curvature

normal curvature  of a surface S at a point p measures its curvature in a specific direction x in the tangent plane, and is defined in terms of the second fundamental form.

principal curvature  the maximum and minimum of the normal curvature  and  at a given point  p on a surface S

radial curvature  the normal curvature of the surface S in the direction of w definded as the (unnormalized) projection of the view vector v onto the tangent plane at p.

G1 Continuity - direction of first derivatives of two surfaces match where they join.
G2 Continuity - G1 plus the radius of curvature matches.
G3 Continuity - G2 plus the change in radius of curvature matches.