I'm still hacking away at the baseball baserunner's problem, where paths/curves in 2D are produced by constant absolute value accelerations. The paths/curves in _velocity_ space are portions of catenaries (y=cosh(x), etc.), but the paths/curves in _position_ space aren't so easy to characterize. If s=asinh(t) (t time), then t=sinh(s) [sinh() is bijective]. acceleration a=[sech(s),tanh(s)] velocity v=[s-vx,cosh(s)-vy] (vx, vy are constants.) position z=[(s-vx)*sinh(s)-cosh(s),((cosh(s)-vy)*sinh(s)+s-vy*sinh(s))/2] Thus, z=[x,y] traces out a curve in 2D space parameterized by s (or alternatively t). Normally, we prefer a parameterized curve to a curve implicitly defined by a function f(x,y)=0, but sometimes it is nice to be able to see the curve w/o the extraneous parameter. If the parameterization utilized only rational functions, then it would be relatively easy to compute the implicit function using resultants, but the parameterization above goes beyond rational functions. I don't expect that the implicit function f(x,y) that I'm looking for is a polynomial, but I'd be interested if there exists an f(x,y) constructed from elementary functions that defines the same curve. Is there any straight-forward way to find such a function or prove that it doesn't exist?