In
differential geometry
Differential geometry is a mathematical discipline that studies the geometry of smooth shapes and smooth spaces, otherwise known as smooth manifolds. It uses the techniques of differential calculus, integral calculus, linear algebra and multilin ...
a translation surface is 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 ...
that is generated by
translations:
* For two
space 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
with a common point
, the curve
is shifted such that point
is moving on
. By this procedure curve
generates a surface: the ''translation surface''.
If both curves are contained in a common plane, the translation surface is planar (part of a plane). This case is generally ignored.
Simple ''examples'':
#
Right circular cylinder
A cylinder (from ) has traditionally been a three-dimensional solid, one of the most basic of curvilinear geometric shapes. In elementary geometry, it is considered a prism with a circle as its base.
A cylinder may also be defined as an infin ...
:
is a
circle
A circle is a shape consisting of all points in a plane that are at a given distance from a given point, the centre. Equivalently, it is the curve traced out by a point that moves in a plane so that its distance from a given point is con ...
(or another cross section) and
is a line.
#The ''elliptic''
paraboloid
In geometry, a paraboloid is a quadric surface that has exactly one axis of symmetry and no center of symmetry. The term "paraboloid" is derived from parabola, which refers to a conic section that has a similar property of symmetry.
Every plane ...
can be generated by
and
(both curves are
parabola
In mathematics, a parabola is a plane curve which is mirror-symmetrical and is approximately U-shaped. It fits several superficially different mathematical descriptions, which can all be proved to define exactly the same curves.
One descri ...
s).
#The ''hyperbolic'' paraboloid
can be generated by
(parabola) and
(downwards open parabola).
Translation surfaces are popular in
descriptive geometry
Descriptive geometry is the branch of geometry which allows the representation of three-dimensional objects in two dimensions by using a specific set of procedures. The resulting techniques are important for engineering, architecture, design and ...
and architecture, because they can be modelled easily.
In
differential geometry
Differential geometry is a mathematical discipline that studies the geometry of smooth shapes and smooth spaces, otherwise known as smooth manifolds. It uses the techniques of differential calculus, integral calculus, linear algebra and multilin ...
minimal surfaces are represented by translation surfaces or as ''midchord surfaces'' (s. below).
The translation surfaces as defined here should not be confused with the
translation surfaces in
complex geometry
In mathematics, complex geometry is the study of geometric structures and constructions arising out of, or described by, the complex numbers. In particular, complex geometry is concerned with the study of spaces such as complex manifolds and ...
.
Parametric representation
For two space curves
and
with
the translation surface
can be represented by:
:(TS)
and contains the origin. Obviously this definition is symmetric regarding the curves
and
. Therefore, both curves are called generatrices (one:
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 ...
). Any point
of the surface is contained in a shifted copy of
and
resp.. The
tangent plane
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
is generated by the tangentvectors of the generatrices at this point, if these vectors are
linearly independent
In the theory of vector spaces, a set of vectors is said to be if there is a nontrivial linear combination of the vectors that equals the zero vector. If no such linear combination exists, then the vectors are said to be . These concepts ar ...
.
If the precondition
is not fulfilled, the surface defined by (TS) may not contain the origin and the curves
. But in any case the surface contains shifted copies of any of the curves
as parametric curves
and
respectively.
The two curves
can be used to generate the so called corresponding midchord surface. Its parametric representation is
: (MCS)
Helicoid as translation surface and midchord surface
A
helicoid
The helicoid, also known as helical surface, after the plane and the catenoid, is the third minimal surface to be known.
Description
It was described by Euler in 1774 and by Jean Baptiste Meusnier in 1776. Its name derives from its similari ...
is a special case of a
generalized helicoid
In geometry, a generalized helicoid is a surface in Euclidean space generated by rotating and simultaneously displacing a curve, the ''profile curve'', along a line, its ''axis''. Any point of the given curve is the starting point of a circular hel ...
and a
ruled surface
In geometry, a surface is ruled (also called a scroll) if through every point of there is a straight line that lies on . Examples include the plane, the lateral surface of a cylinder or cone, a conical surface with elliptical directrix, ...
. It is an example of a
minimal surface
In mathematics, a minimal surface is a surface that locally minimizes its area. This is equivalent to having zero mean curvature (see definitions below).
