Parametrically Defined Circles and Lines

Circles and lines have equations you already know: $(x - h)^2 + (y - k)^2 = r^2$ and $y = mx + b$. Parametric equations describe the same shapes as trips: where you start, which way you go, how fast, and where you stop. That extra information is what you need to model a wheel turning or a drone flying.

The unit circle, as a trip

On the unit circle the point at angle $t$ is $(\cos t, \sin t)$. So the parametric function $x(t) = \cos t$, $y(t) = \sin t$ for $0 \le t \le 2\pi$ starts at $(1, 0)$ when $t = 0$, passes $(0, 1)$ at $t = \frac{\pi}{2}$, $(-1, 0)$ at $t = \pi$, $(0, -1)$ at $t = \frac{3\pi}{2}$, and returns to $(1, 0)$: once around, counterclockwise. The parameter is the angle.

Any circle

Scale by the radius $r$, then shift the center to $(h, k)$:

Every point is $r$ units from $(h, k)$. To check, use the Pythagorean identity: $(x - h)^2$$+ (y - k)^2$$= r^2\cos^2 t$$+ r^2\sin^2 t$$= r^2$.

Changing the trip, not the circle

For the third row, check two points: $t = 0$ gives $(h, k + r)$, the top, and $t = \frac{\pi}{2}$ gives $(h + r, k)$, the right side. Top to right is clockwise.

Segments and lines

To go from $(x_1, y_1)$ to $(x_2, y_2)$ in a straight line, start at the first point and add a fraction $t$ of the change in each coordinate:

At $t = 0$ you are at the start, at $t = 1$ at the end, and at $t = \frac{1}{2}$ at the midpoint. Letting $t$ run over all real numbers gives the whole line. If the trip takes $T$ units of time instead of $1$, divide the change by $T$: $x(t)$$= x_1 + \frac{x_2 - x_1}{T}t$.

In general, $x(t) = x_0 + at$, $y(t) = y_0 + bt$ is a line through $(x_0, y_0)$ with slope $\frac{b}{a}$ when $a \ne 0$, and a vertical line when $a = 0$ and $b \ne 0$.