Polar Coordinates Calculator

The Polar Coordinates Calculator converts between Cartesian coordinates $$(x, y)$$ and polar coordinates $$(r, \theta)$$. Polar coordinates describe a point by its distance from the origin and the angle from the positive x-axis, providing an alternative to the rectangular Cartesian system that is particularly useful for problems with radial symmetry.

The conversion from Cartesian to polar is:

$$r = \sqrt{x^2 + y^2}$$

$$\theta = \text{atan2}(y, x)$$

where $$\text{atan2}$$ is the two-argument arctangent function that correctly handles all four quadrants, returning an angle in the range $$(-180°, 180°]$$ or $$(-\pi, \pi]$$ radians.

The conversion from polar to Cartesian is:

$$x = r \cos\theta$$

$$y = r \sin\theta$$

The polar coordinate system was first used systematically by Isaac Newton and Jacob Bernoulli in the 17th century. It offers elegant descriptions of curves that would be complex in Cartesian form. For example, a circle of radius $$a$$ centered at the origin is simply $$r = a$$ in polar coordinates, while in Cartesian form it requires $$x^2 + y^2 = a^2$$. Spirals, rose curves, cardioids, and limacons all have simple polar equations.

In physics, polar coordinates naturally describe orbital motion. Kepler's laws of planetary motion are most elegantly expressed in polar form, where the radial distance $$r$$ depends on the angle $$\theta$$ through the orbit equation:

$$r = \frac{a(1 - e^2)}{1 + e\cos\theta}$$

In engineering, antenna radiation patterns are plotted in polar coordinates to show signal strength as a function of direction. In fluid dynamics, the flow around a cylinder is described using polar coordinates centered on the cylinder. In navigation, radar systems use polar coordinates with the radar station as the origin.

The Jacobian of the transformation from Cartesian to polar coordinates is $$r$$, meaning area elements transform as $$dx\,dy = r\,dr\,d\theta$$. This is essential for evaluating double integrals in polar coordinates, such as the Gaussian integral $$\int_{-\infty}^{\infty} e^{-x^2}\,dx = \sqrt{\pi}$$, which is most easily solved by converting to polar form.

Complex numbers are naturally represented in polar form as $$z = r e^{i\theta} = r(\cos\theta + i\sin\theta)$$, known as Euler's formula. This representation simplifies multiplication, division, and exponentiation of complex numbers. The magnitude $$r = |z|$$ is the modulus and $$\theta = \arg(z)$$ is the argument.