Parameterizing a path means representing the path as a function with a set of variables, usually denoted by t, that can take on different values to trace out the path.
Parameterizing a path allows for easier manipulation and calculation of the path, as well as the ability to generalize the path to different scenarios.
To parameterize a path, you need to express the coordinates of the path as a function of a parameter. For example, for a path in the xy-plane, the coordinates could be expressed as x(t) and y(t), where t is the parameter.
Some examples of parameterizing a path include expressing a straight line as x(t) = mt + b, where m is the slope and b is the y-intercept, or a circle as x(t) = rcos(t) and y(t) = rsin(t), where r is the radius.
Parameterizing a path is closely related to vector-valued functions, as the function that represents the path is usually a vector function. The vector function takes in the parameter and outputs a vector that corresponds to a point on the path.