3.2: Parametrized curves

We often use the Greek letter gamma for a parameterized curve, i.e.

\[\gamma (t) = (x(t), y(t)). \nonumber\]

We think of this as a moving point tracing out a curve in the plane. The tangent vector

\[\gamma '(t) = (x'(t), y'(t)) \nonumber\]

is tangent to the curve at the point \((x (t), y(t))\). It's length \(|\gamma '(t)|\) is the instantaneous speed of the moving point.

Figure \(\PageIndex{1}\): Parametrized curve \(\gamma (t)\) with some tangent vectors \(\gamma '(t)\). (CC BY-NC; Ümit Kaya)

Example \(\PageIndex{1}\)

Parametrize the straight line from the point \((x_0, y_0)\) to \((x_1, y_1)\).

Solution

There are always many parametrizations of a given curve. A standard one for straight lines is

\[\gamma (t) = (x, y) = (x_0, y_0) + t(x_1 - x_0, y_1 - y_0), \text{ with } 0 \le t \le 1. \nonumber\]

Example \(\PageIndex{2}\)

Parametrize the circle of radius \(r\) around the point \((x_0, y_0)\).

Solution

Again there are many parametrizations. Here is the standard one with the circle traversed in the counterclockwise direction:

\[\gamma (t) = (x, y) = (x_0, y_0) + r(\cos (t), \sin (t)), \text{ with } 0 \le t \le 2\pi. \nonumber\]

Figure \(\PageIndex{2}\): Line from \((x_0, y_0)\) to (\(x_1, y_1\)) and circle around \((x_0, y_0)\). (CC BY-NC; Ümit Kaya)