Radius of Curvature?

"is it the largest radius that encompasses all of the curve?"
No.

"is it just the radius that encompasses most of the curve?"
No.

"lets say i have a parabola or an half-ellipse, these two have a radius of curvature right? "
No, a curve has different radii of curvatures at different points.
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For example, look at this: https://www.intmath.com/applications-differentiation/img/radiusCurvature1.png
The radius of curvature at the point marked on the blue curve is the radius of the ‘snugly fitting’ circle constructed at the point.

Imagine *any* smooth curve. If you pick a point, P, on the curve, you can always construct a circle that closely fits the curve ‘snugly’ at P. If you cut away a tiny (infinitesimal) section of the curve around the P, this tiny section of the curve would be part of a circle; the radius of this circle is the radius of curvature of the curve at P.

As a point of interest, the radius of curvature at any point on a straight line is infinity. (Any finite part of a circle of infinite radius is a straight line.)

A circle i just a special case of an elipse