Andromeda
Note

Polar Coordinate Conversions

Definition

Polar Coordinates represent a point PP in a plane using an ordered pair (r,θ)(r, \theta), where rr is the directed distance from the pole (origin) and θ\theta is the directed angle from the polar axis (positive xx-axis).

Why It Matters

Real-world sensors (radar, LIDAR, eyes) see in Polar, but we build and manufacture in Rectangular. If you can’t convert between them with 100% accuracy, you cannot bridge the gap between “perception” and “action.” This math is the essential translation layer for robotics and autonomous navigation—it’s the reason a self-driving car can turn a “distance and angle” signal into a steering command.

Core Concepts

  • Multiple Representations: Unlike rectangular coordinates, a single point in polar coordinates has infinitely many representations: (r,θ)=(r,θ+2kπ)(r, \theta) = (r, \theta + 2k\pi) or (−r,θ+π+2kπ)(-r, \theta + \pi + 2k\pi) for any integer kk.
  • How to read: “The polar coordinate pair r and theta is equivalent to the pair r and theta plus two times k times pi, and also equivalent to the pair negative r and theta plus pi plus two times k times pi.”
  • Meaning: Same physical point, infinitely many angle/radius pairs—add full rotations or flip radius and rotate by pi.
  • Polar to Rectangular Conversion:
    • x=rcos⁡θx = r \cos \theta
    • y=rsin⁡θy = r \sin \theta
    • How to read: “The coordinate x is equal to the radius r times the cosine of theta; and the coordinate y is equal to the radius r times the sine of theta.”
    • Meaning: Project polar radius onto axes—standard polar-to-rectangular conversion.
  • Rectangular to Polar Conversion:
    • r2=x2+y2r^2 = x^2 + y^2
    • tan⁡θ=yx\tan \theta = \frac{y}{x} (Note: Must check quadrant of (x,y)(x, y) to determine correct θ\theta).
    • How to read: “The radius r squared is equal to x squared plus y squared; and the tangent of theta is equal to the ratio of y to x.”
    • Meaning / when to use: Pythagorean distance for rr; arctangent for angle but quadrant fixes the branch.

Connected Concepts

Connected notes