ways, to conform to more specific cases, such as the equation
<math display="block">r(\varphi)=a </math>
for a circle with a center at the pole and radius ''a''.<ref name="ping">{{Cite web |last=Claeys |first=Johan |title=Polar coordinates |url=http://www.ping.be/~ping1339/polar.htm |url-status=dead |archive-url=https://web.archive.org/web/20060427230725/http://www.ping.be/~ping1339/polar.htm |archive-date=2006-04-27 |access-date=2006-05-25}}</ref>

When {{math|1=''r''<sub>0</sub> = ''a''}} or the origin lies on the circle, the equation becomes
<math display="block">r = 2 a\cos(\varphi - \gamma).</math>

In the general case, the equation can be solved for {{math|''r''}}, giving
<math display="block">r = r_0 \cos(\varphi - \gamma) + \sqrt{a^2 - r_0^2 \sin^2(\varphi - \gamma)}</math>
The solution with a minus sign in front of the square root gives the same curve.<!-- Better rephrasing? -->

===Line===
''Radial'' lines (those running through the pole) are represented by the equation
<math display="block">\varphi = \gamma,</math>
where <math>\gamma</math> is the angle of elevation of the line; that is, <math>\varphi = \arctan m</math>, where <math>m</math> is the [[slope]] of the line in the Cartesian coordinate system. The non-radial line that crosses the radial line <math>\varphi = \gamma</math> [[perpendicular]]ly at the point <math>(r_0, \gamma)</math> has the equation
<math display="block">r(\varphi) = r_0 \sec(\varphi - \gamma).</math>

Otherwise stated <math>(r_0, \gamma)</math> is the point in which the tangent intersects the imaginary circle of radius <math>r_0</math>

===Polar rose===
[[Image:Rose 2sin(4theta).svg|thumb|right|A polar rose with equation {{math|1=''r''(''φ'') {{=}} 2 sin 4''φ''}}]]
A [[rose (mathematics)|polar rose]] is a mathematical curve that looks like a petaled flower, and that can be expressed as a simple polar equation,
<math display="block">r(\varphi) = a\cos\left(k\varphi + \gamma_0\right)</math>

for any constant γ<sub>0</sub> (including 0). If ''k'' is an integer, these equations will produce a ''k''-petaled rose if ''k'' is [[even and odd numbers|odd]], or a 2''k''-petaled rose if ''k'' is even. If ''k'' is rational, but not an integer, a rose-like shape may form but with overlapping petals. Note that these equations never define a rose with 2, 6, 10, 14, etc. petals. The [[variable (math)|variable]] ''a'' directly represents the length or amplitude of the petals of the rose, while ''k'' relates to their spatial frequency. The constant γ<sub>0</sub> can be regarded as a phase angle.
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===Archimedean spiral===
[[File:Spiral of Archimedes.svg|thumb|right|One arm of an Archimedean spiral with equation {{math|1=''r''(''φ'') = ''φ'' / 2''π''}} for {{math|0 < ''φ'' < 6''π''}}]]
The [[Archimedean spiral]] is a spiral discovered by [[Archimedes]] which can also be expressed as a simple polar equation. It is represented by the equation
<math display="block">r(\varphi) = a + b\varphi. </math>
Changing the parameter ''a'' will turn the spiral, while ''b'' controls the distance between the arms, which for a given spiral is always constant. The Archimedean spiral has two arms, one for {{math|''φ'' > 0}} and one for {{math|''φ'' < 0}}. The two arms are smoothly connected at the pole. If {{math|1=''a'' = 0}}, taking the mirror image of one arm across the 90°/270° line will yield the other arm. This curve is notable as one of the first curves, after the [[conic section]]s, to be described in a mathematical treatise, and as a prime example of a curve best defined by a polar equation.
{{Clear}}[[Image:Elps-slr.svg|thumb|Ellipse, showing semi-latus rectum]]

===Conic sections===
A [[conic section]] with one focus on the pole and the other somewhere on the 0° ray (so that the conic's [[semi-major axis|major axis]] lies along the polar axis) is given by:
<math display="block">r = { \ell\over {1 - e \cos \varphi}}</math>
where ''e'' is the [[eccentricity (mathematics)|eccentricity]] and <math>\ell</math> is the [[semi-latus rectum]] (the perpendicular distance at a focus from the major axis to the curve). If {{nowrap|''e'' > 1}}, this equation defines a [[hyperbola]]; if {{math|1=''e'' = 1}}, it defines a [[parabola]]; and if {{math|''e'' < 1}}, it defines an [[ellipse]]. The special case {{math|1=''e'' = 0}} of the latter results in a circle of the radius <math>\ell</math>.
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===Quadratrix===
[[File:Quadratrix animation.gif|thumb]]
{{Main|Quadratrix of Hippias}}

A quadratrix in the first quadrant (''x, y'') is a curve with ''y'' = &rho; sin &theta; equal to the fraction of the quarter circle with radius ''r'' determined by the radius through the curve point. Since this fraction is <math>\frac{2 r \theta}{\pi}</math>, the curve is given by <math>\rho (\theta) = \frac{2 r \theta}{\pi \sin \theta}</math>.<ref>N.H. Lucas, P.J. Bunt & J.D Bedient (1976) ''Historical Roots of Elementary Mathematics'', page 113</ref>

===Intersection of two polar curves===
The graphs of two polar functions <math>r = f(\theta)</math> and <math>r = g(\theta)</math> have possible intersections of three types:
# In the origin, if the equations <math>f(\theta) = 0</math> and <math>g(\theta) = 0</math> have at least one solution each.
# All the points <math>[g(\theta_i),\theta_i]</math> where <math>\theta_i</math> are solutions to the equation <math>f(\theta+2k\pi)=g(\theta)</math> where <math>k</math> is an integer.
# All the points <math>[g(\theta_i),\theta_i]</math> where <math>\theta_i</math> are solutions to the equation <math>f(\theta+(2k+1)\pi)=-g(\theta)</math> where <math>k</math> is an integer.
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