Dupin's theorem

Dupin's theorem is a tool for determining the curvature lines of a surface by intersection with suitable surfaces (see examples), without time-consuming calculation of derivatives and principal curvatures. The next example shows, that the embedding of a surface into a threefold orthogonal system is not unique.

Any point of consideration is contained in exactly one surface of any pencil of the orthogonal system. The three parameters ${\displaystyle u,v,w}$ describing these three surfaces can be considered as new coordinates. Hence any point can be represented by:

${\displaystyle x=x(u,v,w)}$

${\displaystyle y=y(u,v,w)}$

${\displaystyle z=z(u,v,w)\quad }$ or shortly: ${\displaystyle \quad {\vec {x}}={\vec {x}}(u,v,w)\;.}$

For the example (cylinder) in the lead the new coordinates are the radius ${\displaystyle r}$ of the actual cylinder, angle ${\displaystyle \varphi }$ between the vertical plane and the x-axis and ${\displaystyle \zeta }$ the height of the horizontal plane. Hence, ${\displaystyle r,\varphi ,\zeta }$ can be considered as the cylinder coordinates of the point of consideration.

The condition "the surfaces intersect orthogonally" at point ${\displaystyle {\vec {x}}(u,v,w)}$ means, the surface normals ${\displaystyle \;{\vec {x}}_{u}\times {\vec {x}}_{v},\;{\vec {x}}_{v}\times {\vec {x}}_{w},\;{\vec {x}}_{w}\times {\vec {x}}_{u}}$ are pairwise orthogonal. This is true, if

${\displaystyle {\vec {x}}_{u},\;{\vec {x}}_{v},\;{\vec {x}}_{w}\quad }$ are pairwise orthogonal. This property can be checked with help of Lagrange's identity.

Hence

(1)${\displaystyle \quad {\vec {x}}_{u}\cdot {\vec {x}}_{v}=0,\quad {\vec {x}}_{v}\cdot {\vec {x}}_{w}=0,\quad {\vec {x}}_{w}\cdot {\vec {x}}_{u}=0\;.}$

Deriving these equations for the variable, which is not contained in the equation, one gets

${\displaystyle {\vec {x}}_{uw}\cdot {\vec {x}}_{v}+{\vec {x}}_{u}\cdot {\vec {x}}_{vw}=0\;,}$

${\displaystyle {\vec {x}}_{uv}\cdot {\vec {x}}_{w}+{\vec {x}}_{v}\cdot {\vec {x}}_{uw}=0\;,}$

${\displaystyle {\vec {x}}_{vw}\cdot {\vec {x}}_{u}+{\vec {x}}_{w}\cdot {\vec {x}}_{vw}=0\;.}$

Solving this linear system for the three appearing scalar products yields:

(2)${\displaystyle \quad {\vec {x}}_{uv}\cdot {\vec {x}}_{w}=0,\quad {\vec {x}}_{vw}\cdot {\vec {x}}_{u}=0,\quad {\vec {x}}_{uw}\cdot {\vec {x}}_{v}=0\;.}$

From (1) and (2): The three vectors ${\displaystyle {\vec {x}}_{u},{\vec {x}}_{v},{\vec {x}}_{uv}}$ are orthogonal to vector ${\displaystyle {\vec {x}}_{w}}$ and hence are linear dependent (are contained in a common plane), which can be expressed by:

(3)${\displaystyle \quad \det({\vec {x}}_{uv},{\vec {x}}_{u},{\vec {x}}_{v})=0\;.}$

From equation (1) one gets ${\displaystyle F=0}$ (coefficient of the first fundamental form) and
from equation (3): ${\displaystyle M=0}$ (coefficient of the second fundamental form) of the surface ${\displaystyle {\vec {x}}={\vec {x}}(u,v,w_{0})}$.

Consequence: The parameter curves are curvature lines.

The analogous result for the other two surfaces through point ${\displaystyle {\vec {x}}(u_{0},v_{0},w_{0})}$ is true, too.

• H.S.M. Coxeter: Introduction to geometry, Wiley, 1961, pp. 11, 258.
• Ch. Dupin: Développements de géométrie, Paris 1813.
• F. Klein: Vorlesungen über Höhere Geometrie, Springer-Verlag, 2013, ISBN 3642886744, p. 9.
• Ludwig Schläfli: Über die allgemeinste Flächenschar zweiten Grades, die mit irgend zwei anderen Flächenscharen ein orthogonales System bildet, in L. Schläfli: Gesammelte mathematische Abhandlungen p. 163, Springer-Verlag, 2013, ISBN 3034841167.
• J. Weingarten: Über die Bedingung, unter welcher eine Flächenfamilie einem orthogonalen Flächensystem angehört., Journal für die reine und angewandte Mathematik (Crelles Journal), Band 1877, Heft 83, pp. 1–12, ISSN (Online) 1435–5345, ISSN (Print) 0075–4102.
• T. J. Willmore: An Introduction to Differential Geometry, Courier Corporation, 2013, ISBN 0486282104, p. 295.