Are algorithms already known, that generate (arbitrarily good approximations of) random curves, w.l.o.g. with unit length, and joining endpoints $(0,0)$ and $(\alpha,0)$ with $\alpha \lt1$ given?

The fixed distance between the endpoints is essential for the question, because otherwise a simple rescaling of an arbitrary curve would work.

edit

In view of the comments and the answer of Bjørn Kjos-Hanssen, I see the need for some clarification:

• By random curve of unit length connecting $(0,0)$ and $(\alpha,0)$, I mean a random sample from the space of all such curves; that means, that the algorithm should be capable to approximate every such curve to arbitrary precision with a finite number of steps.
  So "random" is not restricted to the appearance of the curve.

• Being able to generate Brownian Bridges is not sufficient, because I would like the algorithm to be able to generate curves (ideally in any $\mathbb{R}^n$) and not only functions.

So my apologies for not being precise enough.

I have used the formulation "are algorithms already known", because I have found one, that seems to be able to produce all those curves.

I will provide details in a later edit.