What is polar curve? — Mathematics

Definition 24.6 University Mathematics — Year 1 · Chapter 24 — Plane Curves

A polar curve is given by $r = r(\theta)$ : the point of parameter $\theta$ is

M(\theta) = r(\theta)\,\vec u(\theta), \qquad \vec u(\theta) = (\cos\theta, \sin\theta).

With $\vec v(\theta) = (-\sin\theta, \cos\theta) = \vec u\,'(\theta)$ , the velocity is

M'(\theta) = r'(\theta)\, \vec u(\theta) + r(\theta)\, \vec v(\theta) .

Consequences: where $r \neq 0$ , the point is regular, and the tangent makes with the ray the angle $V$ given by $\tan V = \frac{r}{r'}$ (angle between $M'$ and $\vec u$ ); where $r(\theta_0) = 0$ , the curve passes through the origin with tangent the ray $\theta = \theta_0$ (direction $\vec u(\theta_0)$ , read off $M'(\theta_0) = r'(\theta_0)\vec u(\theta_0)$ , or from the limit chord: the chord from $O$ to $M(\theta)$ is carried by $\vec u(\theta)$ itself, which tends to $\vec u(\theta_0)$ — so the rule holds even when $r'(\theta_0) = 0$ and the point is singular, which is why polar cusps at the origin, like the cardioid’s, get their tangent for free).

The four-petaled rose r = 2 (). The petals along the y-axis are traced with r < 0 (the point plots on the ray opposite to ); the dashed diagonals = ± π4 are the tangents at the origin, where r vanishes. — The four-petaled rose $r = \cos 2\theta$ (Exercise 24.4). The petals along the $y$ -axis are traced with $r < 0$ (the point plots on the ray opposite to $\theta$ ); the dashed diagonals $\theta = \pm\frac\pi4$ are the tangents at the origin, where $r$ vanishes.

The cardioid r = 1 +. Polar curves are read by sweeping the angle: the radius swells and shrinks as turns. — The cardioid $r = 1 + \cos\theta$ . Polar curves are read by sweeping the angle: the radius swells and shrinks as $\theta$ turns.

Examples

Example 24.7 (The cardioid)

$r(\theta) = 1 + \cos\theta$ . Symmetry: $r(-\theta) = r(\theta)$ : reflection in the $x$ -axis; study $\theta \in \intcc{0}{\pi}$ . $r$ decreases from $2$ to $0$ ; at $\theta = \pi$ , $r = 0$ : the curve reaches the origin tangentially to the ray $\theta = \pi$ (the $x$ -axis), forming a cusp there — the heart’s point. Tangent at $\theta = 0$ : $r' = 0$ , so $\tan V = \infty$ : perpendicular to the axis.

The angle $V$ deserves one more reading. At $\theta = \frac\pi2$ : $r = 1$ and $r' = -1$ , so $\tan V = \frac{r}{r'} = -1$ : the tangent makes three-quarters of a right angle with the outgoing ray — the curve is already bending back toward its cusp. The formula $\tan V = r/r'$ delivers tangent directions along the whole curve with no computation of $M'(\theta)$ whatsoever: it is the polar analogue of reading a slope.

Example 24.8 (Polar to cartesian: a hidden circle)

What is the polar curve $r = 2\cos\theta$ ? Multiply by $r$ : $r^2 = 2r\cos\theta$ , i.e. $x^2 + y^2 = 2x$ , i.e.

(x - 1)^2 + y^2 = 1 :

the circle of center $(1, 0)$ and radius $1$ , passing through the origin. Bookkeeping matters: as $\theta$ runs over $\intoc{-\frac\pi2}{\frac\pi2}$ the whole circle is swept exactly once ( $r$ vanishes at both ends), and at $\theta = \pm\frac\pi2$ the rule “tangent at the origin along the ray $\theta = \theta_0$ ” gives a vertical tangent there — matching the geometry, since the vertical axis is indeed tangent to this circle at $O$ . For $\theta$ beyond that range, $r < 0$ retraces the same circle: a reminder that a polar curve is a parametrized object, allowed to pass over itself.

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