3.2: Parametrized curves
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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.
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
\[\gamma (t) = (x, y) = (x_0, y_0) + t(x_1 - x_0, y_1 - y_0), \text{ with } 0 \le t \le 1. \nonumber \]
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:
\[\gamma (t) = (x, y) = (x_0, y_0) + r(\cos (t), \sin (t)), \text{ with } 0 \le t \le 2\pi. \nonumber \]