In differential geometry, 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 t ...

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 the straight line that "just touches" the curve at that point. Leibniz defined it as the line through a pair of infinitely close points on the curve. More ...

at ''p'' also have the same

normal curvature In the differential geometry of surfaces, a Darboux frame is a natural moving frame constructed on a surface. It is the analog of the Frenet–Serret frame as applied to surface geometry. A Darboux frame exists at any non-umbilic point of a ...

at ''p'' and their

osculating circle In differential geometry of curves, the osculating circle of a sufficiently smooth plane curve at a given point ''p'' on the curve has been traditionally defined as the circle passing through ''p'' and a pair of additional points on the curve ...

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 curvatur ...

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 (, , ; ) 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 Place de l'Étoile—the ''étoile'' ...

Paris Paris () is the capital and most populous city of France, with an estimated population of 2,165,423 residents in 2019 in an area of more than 105 km² (41 sq mi), making it the 30th most densely populated city in the world in 2020. ...

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