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 ...

, the Dupin indicatrix is a method for characterising the local shape of 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 ...

. Draw a

plane Plane most often refers to: * Aero- or airplane, a powered, fixed-wing aircraft * Plane (geometry), a flat, 2-dimensional surface * Plane (mathematics), generalizations of a geometrical plane Plane or planes may also refer to: Biology * Plane ...

parallel Parallel may refer to: Mathematics * Parallel (geometry), two lines in the Euclidean plane which never intersect * Parallel (operator), mathematical operation named after the composition of electrical resistance in parallel circuits Science a ...

to the

tangent plane 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 ...

and a small distance away from it. Consider the intersection of the surface with this plane. The shape of the intersection is related to the

Gaussian curvature In differential geometry, the Gaussian curvature or Gauss curvature of a smooth Surface (topology), surface in three-dimensional space at a point is the product of the principal curvatures, and , at the given point: K = \kappa_1 \kappa_2. For ...

. The Dupin indicatrix is the result of the limiting process as the plane approaches the tangent plane. The indicatrix was introduced by

Charles Dupin Baron Pierre Charles François Dupin (; 6 October 1784, Varzy, Nièvre – 18 January 1873, Paris, France) was a French Catholic mathematician, engineer, economist and politician, particularly known for work in the field of mathematics, where t ...

. Equivalently, one can construct the Dupin indicatrix at point ''p'', by first rotating and translating the surface, so that ''p'' is at origin, and the tangent plane is the ''xy''-plane. Now the

contour plot A contour line (also isoline, isopleth, isoquant or isarithm) of a function of two variables is a curve along which the function has a constant value, so that the curve joins points of equal value. It is a plane section of the three-dimensi ...

of the surface are the Dupin indicatrices. Asymmetricwave2

Classification

For elliptical points where the Gaussian curvature is positive the intersection will either be empty or form a closed curve. In the limit this curve will form an

ellipse In mathematics, an ellipse is a plane curve surrounding two focus (geometry), focal points, such that for all points on the curve, the sum of the two distances to the focal points is a constant. It generalizes a circle, which is the special ty ...

aligned with the principal directions. The curvature lines make up the major and minor axes of the ellipse. In particular, the indicatrix of an

umbilical point In the differential geometry of surfaces in three dimensions, umbilics or umbilical points are points on a surface that are locally spherical. At such points the normal curvatures in all directions are equal, hence, both principal curvatures are e ...

is a circle. For hyperbolic points, where the Gaussian curvature is negative, the intersection will form a

hyperbola In mathematics, a hyperbola is a type of smooth function, smooth plane curve, curve lying in a plane, defined by its geometric properties or by equations for which it is the solution set. A hyperbola has two pieces, called connected component ( ...

. Two different hyperbolas will be formed on either side of the tangent plane. These hyperbolas share the same axis and asymptotes. The directions of the asymptotes are the same as the

asymptotic direction In the differential geometry of surfaces, an asymptotic curve is a curve always tangent to an asymptotic direction of the surface (where they exist). It is sometimes called an asymptotic line, although it need not be a line. Definitions There are ...

s. In particular, the indicatrix of each point on a minimal surface is two lines intersecting at right angles, which each make a 45-degree angle with the two curvature lines. For parabolic points, where the Gaussian curvature is zero, the intersection will form two parallel lines. The direction of those two lines are the same as the

s. In particular, the indicatrix of each point on a

developable surface In mathematics, a developable surface (or torse: archaic) is a smooth surface with zero Gaussian curvature. That is, it is a surface that can be flattened onto a plane without distortion (i.e. it can be bent without stretching or compression). ...

is a pair of lines parallel to the

generatrix In geometry, a generatrix () or describent is a point, curve or surface that, when moved along a given path, generates a new shape. The path directing the motion of the generatrix motion is called a directrix or dirigent. Examples A cone can b ...

. For more complex cases where all the second-degree derivatives are zero, but higher-degree derivatives are nonzero, the Dupin indicatrix is more complex. For example, the

monkey saddle In mathematics, the monkey saddle is the surface defined by the equation : z = x^3 - 3xy^2, \, or in cylindrical coordinates :z = \rho^3 \cos(3\varphi). It belongs to the class of saddle surfaces, and its name derives from the observation that ...

has Dupin indicatrix in the shape of six-pointed hyperbola.

References

Full 1909 text
(now out of copyright) Differential geometry of surfaces Surfaces {{differential-geometry-stub

Classification

See also

References