
# 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.

## 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:

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