osculating circle vs circle of curvature

osculating circle and circle of curvature both are nouns.

Word	Noun	Adjective	Verb	Adverb
osculating circle	Yes	No	No	No
circle of curvature	Yes	No	No	No

As nouns, osculating circle and circle of curvature are synonyms defined as:

osculating circle and circle of curvature: the circle that touches a curve (on the concave side) and whose radius is the radius of curvature

osculating circle (noun)	circle of curvature (noun)
the circle that touches a curve (on the concave side) and whose radius is the radius of curvature	the circle that touches a curve (on the concave side) and whose radius is the radius of curvature