The easiest example I can think of for mental visualisation purposes is a circular cone at the bottom and a triangle at the top.
As you shift closer to the triangular tip either side of the circle shifts with a straightening aspect to a elliptoid along with it's other aspects until by the time it reaches the tip it is a straight line that has folded to a point. also the elliptoid becomes more hyperbolic as then it phases to straight.
I think this type of geometry is real however in maths it could lead to faster function's like working with the exponent of an N-sphere fast or maybe it could assist in modelling the output better of complex iterative functions and other functions.
Humanity could move from a nice little range of curve dynamic all the way to modelling some serious curved spaces with some serious degrees of data dimension and with serious mathematical efficiency.