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, 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.