Dupin's theorem

Orthogonal surfaces through a point

Two planes (purple, blue) cut out the lines of curvature in a point (red) as elements of a threefold orthogonal system from a cylinder

The set of Dupin , named after the French mathematician Charles Dupin , is in differential geometry , the statement

• The surfaces of a triple orthogonal system intersect in pairs in lines of curvature .

A triple orthogonal system is understood to mean three sets of surfaces , in which each surface of one set intersects orthogonally with each surface of the other sets .

The simplest example of a three-fold orthogonal system are the coordinate planes and the planes parallel to them. This example is of no interest here because a plane has no lines of curvature.

A simple example with at least one set of curved surfaces: 1) all circular cylinders with the z-axis as the axis, 2) all planes that contain the z-axis and 3) all horizontal planes.

A line of curvature of a surface is a curve on the surface, the direction of which is a main direction of curvature (maximum or minimum curvature) at each point. In the case of a vertical circular cylinder, these are the horizontal circles and the straight lines (generators) of the cylinder. The cylinder has its maximum curvature along a horizontal circle; along a generating line the curvature is minimal, namely zero. A plane has no lines of curvature, since every normal curvature is zero at every point. In the above example it only makes sense to consider sections of the cylinders with a plane from the given families. In one case (horizontal planes) there are the circles of the cylinder and in the other case the generators.

The idea of ​​the triple orthogonal systems of surfaces can be seen as a generalization of the planar concept of orthogonal trajectory. The confocal conic sections are special orthogonal systems of plane curves .

application

Dupin's theorem makes it possible to describe the lines of curvature through a point on a surface as intersection curves with two suitable surfaces. So without having to laboriously calculate main curvatures. The next example shows that the embedding of a surface in an orthogonal system is not unique

Examples

cone

Given: A cone, green in the picture.
Wanted: the lines of curvature.

Orthogonal system (purple, green, blue) of surfaces for a cone (green), lines of curvature: green, red

1. Flock : By moving the given cone K (tip S) in the direction of its axis, a surface array is created that contains the cone K itself (in the picture: green).
2. Family : cones with points on the given axis of the cone, whose generatrices are perpendicular to the generators of K (blue).
3rd group : The levels through the cone axis (purple).

These three groups of surfaces form an orthogonal system. The blue cones cut out circles (red) from the cone K. The purple layers cut out the generators (green).

Alternative with balls

The points of space are described by the usual spherical coordinates . It is S = M = zero point. ${\displaystyle (r,\varphi ,\theta )}$

1.Schar: All cone with vertex S and the axis of the given cone K (Green) . 2nd character: All spheres with center M = S (blue): 3rd character : All levels through the cone axis (purple): ( ). ${\displaystyle (r,\varphi ,\theta _{0})}$
${\displaystyle (r_{0},\varphi ,\theta )}$
${\displaystyle (r,\varphi _{0},\theta )}$

Torus

Orthogonal system (purple, green, blue) of surfaces for a torus (green), lines of curvature: green, red

1.Schar : Tori with the same guide circle (green).
2.Schar : cone through the guide circle of the torus with the tip on the torus axis (blue).
3rd character : planes through the torus axis (purple).

The blue cones cut out the horizontal circles (red). The purple layers cut out the vertical circles (green). The lines of curvature of a torus thus form a network of orthogonally intersecting circles.

Surface of revolution

Orthogonal system to a surface of revolution (green)

A surface of revolution is usually given by a meridian . The rotation surface is created by rotating the meridian around the axis of rotation. The method for a cone and a torus can also be used here. ${\displaystyle m_{0}}$

1.Schar : parallel surfaces to the given surface of revolution .
2. Family : cones with points on the axis of rotation, the generators of which are perpendicular to the surface of revolution (blue).
3rd family : planes through the axis of rotation (purple).

The cones cut out the horizontal circle (red). The purple levels cut out the meridians (green). So:

• The lines of curvature of a surface of revolution are the meridians and the circles perpendicular to the axis of rotation.

Confocal Quadrics

Ellipsoid with lines of curvature

Hyperboloid with lines of curvature

In the article on confocal conics be confocal quadrics explained. They form a threefold orthogonal system. According to Dupin's theorem, the lines of curvature on each of the quadrics involved can be understood as intersection curves with the other quadrics (see picture), which means that they are much easier to visualize and examine. Confocal quadrics are always 3-axis , so no surfaces of revolution . The lines of curvature are thus fourth degree curves. (In the case of squares of revolution, the lines of curvature are conic sections (see above).)

