Polar Curves Plotter

Introduction to Polar Coordinates

In mathematics, the polar coordinate system is a two-dimensional coordinate system in which each point on a plane is determined by a distance from a reference point and an angle from a reference direction.

• The reference point (analogous to the origin of a Cartesian system) is called the pole.
• The ray from the pole in the reference direction is the polar axis (usually drawn horizontally to the right, corresponding to the positive x-axis in Cartesian coordinates).
• The distance from the pole is called the radial coordinate, or radius, usually denoted by $r$.
• The angle, measured counterclockwise from the polar axis, is called the angular coordinate, polar angle, or azimuth, usually denoted by $\theta$.

Conversion to Cartesian Coordinates

The polar coordinates $r$ and $\theta$ can be converted to the Cartesian coordinates $x$ and $y$ by using the trigonometric functions sine and cosine:

$$ x = r \cos \theta $$

$$ y = r \sin \theta $$

Conversely, Cartesian coordinates $(x, y)$ can be converted to polar coordinates $(r, \theta)$ by:

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

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

(The $\operatorname{atan2}$ function correctly handles all quadrants for $\theta$).

Polar coordinates are useful in situations where the relationship between two points is most easily expressed in terms of angles and distances; in many cases, equations can be expressed more simply in polar form than in Cartesian form.

Common Polar Curves

Many fascinating shapes can be described by simple polar equations. Here are a few common types:

Circles

• $r = a$: A circle centered at the pole with radius $|a|$.
• $r = a \sin \theta$: A circle passing through the pole, symmetric about the line $\theta = \pi/2$. If $a > 0$, it's above the polar axis. Diameter is $|a|$.
• $r = a \cos \theta$: A circle passing through the pole, symmetric about the polar axis. If $a > 0$, it's to the right of the pole. Diameter is $|a|$.

Limacons: $r = a + b \cos \theta$ or $r = a + b \sin \theta$

The shape of a limacon depends on the ratio $|a/b|$:

• $|a/b| < 1$: Limacon with an inner loop.
• $|a/b| = 1$: Cardioid (heart-shaped, cusp at the pole). Example: $r = a(1 + \cos \theta)$.
• $1 < |a/b| < 2$: Dimpled limacon (no inner loop, but indented).
• $|a/b| \ge 2$: Convex limacon (no inner loop, no dimple).

Rose Curves: $r = a \cos(n\theta)$ or $r = a \sin(n\theta)$

These curves have petal-like shapes:

• If $n$ is an odd integer, the rose has $n$ petals.
• If $n$ is an even integer, the rose has $2n$ petals.
• $|a|$ determines the length of the petals.

Spirals

• Archimedean Spiral: $r = a\theta$. The distance between successive turns is constant.
• Logarithmic (Equiangular) Spiral: $r = ae^{b\theta}$ or $r = ab^\theta$. The angle between the tangent and the radial line is constant. These spirals appear in nature (e.g., nautilus shells, spiral galaxies).

Lemniscates

• Lemniscate of Bernoulli: $r^2 = a^2 \cos(2\theta)$ or $r^2 = a^2 \sin(2\theta)$. These are figure-eight shaped curves.

Lines

• $\theta = c$: A line passing through the pole with angle $c$ to the polar axis.
• $r \cos \theta = a$ (or $x=a$): A vertical line. In polar, $r = a \sec \theta$.
• $r \sin \theta = b$ (or $y=b$): A horizontal line. In polar, $r = b \csc \theta$.

Applications of Polar Coordinates

Polar coordinates are used in a variety of fields:

• Navigation: Aircraft and ship navigation systems often use a system similar to polar coordinates (distance and bearing).
• Robotics: Robot arm movements are often described using angles and segment lengths.
• Physics and Engineering: Describing phenomena with central symmetry, like gravitational fields, electromagnetic radiation patterns (e.g., antenna radiation), and circular or spiral motion.
• Computer Graphics: Generating circular or spiral patterns, and for certain types of image transformations.
• Mathematics: Evaluating certain types of double integrals, especially over circular regions, becomes much simpler using polar coordinates. They are also fundamental in complex analysis (polar form of complex numbers).
• Astronomy: Describing the orbits of planets and other celestial bodies.