Polar Coordinate Conversions

Concept Explanation

In rectangular coordinates, a point is positioned by two orthogonal linear displacements: horizontally by $x$ and vertically by $y$. In polar coordinates, that same physical position is specified by a radius $r$ and an angle $\theta$.

                     y ^
                       |       (x, y) = (r cos θ, r sin θ)
                       |        *
                       |       /|
                       |   r  / |
                       |     /  | y = r sin θ
                       |    / θ |
                       +---/----+--------> x
                          O   x = r cos θ

1. Polar-to-Cartesian Conversion ($r, \theta \to x, y$)

Given a radial distance $r$ and angle $\theta$, projection onto orthogonal Cartesian axes is given directly by right-triangle trigonometry:

2. Cartesian-to-Polar Conversion ($x, y \to r, \theta$)

Given Cartesian coordinates $(x, y)$:

1. Compute the Radial Distance $r$: Applying the Pythagorean theorem yields:
   $$r = \sqrt{x^2 + y^2}$$
2. Determine the Angle $\theta$ with Quadrant Awareness: The reference angle satisfies $\tan \theta = \frac{y}{x}$ (for $x \neq 0$). Because the tangent function has a period of $\pi$, relying solely on $\arctan(y/x)$ only returns values in Quadrants I and IV $\left(-\frac{\pi}{2} < \theta < \frac{\pi}{2}\right)$. You must inspect the signs of $x$ and $y$ to place $\theta$ in the correct quadrant:
   $$\theta = \begin{cases} \arctan(y/x) & \text{if } x > 0 \text{ (Quadrant I or IV)} \\ \arctan(y/x) + \pi & \text{if } x < 0 \text{ (Quadrant II or III)} \\ \frac{\pi}{2} & \text{if } x = 0 \text{ and } y > 0 \\ \frac{3\pi}{2} \text{ (or } -\frac{\pi}{2}\text{)} & \text{if } x = 0 \text{ and } y < 0 \end{cases}$$

3. Converting Equations Between Coordinate Systems

• Circles Centered at the Origin:
  
  $$x^2 + y^2 = a^2 \implies r^2 = a^2 \implies r = a$$
  Notice that a curved quadratic equation in Cartesian coordinates collapses into a constant line $r = a$ in polar coordinates.
• Rays and Straight Lines Through the Origin:
  
  $$y = mx \implies \frac{y}{x} = m \implies \tan \theta = m \implies \theta = \arctan(m)$$
• Vertical and Horizontal Lines:
  
  $$x = c \implies r \cos \theta = c \implies r = c \sec \theta$$
  $$y = d \implies r \sin \theta = d \implies r = d \csc \theta$$
• Offset Circles Passing Through the Origin:
  A circle of radius $a$ centered at $(a, 0)$ on the $x$-axis has Cartesian equation:
  $$(x - a)^2 + y^2 = a^2 \implies x^2 - 2ax + a^2 + y^2 = a^2 \implies x^2 + y^2 = 2ax$$
  Substituting $x^2 + y^2 = r^2$ and $x = r \cos \theta$:
  $$r^2 = 2ar \cos \theta \implies r = 2a \cos \theta \quad \left(-\frac{\pi}{2} \le \theta \le \frac{\pi}{2}\right)$$
  Similarly, a circle of radius $b$ centered at $(0, b)$ on the $y$-axis simplifies to:
  $$r = 2b \sin \theta \quad (0 \le \theta \le \pi)$$

When to Use It / Method Comparison

Why We Use It in Calculus (Calculus III Context)

In multivariable calculus (MATH-283), polar coordinate conversion is the cornerstone of Double Integrals in Polar Coordinates (Session 15) and serves as the 2D foundation for Cylindrical Coordinates in 3D integration:

1. Eliminating Radical Boundary Limits: Setting up double integrals over circular domains in Cartesian coordinates forces radicals into inner bounds: $\int_{-a}^a \int_{-\sqrt{a^2-x^2}}^{\sqrt{a^2-x^2}} f(x, y) \, dy \, dx$. In polar coordinates, the exact same domain becomes a simple constant box: $\int_0^{2\pi} \int_0^a f(r\cos\theta, r\sin\theta) \, r \, dr \, d\theta$.
2. Chain-Rule Multiplier via the Jacobian: When transforming $dx \, dy \to r \, dr \, d\theta$, the Jacobian area scaling factor $r$ naturally pairs with terms like $e^{-(x^2+y^2)} = e^{-r^2}$, enabling standard single-variable $u$-substitution ($u = -r^2, du = -2r\,dr$) on functions that have no closed-form Cartesian antiderivative (such as the Gaussian distribution).
3. Offset Circular Domains in Engineering: Engineering models of cylindrical fluid pipes, offset drilling shafts, and turbine housing clearances require bounding offset circles ($r = 2a\cos\theta$) to compute net fluid flux and torsional stress distributions.

Example Problems

Example Problem 1: Point Conversion with Quadrant Awareness

Convert the Cartesian coordinate point $P(-2\sqrt{3}, 2)$ into polar coordinates $(r, \theta)$ with $r > 0$ and $0 \le \theta < 2\pi$.

Example Problem 2: Transforming a Cartesian Boundary into Polar Form

Transform the Cartesian equation $(x - 3)^2 + y^2 = 9$ into polar coordinate form $r = g(\theta)$, and determine the appropriate angular sweep $\theta$ that traces the entire boundary.

Connections & References

• Parent Concepts: Calculus Foundations, Vectors in 2D and 3D
• Related Concepts: Double Integrals in Polar Coordinates, Polar Rectangles and Annular Sectors, Trigonometric Identities, Trigonometric Power-Reduction Formulas
• Course Links: Calc-3 Session 15, Calc-3 Discussion 08