A point $P$ in the polar coordinate system is uniquely represented by the ordered pair $(r, \theta)$.

• Pole (Origin): The fixed reference point in the plane, typically denoted as $O$.
• Polar Axis: A fixed ray extending outward from the pole, usually corresponding to the positive $x$-axis in the rectangular system.
• Radial Distance ($r$): The directed distance from the pole to the point $P$.
• Polar Angle ($\theta$): The directed angle measured from the polar axis to the line segment connecting the pole and the point $P$.

Finding the absolute straight-line Euclidean geometric distance exactly between two entirely unique point coordinates purely defined within the mathematical 2D polar system, specifically $(r_1, \theta_1)$ and completely $(r_2, \theta_2)$, does not rigidly require converting them back to traditional rectangular $(x, y)$ coordinates. Instead, the precise linear distance $d$ across the continuous plane can be calculated directly by algebraically employing the foundational trigonometric Law of Cosines explicitly onto the theoretical triangle perfectly formed by the central pole (origin) and the dual radial distance vectors pointing exactly to the two distinct points.

Because both coordinate systems map the exact same underlying 2D geometric plane, we can seamlessly translate mathematical points and complex equations back and forth between them. To properly convert between polar coordinates $(r, \theta)$ and rectangular coordinates $(x, y)$, we rigorously apply basic trigonometric relationships. We achieve this by conceptually dropping a perpendicular line from the point to the $x$-axis, thereby forming a standard right triangle where the origin acts as the primary vertex, $r$ represents the hypotenuse, and $\theta$ is the primary reference angle.

Given a polar coordinate $(r, \theta)$, find the rectangular coordinate $(x, y)$:

Given a rectangular coordinate $(x, y)$, find the polar coordinate $(r, \theta)$:

Equations in polar coordinates often graph as beautiful and complex curves that would be immensely difficult to express or manipulate algebraically in standard rectangular coordinates.

• Circles: $r = a$ (centered at origin), $r = a \cos \theta$ (touches origin, lies on $x$-axis), $r = a \sin \theta$ (touches origin, lies on $y$-axis).
• Straight Lines: $\theta = \alpha$ (passes through origin), $r = \frac{a}{\cos \theta}$ (vertical line $x = a$), $r = \frac{a}{\sin \theta}$ (horizontal line $y = a$).
• Limaçons: Formed by equations $r = a \pm b \cos \theta$ or $r = a \pm b \sin \theta$. If $a \lt b$, it has an inner loop. If $a = b$, it forms a heart-shape called a Cardioid. If $a \gt b$, it is a convex or dimpled limaçon.
• Rose Curves: Formed by equations $r = a \cos(n\theta)$ or $r = a \sin(n\theta)$. If the multiplier $n$ is an odd number, the rose has exactly $n$ petals. If $n$ is an even number, the rose has exactly $2n$ petals.
• Spirals: The Archimedean Spiral ($r = a\theta$) where distance grows linearly with rotation, and the Logarithmic Spiral ($r = e^{a\theta}$) which appears often in nature (e.g., nautilus shells).

Before attempting to plot a complex polar curve by hand, identifying its innate symmetry can drastically reduce the number of points that need to be calculated.