Ellipsoid (see picture)

Semi-axes: . The lines of curvature are sections with single- and double-shell hyperboloids. The red points are umbilical points . (For an ellipsoid of revolution, the lines of curvature are circles and ellipses, see section Surface of revolution.) ${\displaystyle \;a=1,\;b=0{,}8,\;c=0{,}6\;}$

single-shell hyperboloid (see picture)

Semi-axes: . The lines of curvature are sections with ellipsoids (blue) and double-shell hyperboloids (purple). ${\displaystyle \;a=0{,}67,\;b=0{,}3,\;c=0{,}44\;}$

Dupin's cyclides

Ring cyclides with their focal conic sections (dark red: ellipse, dark blue: hyperbola). Purple: surface normal and common straight line of the cone at point P.

A Dupin's cyclid and its parallel surfaces are defined by a pair of focal conics . The picture shows a ring cyclide with its focal conic sections (ellipse: dark red, hyperbola: dark blue). The cyclides can be understood as an element of an orthogonal system:

1st set : parallel surfaces of the cyclides.
2nd set: vertical circular cones through the ellipse (their tips lie on the hyperbola)
3rd set: vertical circular cones through the hyperbola (their tips lie on the ellipse)

The two circular cones through the point P intersect orthogonally in the surface normal (purple). Each cone cuts the cyclide in a circle (red or blue)

The special property of a cyclide is:

The lines of curvature of a Dupin's cyclide are circles .

For proof

It is only the point set of which is of interest for which a surface from each group of surfaces passes through each point. If the family parameters are, these three numbers can be thought of as new coordinates. So each point can be described as follows: ${\displaystyle \mathbb {R} ^{3}}$ ${\displaystyle u,v,w}$

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

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

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

In the example above: The cylinders are described by the respective radius , the vertical planes by the angle with the x-axis and the horizontal planes by their z-height . can be thought of as the cylindrical coordinates of a point. ${\displaystyle r}$ ${\displaystyle \varphi }$ ${\displaystyle \zeta }$ ${\displaystyle r,\varphi ,\zeta }$

So that the three surfaces through this point intersect perpendicularly at one point , the three surface normals must be orthogonal in pairs there. This is the case if and only if ${\displaystyle {\vec {x}}(u,v,w)}$ ${\displaystyle \;{\vec {x}}_{u}\times {\vec {x}}_{v},\;{\vec {x}}_{v}\times {\vec {x}}_{w},\;{\vec {x}}_{w}\times {\vec {x}}_{u}}$

${\displaystyle {\vec {x}}_{u},\;{\vec {x}}_{v},\;{\vec {x}}_{w}\quad }$ are orthogonal in pairs.

(This is checked with the help of the Lagrange identity and the permissible simplification that the tangent vectors are normalized beforehand (length 1)).

That means it must apply

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

To prove the theorem, one derives these equations further according to the parameter missing in the derivatives. The first after , the second after and the third after : ${\displaystyle w}$ ${\displaystyle u}$ ${\displaystyle v}$

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

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

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

Solving this linear system of equations for the three occurring scalar products results in:

(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) it follows: the three vectors are perpendicular to the vector and are therefore linearly dependent (lie in one plane). Ie: ${\displaystyle {\vec {x}}_{u},{\vec {x}}_{v},{\vec {x}}_{uv}}$ ${\displaystyle {\vec {x}}_{w}}$

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

For the coefficient of the first fundamental form or the second fundamental form of the surface and its parameter lines (= intersection curves with surfaces of the other families) it follows from (1) and (3) : ${\displaystyle F}$ ${\displaystyle M}$ ${\displaystyle {\vec {x}}={\vec {x}}(u,v,w_{0})}$

${\displaystyle F=0\;,\quad M=0\;.}$

As a consequence:

The parameter lines are lines of curvature.

The same result applies to the other surfaces.

literature

1. ↑ W. Blaschke: Lectures on differential geometry 1 , Springer-Verlag, 1921, p. 63

• HSM Coxeter : Introduction to geometry , Wiley, 1961, pp. 11, 258.
• Ch.Dupin: Développements de géométrie , Paris 1813.
• F. Klein : Lectures on Higher Geometry , Springer-Verlag, 2013, ISBN 3642886744 , p. 9.
• Ludwig Schläfli : About the most general family of surfaces of the second degree, which forms an orthogonal system with any two other families of surfaces , in L. Schläfli: Collected mathematical treatises p. 163, Springer-Verlag, 2013, ISBN 3034841167 .
• F. Schleicher: Paperback for civil engineers: first volume , Springer-Verlag, 2013, ISBN 3642883486 , p. 149.
• J. Weingarten : About the condition under which a surface family belongs to an orthogonal surface system. , Journal for pure and applied mathematics (Crelles Journal), Volume 1877, Issue 83, pages 1-12, ISSN (Online) 1435-5345, ISSN (Print) 0075-4102.
• TJ Willmore: An Introduction to Differential Geometry , Courier Corporation, 2013, ISBN 0486282104 , p. 295.