The simplest of all parametrized curves is a parametrization of an interval on the real line.

The simplest of all parametrized curves is a parametrization of the y-axis, given by a single function y(t). Simple examples of such curves are t->[0,t²] or t->[0,t³]. Such curves can be obtained by projecting a function graph, t->[t,y(t)], to the vertical axis.

Such a function is called a reparametrization if it is one-to-one, so either strictly increasing or strictly decreasing. If it is strictly increasing then y(t+h) > y(t) for all t and all h > 0, in which case we say that the curve preserves the orientation of the y-axis. If x(t+h)<x(t) for all t and all h > 0, then the function is strictly decreasing and we say that the curve reverses the orientation of the y-axis. Examples of orientation-preserving reparametrizations are y(t) = t³ + t defined for all real t, or y(t) = sin(t) defined over the interval 0<t<p , or x(t) = |t| + 2t defined for all real t. Any orientation-reversing reparametrization is just the negative of an orientation-preserving reparametrization, and conversely.

Demonstration 1: Line Curves

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This demo allows you to reparametrize the y-axis and watch it being traced out.

For which n does t->[0,tⁿ] give a reparametrization of the whole line.

For which intervals a≤t≤b does the function y=t² give a reparamtrization of the interval? When does the reparametrization preserve orientation? What about y(t)=sin(t)?

For which c does y(t)=t³+ct give a reparametrization of the line? What about y(t)=Abs(t)+ct?