Polar coordinate system
From Free net encyclopedia
The polar coordinate system is a two-dimensional coordinate system in which points are given by an angle and a distance from the pole, called the origin in the Cartesian coordinate system.
The polar coordinates r (the radial coordinate) and θ (the angular coordinate, often called the polar angle), sometimes represented as φ or t, are defined in terms of Cartesian coordinates by
- <math>x = r \cos \theta \,</math>
- <math>y = r \sin \theta \,</math>
- and therefore <math>r = \sqrt{x^2 + y^2} \,</math>
where r is the radial distance from the pole, and θ is the counterclockwise angle from the 0° ray, which is the section of the Cartesian x-axis from the origin eastward.
For example, if you had the coordinates (3, 60°), the point would be plotted 3 units from the origin on the 60° ray. If you had the coordinates (-3, 240°), the point would be in the same location, because -3 units on the 240° ray is the same as 3 units on the 60° ray.
The equation of a curve expressed in polar coordinates is known as a polar equation, and is usually written with r as a function of θ.
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Common polar equations
Image:Circle r=1.PNG [[Image:limacon_r=.75+1.5cos(theta).PNG|thumb|right|A limacon with equation r(θ) = 3/4 + 3/2 cosθ]] Image:Cardioid r=1-sin(t).PNG Image:Lemniscate r=sqrt(cos(2theta)).PNG Image:Rose r=2sin(4theta).PNG Image:Archimedian spiral.PNG
Line
A line can be expressed as a polar equation in two different ways, depending on whether it runs through the pole.
If a line does run through the pole, it's equation can be represented by the equation
- <math>\theta = \phi \,</math>, where φ is the angle of elevation of the line, or
- <math>\theta = \arctan(m) \,</math>, where m is the slope of the line in the Cartesian coordinate system.
If a line does not run through the pole, but runs through the point (r0, φ), its equation is
- <math>r(\theta) = \frac{r_0}{\cos(\theta-\phi)} \,</math>,
which will generate a line perpendicular to the line θ = φ.
Circle
The are several ways to write the polar equation of a circle, which conform to circles at different locations and of different sizes.
For a circle with a center at the pole and radius a the equation is
- <math>r(\theta)=a \,</math>
For a circle with a center at (r0, φ) and radius |r1| the equation is
- <math>r(\theta)=2r_0 \cos(\theta-\phi) \,</math>
For any circle with a center at (r0, φ) and radius a the equation is
- <math>r^2 - 2 r r_0 \cos(\theta - \phi) + r_0^2 = a^2 \,</math>
Limaçon
A limaçon (pronounced leem-ah-son), also known as a limaçon of Pascal, is a heart-shaped mathematical curve. It is given by the equations
- <math>r(\theta) = a \pm b \cos \theta \,</math> OR
- <math>r(\theta) = a \pm b \sin \theta \,</math>
Cardioid
A cardioid is a special limaçon where a and b are equal. It is it given by the equations
- <math>r(\theta) = a \pm a \cos \theta \,</math> OR
- <math>r(\theta) = a \pm a \sin \theta \,</math>
Lemniscate
A lemniscate is a mathematical curve which looks like a figure eight. It is it given by the equations
- <math>r^2 = a \cos \theta \,</math> OR
- <math>r^2 = a \sin \theta \,</math>
Polar Rose
A polar rose is a mathematical curve which looks like a petalled flower. It is given by the equations
- <math>r(\theta) = a \cos(k\theta) \,</math> OR
- <math>r(\theta) = a \sin(k\theta) \,</math>.
These equations will produce a k-petalled rose if k is odd, or a 2k-petalled rose if k is even. Note that it is impossible to make a rose with 2 more than a multiple of 4 (2, 6, 10, etc.) petals.
Spiral of Archimedes
The Archimedean spiral is a spiral that was discovered by Archimedes. It is represented by the equation:
- <math>r(\theta) = a+b\theta \,</math>.
Changing the parameter a will turn the spiral, while b controls the distance between the arms.
Complex numbers
Complex numbers, written in rectangular form as <math>a + bi \,</math>, can also be expressed in polar form in two different ways:
- <math>r(\cos\theta+i\sin\theta) \,</math>, abbreviated <math>r \mbox{ cis } \theta \,</math> or <math>(r \angle \theta) \,</math>
- <math>r e^{i\theta} \,</math>
of which both are equivalent as per Euler's formula. To convert between rectangular and polar complex numbers, the following conversion formulas are used:
- <math>a = r \cos \theta \,</math>
- <math>b = r \sin \theta \,</math>
- and therefore <math>r = \sqrt{a^2 + b^2} \,</math>
For the operations of multiplication, division, and exponentiation, and finding roots of complex numbers, it is much easier to use polar complex numbers than rectangular complex numbers. In abbreviated form:
- Multiplication: <math>(r \mbox{ cis } \theta) * (R \mbox{ cis } \phi) = rR \mbox{ cis } (\theta+\phi) \,</math>
- Division: <math>\frac{r \mbox{ cis } \theta}{R \mbox{ cis } \phi} = \frac{r}{R} \mbox{ cis } (\theta-\phi) \,</math>
- Exponentiation (De Moivre's formula): <math>(r \mbox{ cis } \theta)^n = r^n \mbox{ cis } (n\theta) \,</math>