Polar coordinates and parametric curves

Intuition [Beginner]

A point in the plane can be located in two ways. The familiar way uses horizontal and vertical distances: $(x,y)$. The second way uses a distance and a direction: walk $r$ steps from the origin at angle $\theta$, and you arrive at the same point. The pair $(r,\theta )$ is called a polar coordinate of the point.

Think of a radar screen. The sweep arm rotates around the centre, and a blip appears at a distance and a direction from the centre. That is polar coordinates: the distance $r$ is how far out the blip is, and the angle $\theta$ is which direction the arm was pointing when the blip appeared. The radar does not need an $x$-axis and a $y$-axis — it uses a single angle and a single distance.

Polar coordinates exist because some curves are much easier to describe with a distance and an angle than with two separate distances. A circle centred at the origin is just $r=5$ in polar form, a single number, whereas in Cartesian form it is $x^2+y^2=25$.

Visual [Beginner]

A polar grid: concentric circles centred at the origin representing constant distance $r$, and radial lines from the origin representing constant angle $\theta$. A point $P$ is marked at distance $r=3$ from the origin at angle $\theta =60$ degrees.

The polar grid replaces the rectangular grid of horizontal and vertical lines with circles and rays. Every point in the plane has a polar address $(r,\theta )$.

Worked example [Beginner]

Convert the polar point $(r,\theta )=(4,30^{\circ })$ to Cartesian coordinates $(x,y)$.

Step 1. The conversion formulas are $x=r\cos \theta$ and $y=r\sin \theta$.

Step 2. Using the special-angle values from 00.07.01: $\cos 30^{\circ }=\sqrt{3}/2$ and $\sin 30^{\circ }=1/2$.

Step 3. Compute: $x=4\cdot \sqrt{3}/2=2\sqrt{3}$ and $y=4\cdot 1/2=2$.

What this tells us: polar and Cartesian coordinates name the same point differently. The conversion is a direct application of sine and cosine from 00.07.01.

Check your understanding [Beginner]

Formal definition [Intermediate+]

The polar coordinates of a point $P$ in the plane are an ordered pair $(r,\theta )$ where $r\ge 0$ is the distance from the origin $O$ to $P$, and $\theta$ is the angle measured counter-clockwise from the positive $x$-axis to the ray $OP$ ^{[Lang — Basic Mathematics Ch. 9]}.

Conversion formulas. The polar pair $(r,\theta )$ and the Cartesian pair $(x,y)$ are related by

$$x=r\cos \theta ,y=r\sin \theta ,$$

and inversely

$$r=\sqrt{x^2+y^2},\tan \theta =\frac{y}{x}(x\ne 0).$$

The angle $\theta$ is determined by $\theta =atan2(y,x)$ (the two-argument arctangent), which accounts for the quadrant.

Parametric curves. A parametric curve in ${\mathbb{R}}^2$ is a map $t\mapsto (x(t),y(t))$ from an interval of real numbers to the plane. The parameter $t$ traces the curve as it varies.

Polar curves. A polar curve is the set of points $(r,\theta )$ satisfying an equation $r=f(\theta )$ for $\theta$ in some interval.

Counterexamples to common slips

• Polar coordinates are not unique. The point $(r,\theta )$ is the same as $(r,\theta +2\pi )$ and, when $r>0$, also the same as $(-r,\theta +\pi )$. The origin has $r=0$ with $\theta$ arbitrary.
• The formula $\theta =\arctan (y/x)$ fails when $x=0$ (vertical line through the origin). The two-argument arctangent $atan2(y,x)$ handles all cases by checking the signs of $x$ and $y$.
• A parametric curve can cross itself. The curve $t\mapsto (\cos t,\sin 2t)$ passes through the same point at different $t$-values, but a function $y=f(x)$ cannot.

Key theorem with proof [Intermediate+]

Theorem (polar-rectangular equivalence). The map $(r,\theta )\mapsto (r\cos \theta ,r\sin \theta )$ is a surjection from $[0,\infty )\times [0,2\pi )$ onto ${\mathbb{R}}^2$, and is injective on $(0,\infty )\times [0,2\pi )$. Every point $(x,y)\ne (0,0)$ has a unique polar representation with $r>0$ and $0\le \theta <2\pi$.

Proof. Surjectivity: for any $(x,y)\in {\mathbb{R}}^2$, set $r=\sqrt{x^2+y^2}$ and $\theta =atan2(y,x)$ adjusted to $[0,2\pi )$. Then $r\cos \theta =\sqrt{x^2+y^2}\cdot x/\sqrt{x^2+y^2}=x$ and $r\sin \theta =y$, verifying $(r,\theta )\mapsto (x,y)$.

