2.2 Parametric Playground

Parametric curves earn their keep in the plane long before they need a third dimension. The pair $x = f (t)$ , $y = g (t)$ describes a motion, and curves that are awkward or impossible to write as $y = F (x)$ often have short, transparent parametric stories. This section is a playground with two classic inhabitants, each carrying a slider so you can interrogate the formula by moving it.

First, the cycloid: the path traced by a pebble stuck to the rim of a wheel of radius $a$ rolling along a line. Rolling means the wheel's rotation and its forward travel are locked together — after turning through angle $t$ , the axle has advanced exactly $a t$ . Subtracting the pebble's position on the rim from the axle's motion gives

$x = a (t - \sin t), y = a (1 - \cos t) .$

A thick red curve of three identical arches resting on a dashed gray horizontal line, like the path traced by a pebble stuck to a rolling wheel: each arch rises smoothly from a sharp downward point (a cusp) on the line, crests, and returns to the next cusp. A slider named a sets the rolling wheel's radius: dragging it up makes the arches both taller and wider in the same proportion, and dragging it down shrinks them, while the number of arches shown stays three. — Three arches of the cycloid $(a (t - \sin t), a (1 - \cos t))$ over a dashed baseline. Drag the slider for $a$ — the wheel's radius — and watch the arches scale. Orbit to look straight down for the cleanest view.

Explore the cycloid

Each arch ends in a sharp point — a cusp — where the curve touches the baseline. Physically, the pebble is at the bottom of the wheel there. At which parameter values $t$ does that happen, and why does the motion momentarily stall?
Drag $a$ from 0.5 to 2. Height and width scale together — read the arch height and the arch width off the grid as multiples of $a$ . (You should find height $2 a$ and width $2 π a$ .)
The curve never dips below the baseline and never doubles back in $x$ . Which parts of the two formulas guarantee each fact?

Second, the polar rose. A polar curve $r = g (θ)$ is secretly parametric: since $x = r \cos θ$ and $y = r \sin θ$ , the graph is the parametric curve

$x = g (t) \cos t, y = g (t) \sin t,$

with the polar angle as parameter. Take $g (t) = \cos (b t)$ for a whole number $b$ and you get a rose.

A violet rose-shaped curve of identical petals radiating from the origin, drawn flat in the plane and inscribed in a dashed gray unit circle, viewed from almost straight overhead. A slider named b sets the frequency in r = cos(b theta): odd values of b draw exactly b petals, while even values draw twice as many, 2b, which is the parity surprise the scene is built to expose. — The rose $r = \cos (b θ)$ , drawn inside a dashed unit circle. The slider sets the frequency $b$ from 1 to 6 in whole-number steps. Count petals at each value before you trust any rule you have heard.

Explore the rose

Set $b equals 1$ and identify the curve (it is a familiar shape in disguise — a circle of radius one half through the origin). Then count petals at $b equals 2, 3, 4, 5, 6$ and record the counts.
The pattern splits by parity: odd $b$ gives $b$ petals, even $b$ gives $2 b$ . Watch the curve being traced near a value where $\cos (b t)$ is negative: a negative radius plots the point on the opposite side of the origin. For odd $b$ , where do those negative-radius petals land?
Every petal tip touches the dashed circle. Why must the whole rose live inside it?

Why parity decides the petal count

A petal is traced each time $\cos (b t)$ runs from 0 up to $± 1$ and back to 0, which happens $2 b$ times as $t$ crosses $[ 0, 2 π ]$ . When $b$ is even, those $2 b$ passes land in $2 b$ distinct directions: $2 b$ petals. When $b$ is odd, the negative-radius passes retrace petals already drawn on the opposite side — each petal is drawn exactly twice — leaving only $b$ visible. The slider makes this a two-minute experiment; the algebra behind it is the single identity that $(r, θ)$ and $(- r, θ + π)$ name the same point.

An original work of XYZ Homework, built around interactive XYZ 3D figures. Aligned to OpenStax Calculus Volume 3 (Strang & Herman), © OpenStax (Rice University), licensed CC BY-NC-SA 4.0; no OpenStax content is reproduced, and this work is not affiliated with or endorsed by OpenStax or Rice University. License: CC-BY-NC-SA-4.0.