Parametric functions

Just out of interest, here are the standard functions we are used to, but converting the second differential of the function to curvature instead. Or equivalently, converting gradient to path angle:

x

x^2

x^3

x^4

tan x

1/x

exp x

log x

x^x

normal dist

8.5 * normal dist

But one of the more interesting is probably the sine function, which looks quite different just by scaling up:

sin x

pi/2 sin x

2.11 sin x

2.405 sin x

2.66 sin x

pi sin x

4.7 sin x

5.24 sin x

The coefficient in the figure eight sine wave is more precisely the first zero of the Bessel function J_0. Credit to Greg Egan for pointing that out.

You can also render the factorial function as a continuous curve. However to prevent lots of overlap it helps to divide the angle by four:

In addition to standard asymptotes, this depiction of functions can also in somes cases visualise functions with a domain larger than the Reals. 

For instance, if we take a standard cubic function:

and make it periodic every omega on the Surreal domain by setting:

where r_0 is the cofficient of omega^0 in the Conway normal form of x:

then we can visualise it as:

The green lines are values outside of the Reals.

This example is rather artificial as it explicitly creates an omega-periodic function. It is not clear whether any "standard" function on the Reals that is non-periodic can become periodic on the Surreals. It definitely isn't the case for polynomial functions.