>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.  
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**edit**  

In view of the comments and the answer of [Bjørn Kjos-Hanssen](https://mathoverflow.net/users/4600/bj%c3%b8rn-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.  
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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.