Injectivity on $(0,\infty )\times [0,2\pi )$: suppose $(r_1,{\theta }_1)$ and $(r_2,{\theta }_2)$ in this domain map to the same $(x,y)$. Then $r_1=\sqrt{x^2+y^2}=r_2$. If $r_1=r_2>0$, then $\cos {\theta }_1=x/r_1=\cos {\theta }_2$ and $\sin {\theta }_1=y/r_1=\sin {\theta }_2$. Two angles in $[0,2\pi )$ with the same sine and cosine are equal. The origin $(r=0)$ has all $\theta$ mapping to $(0,0)$, so injectivity fails only at $r=0$. $\square$

Bridge. The polar-rectangular conversion builds toward 00.10.01 where the polar form $r=l/(1+e\cos \theta )$ unifies the conic sections, and appears again in 00.11.02 where parametric descriptions of conics use the angular parameter. The foundational reason polar coordinates work is that the pair $(\cos \theta ,\sin \theta )$ is a unit vector, and this is exactly the content from 00.07.01 that identifies the trigonometric functions with the coordinates of a point on the unit circle. The central insight is that $(r,\theta )$ encodes the same information as $(x,y)$, and putting these together, the conversion formulas are simply the multiplication of a unit direction vector by a scalar length. The pattern generalises to cylindrical and spherical coordinates in three dimensions, and the bridge is between the rectangular and angular viewpoints on the plane.

Exercises [Intermediate+]

Advanced results [Master]

Theorem 1 (polar form of conics). Every conic section with eccentricity $e$, one focus at the origin, and directrix $x=l/e$ to the right of the origin has polar equation

$$r=\frac{l}{1+e\cos \theta },$$

where $l$ is the semi-latus rectum ^{[Apostol — Calculus Vol. 1]}. The three types correspond to $e<1$ (ellipse), $e=1$ (parabola), $e>1$ (hyperbola).

Theorem 2 (area in polar coordinates). The area enclosed by the polar curve $r=f(\theta )$ for $\alpha \le \theta \le \beta$ is

$$A=\frac{1}{2}{\int }_{\alpha }^{\beta }[f(\theta ){]}^{2}d\theta .$$

Theorem 3 (arc length in polar coordinates). The arc length of the polar curve $r=f(\theta )$ for $\alpha \le \theta \le \beta$ is

$$L={\int }_{\alpha }^{\beta }\sqrt{r^2+{\left(\frac{dr}{d\theta }\right)}^2}d\theta .$$

Theorem 4 (the cycloid). The curve traced by a point on the rim of a circle of radius $a$ rolling along the $x$-axis has parametric equations $x=a(\theta -\sin \theta )$, $y=a(1-\cos \theta )$. The cycloid is the solution to the brachistochrone problem (Johann Bernoulli 1696): the curve of fastest descent between two points under gravity.

Theorem 5 (rose curves). The polar curve $r=a\cos (n\theta )$ or $r=a\sin (n\theta )$ has $n$ petals if $n$ is odd and $2n$ petals if $n$ is even. Each petal has length $a$.

Theorem 6 (lemniscate of Bernoulli). The polar curve $r^2=a^2\cos (2\theta )$ is a figure-eight curve, the first curve described by its polar equation (Bernoulli 1694). Its total area is $a^2$.

Theorem 7 (polar-rectangular equivalence as a coordinate chart). The map $\Phi :(0,\infty )\times (0,2\pi )\to {\mathbb{R}}^2\setminus \{(x,0):x\ge 0\}$ given by $\Phi (r,\theta )=(r\cos \theta ,r\sin \theta )$ is a diffeomorphism onto its image, with Jacobian determinant $r$. The factor $r$ in the Jacobian is the foundational reason the area formula picks up a factor $rdrd\theta$ in double integrals.

Synthesis. The foundational reason polar coordinates are effective is that the angular coordinate $\theta$ is the natural parameter for curves with radial symmetry. The central insight is that the conversion $(r,\theta )\mapsto (r\cos \theta ,r\sin \theta )$ is exactly the polar decomposition of a vector into magnitude and direction, and the bridge is between the algebraic operations on coordinates and the geometric operations of rotation and scaling. This is exactly the content from 00.07.01 where the point $(\cos \theta ,\sin \theta )$ on the unit circle represents a direction. Putting these together, the area formula $A=\frac{1}{2}\int r^{2}d\theta$ identifies the area element in polar coordinates with the Jacobian factor $r$, the arc-length formula generalises the Pythagorean theorem from 00.09.01 to curved paths, and the polar form of conics $r=l/(1+e\cos \theta )$ unifies the three types from 00.10.01 into a single equation. The pattern recurs throughout calculus: the cycloid, the lemniscate, and the rose curves are the canonical polar curves that motivate multivariable integration and the change-of-variables theorem.

Full proof set [Master]

Proposition 1 (area in polar coordinates). The area enclosed by $r=f(\theta )$, $\alpha \le \theta \le \beta$, is $A=\frac{1}{2}{\int }_{\alpha }^{\beta }f(\theta {)}^{2}d\theta$.