The term "minimal surface" is used because these surfaces originally arose as surfaces tha ...
and can be represented as a translation surface.
The helicoid with the parametric representation
:
has a ''turn around shift'' (German: Ganghöhe)
.
Introducing new parameters
[J.C.C. Nitsche: ''Vorlesungen über Minimalflächen'', Springer-Verlag, 2013, , 9783642656194, p. 59] such that
:
and
a positive real number, one gets a new parametric representation
*
:::
which is the parametric representation of a translation surface with the two ''identical'' (!) generatrices
:
and
:
The common point used for the diagram is
.
The (identical) generatrices are helices with the turn around shift
which lie on the cylinder with the equation
. Any parametric curve is a shifted copy of the generatrix
(in diagram: purple) and is contained in the right circular cylinder with radius
, which contains the ''z''-axis.
The new parametric representation represents only such points of the helicoid that are within the cylinder with the equation
.
From the new parametric representation one recognizes, that the helicoid is a midchord surface, too:
:
where
:
and
:
are two identical generatrices.
In diagram:
lies on the helix
and
on the (identical) helix
. The midpoint of the chord is
.
Advantages of a translation surface
; Architecture:
A surface (for example a roof) can be manufactured using a
jig
The jig ( ga, port, gd, port-cruinn) is a form of lively folk dance in compound metre, as well as the accompanying dance tune. It is most associated with Irish music and dance. It first gained popularity in 16th-century Ireland and parts o ...
for curve
and several identical jigs of curve
. The jigs can be designed without any knowledge of mathematics. By positioning the jigs the rules of a translation surface have to be respected only.
; Descriptive geometry:
Establishing a
parallel projection
In three-dimensional geometry, a parallel projection (or axonometric projection) is a projection of an object in three-dimensional space onto a fixed plane, known as the '' projection plane'' or '' image plane'', where the '' rays'', known as ...
of a translation surface one 1) has to produce projections of the two generatrices, 2) make a jig of curve
and 3) draw with help of this jig copies of the curve respecting the rules of a translation surface. The contour of the surface is the envelope of the curves drawn with the jig. This procedure works for orthogonal and oblique projections, but not for
central projection
In mathematics, a projection is a mapping of a set (or other mathematical structure) into a subset (or sub-structure), which is equal to its square for mapping composition, i.e., which is idempotent. The restriction to a subspace of a project ...
s.
; Differential geometry:
For a translation surface with parametric representation
the
partial derivative
In mathematics, a partial derivative of a function of several variables is its derivative with respect to one of those variables, with the others held constant (as opposed to the total derivative, in which all variables are allowed to vary). Par ...
s of
are simple derivatives of the curves. Hence the mixed derivatives are always
and the coefficient
of the
second fundamental form
In differential geometry, the second fundamental form (or shape tensor) is a quadratic form on the tangent plane of a smooth surface in the three-dimensional Euclidean space, usually denoted by \mathrm (read "two"). Together with the first fundamen ...
is
, too. This is an essential facilitation for showing that (for example) a helicoid is a minimal surface.
References
* G. Darboux: ''Leçons sur la théorie générale des surfaces et ses applications géométriques du calcul infinitésimal'' , 1–4 , Chelsea, reprint, 972, pp. Sects. 81–84, 218
* Georg Glaeser: ''Geometrie und ihre Anwendungen in Kunst, Natur und Technik'', Springer-Verlag, 2014, , p. 259
* W. Haack: ''Elementare Differentialgeometrie'', Springer-Verlag, 2013, , p. 140
* C. Leopold: ''Geometrische Grundlagen der Architekturdarstellung.''
Kohlhammer Verlag
W. Kohlhammer Verlag GmbH, or Kohlhammer Verlag, is a German publishing house headquartered in Stuttgart.
History
Kohlhammer Verlag was founded in Stuttgart on 30 April 1866 by . Kohlhammer had taken over the businesses of his late father-in-law ...
, Stuttgart 2005, {{ISBN, 3-17-018489-X, p. 122
* D.J. Struik: ''Lectures on classical differential geometry'' , Dover, reprint ,1988, pp. 103, 109, 184
External links
Encyclopedia of Mathematics
Surfaces
Differential geometry
Differential geometry of surfaces
Analytic geometry