Method 1.
You can draw a circle with logarithmic curves accounting for infinitives.

Method 2
you can draw a circle with curved vertex's derived from quadratic equations.

Method 3.
You can draw circles with fractal/algorithmic pixel referencing.

Method 4.
You can draw a circle on a Cartesian graph with the circular equation x^2 + y ^ 2 = r ^ 2

Method 5.
You can derive a circle on a complex plane as I reaches infinity.

Method 6.
You can draw a circle with basic trigonometry.

When expressing curves things get quite interesting.

Method 1.
On a Cartesian graph you can simply change the radius to obtain various different curves.

Method 2.
You can derive curves with logarithms and infinitives.

Method 3.
In infinity space there are many cone which can be found which can be used to derive curves.

method 4.
you can derive with trigonometric functions for both parabolic and circular curvature compounded.

method 5.
You can derive curves with curved triangular slices of a sphere (this method can be very efficient).

Method 6.
You can derive curves with n-sphere exponential functions.

method 7.
I've never seen this done but you should be able to derive curves with N-Sphere logarithms.

method 8.
you can derive curves with curved vertex quadratics.

method 9.
You can use conic approximations.

method 10.
I've never seen this done but you could use logarithms and exponential derived from conic spirals.

method 11.
fractal maths can be used to derive curves.

method 12.
voxel's can be made to work more efficiently to exhibit better curvature. .

each method produces curves which can approximately mimic each others advanced curve paths. However circulated functions as opposed basic square and compass derived functions do seem to exhibit a more natural look dependent on the method used.