Proof. Subdivide $[\alpha ,\beta ]$ into $n$ subintervals of width $\Delta \theta$. The sector of the circle of radius $r=f({\theta }_i^{\ast })$ between angles ${\theta }_i$ and ${\theta }_{i+1}$ has area $\frac{1}{2}f({\theta }_i^{\ast }{)}^2\Delta \theta$. Summing and taking the limit gives the Riemann integral $\frac{1}{2}{\int }_{\alpha }^{\beta }f(\theta {)}^{2}d\theta$. $\square$

Proposition 2 (arc length in polar coordinates). The arc length of $r=f(\theta )$ is $L={\int }_{\alpha }^{\beta }\sqrt{f(\theta {)}^2+f^{\prime }(\theta {)}^2}d\theta$.

Proof. The parametric form is $x(\theta )=f(\theta )\cos \theta$, $y(\theta )=f(\theta )\sin \theta$. Compute $dx/d\theta =f^{\prime }\cos \theta -f\sin \theta$ and $dy/d\theta =f^{\prime }\sin \theta +f\cos \theta$. Then

$${\left(\frac{dx}{d\theta }\right)}^2+{\left(\frac{dy}{d\theta }\right)}^2=f^{\prime 2}{\cos }^2\theta -2f{f}^{\prime }\cos \theta \sin \theta +f^2{\sin }^2\theta +f^{\prime 2}{\sin }^2\theta +2f{f}^{\prime }\sin \theta \cos \theta +f^2{\cos }^2\theta =f^{\prime 2}+f^2.$$

Taking the square root and integrating gives $L={\int }_{\alpha }^{\beta }\sqrt{f^2+f^{\prime 2}}d\theta$. $\square$

Proposition 3 (polar form of conics). A conic with focus at the origin, eccentricity $e$, and directrix $x=d$ (with $d>0$) has polar equation $r=ed/(1+e\cos \theta )$.

Proof. The focus-directrix condition is $d(P,F)=e\cdot d(P,\ell )$. With $F$ at the origin and the directrix the line $x=d$, a point $P=(r\cos \theta ,r\sin \theta )$ satisfies $r=e|d-r\cos \theta |$. For points with $r\cos \theta <d$: $r=e(d-r\cos \theta )$, so $r(1+e\cos \theta )=ed$, giving $r=ed/(1+e\cos \theta )$. $\square$

Connections [Master]

• Conic sections 00.10.01. The polar form $r=l/(1+e\cos \theta )$ unifies the three conic types from that unit into a single equation parametrised by eccentricity $e$. The polar-rectangular conversion is the tool that transforms the Cartesian standard forms (ellipse, parabola, hyperbola) into their polar counterparts. The classification by eccentricity is the same classification that governs the focus-directrix definition.

• Unit-circle trigonometry 00.07.01. The polar-rectangular conversion $x=r\cos \theta$, $y=r\sin \theta$ is the content from that unit scaled by $r$: the point $(\cos \theta ,\sin \theta )$ on the unit circle is the direction, and $r$ is the magnitude. The periodicity and parity of $\sin$ and $\cos$ govern the symmetry of polar curves.

• Cartesian coordinates and distance 00.09.01. The arc-length formula in polar coordinates generalises the distance formula from that unit: for a straight line $r=c\sec \theta$ the arc-length integral recovers the linear distance, and the area formula $A=\frac{1}{2}\int r^{2}d\theta$ is the polar counterpart of the Cartesian area formulas that build on the Cartesian grid.

Historical & philosophical context [Master]

Isaac Newton 1671 Method of Fluxions ^{[Newton 1671]} introduced polar coordinates as a systematic tool for describing curves, though the idea of specifying points by distance and angle predates him in the work of Ptolemy on the stereographic projection. Jakob Bernoulli 1691, in a series of papers in Acta Eruditorum, developed the systematic use of polar coordinates for studying curves, including the lemniscate ($r^2=a^2\cos (2\theta )$) and the logarithmic spiral ($r=e^{a\theta }$) ^{[Bernoulli 1691]}. Johann Bernoulli 1696 posed the brachistochrone problem, whose solution — the cycloid — demonstrated the power of parametric curve descriptions. The area and arc-length formulas in polar coordinates were developed by Euler in his 1748 Introductio in analysin infinitorum, which gave the first comprehensive treatment of polar curves in the modern analytical framework.

Bibliography [Master]

@book{Newton1671,
  author = {Newton, Isaac},
  title = {Method of Fluxions},
  note = {Written 1671; published posthumously 1736 by Woodburn},
  year = {1671}
}

@article{Bernoulli1691,
  author = {Bernoulli, Jakob},
  title = {Specimen calculi differentialis in dimensione paraboloidis et ellipsoidis},
  journal = {Acta Eruditorum},
  year = {1691}
}

@book{Euler1748,
  author = {Euler, Leonhard},
  title = {Introductio in analysin infinitorum},
  publisher = {Bousquet},
  address = {Lausanne},
  year = {1748}
}

@book{Lang1988,
  author = {Lang, Serge},
  title = {Basic Mathematics},
  publisher = {Springer},
  year = {1988}
}