differential geometry Differential geometry is a Mathematics, mathematical discipline that studies the geometry of smooth shapes and smooth spaces, otherwise known as smooth manifolds. It uses the techniques of Calculus, single variable calculus, vector calculus, lin ...

, Meusnier's theorem states that all

curve In mathematics, a curve (also called a curved line in older texts) is an object similar to a line, but that does not have to be straight. Intuitively, a curve may be thought of as the trace left by a moving point. This is the definition that ...

s on a

surface A surface, as the term is most generally used, is the outermost or uppermost layer of a physical object or space. It is the portion or region of the object that can first be perceived by an observer using the senses of sight and touch, and is ...

passing through a given point ''p'' and having the same

tangent line In geometry, the tangent line (or simply tangent) to a plane curve at a given point is, intuitively, the straight line that "just touches" the curve at that point. Leibniz defined it as the line through a pair of infinitely close points o ...

at ''p'' also have the same

normal curvature In the differential geometry of surface (differential geometry), surfaces, a Darboux frame is a natural moving frame constructed on a surface. It is the analog of the Frenet–Serret formulas, Frenet–Serret frame as applied to surface geo ...

at ''p'' and their

osculating circle An osculating circle is a circle that best approximates the curvature of a curve at a specific point. It is tangent to the curve at that point and has the same curvature as the curve at that point. The osculating circle provides a way to unders ...

s form a sphere. The theorem was first announced by

Jean Baptiste Meusnier Jean Baptiste Marie Charles Meusnier de la Place (Tours, 19 June 1754 — le Pont de Cassel, near Mainz, 13 June 1793) was a French mathematician, engineer and Revolutionary general. He is best known for Meusnier's theorem on the curvature o ...

in 1776, but not published until 1785. At least prior to 1912, several writers in English were in the habit of calling the result ''Meunier's theorem'', although there is no evidence that Meusnier himself ever spelt his name in this way.R. C. Archibald
Query 76
''Mathematical Gazette'', 6 (May, 1912), p. 297 This alternative spelling of Meusnier's name also appears on the

Arc de Triomphe The Arc de Triomphe de l'Étoile, often called simply the Arc de Triomphe, is one of the most famous monuments in Paris, France, standing at the western end of the Champs-Élysées at the centre of Place Charles de Gaulle, formerly named Plac ...

Paris Paris () is the Capital city, capital and List of communes in France with over 20,000 inhabitants, largest city of France. With an estimated population of 2,048,472 residents in January 2025 in an area of more than , Paris is the List of ci ...

References

Further references

Meusnier's theorem Johannes Kepler University Linz, Institute for Applied GeometryMeusnier's theorem in Springer Online
* Theorems in differential geometry {{differential-geometry-stub