In
three-dimensional geometry
In mathematics, solid geometry or stereometry is the traditional name for the geometry of three-dimensional, Euclidean spaces (i.e., 3D geometry).
Stereometry deals with the measurements of volumes of various solid figures (or 3D figures), inc ...
, the girth of a geometric object, in a certain direction, is the
perimeter
A perimeter is a closed path that encompasses, surrounds, or outlines either a two dimensional shape or a one-dimensional length. The perimeter of a circle or an ellipse is called its circumference.
Calculating the perimeter has several pract ...
of its
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 '' li ...
in that direction.
[.][.] For instance, the girth of a
unit cube
A unit cube, more formally a cube of side 1, is a cube whose sides are 1 unit long.. See in particulap. 671. The volume of a 3-dimensional unit cube is 1 cubic unit, and its total surface area is 6 square units..
Unit hypercube
The term '' ...
in a direction parallel to one of the three coordinate axes is four: it projects to a
unit square
In mathematics, a unit square is a square whose sides have length . Often, ''the'' unit square refers specifically to the square in the Cartesian plane with corners at the four points ), , , and .
Cartesian coordinates
In a Cartesian coordinate ...
, which has four as its perimeter.
Surfaces of constant girth
The girth of a
sphere
A sphere () is a Geometry, geometrical object that is a solid geometry, three-dimensional analogue to a two-dimensional circle. A sphere is the Locus (mathematics), set of points that are all at the same distance from a given point in three ...
in any direction equals the
circumference
In geometry, the circumference (from Latin ''circumferens'', meaning "carrying around") is the perimeter of a circle or ellipse. That is, the circumference would be the arc length of the circle, as if it were opened up and straightened out to ...
of its
equator
The equator is a circle of latitude, about in circumference, that divides Earth into the Northern and Southern hemispheres. It is an imaginary line located at 0 degrees latitude, halfway between the North and South poles. The term can als ...
, or of any of its
great circle
In mathematics, a great circle or orthodrome is the circular intersection of a sphere and a plane passing through the sphere's center point.
Any arc of a great circle is a geodesic of the sphere, so that great circles in spherical geomet ...
s. More generally,
if is a
surface of constant width
In geometry, a surface of constant width is a convex form whose width, measured by the distance between two opposite parallel planes touching its boundary, is the same regardless of the direction of those two parallel planes. One defines the wi ...
, then every projection of is a
curve of constant width
In geometry, a curve of constant width is a simple closed curve in the plane whose width (the distance between parallel supporting lines) is the same in all directions. The shape bounded by a curve of constant width is a body of constant width or ...
, with the same width . All curves of constant width have the same perimeter, the same value as the circumference of a circle with that width (this is
Barbier's theorem
In geometry, Barbier's theorem states that every curve of constant width has perimeter times its width, regardless of its precise shape. This theorem was first published by Joseph-Émile Barbier in 1860.
Examples
The most familiar examples of c ...
). Therefore, every surface of constant width is also a surface of constant girth: its girth in all directions is the same number .
Hermann Minkowski
Hermann Minkowski (; ; 22 June 1864 – 12 January 1909) was a German mathematician and professor at Königsberg, Zürich and Göttingen. He created and developed the geometry of numbers and used geometrical methods to solve problems in number t ...
proved, conversely, that every convex surface of constant girth is also a surface of constant width.
Projection versus cross-section
For a
prism
Prism usually refers to:
* Prism (optics), a transparent optical component with flat surfaces that refract light
* Prism (geometry), a kind of polyhedron
Prism may also refer to:
Science and mathematics
* Prism (geology), a type of sedimentary ...
or
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 ...
, its projection in the direction parallel to its axis is the same as its
cross section
Cross section may refer to:
* Cross section (geometry)
** Cross-sectional views in architecture & engineering 3D
*Cross section (geology)
* Cross section (electronics)
* Radar cross section, measure of detectability
* Cross section (physics)
**Abs ...
, so in these cases the girth also equals the perimeter of the cross section. In some application areas such as
shipbuilding
Shipbuilding is the construction of ships and other floating vessels. It normally takes place in a specialized facility known as a shipyard. Shipbuilders, also called shipwrights, follow a specialized occupation that traces its roots to befor ...
this alternative meaning, the perimeter of a cross section, is taken as the definition of girth.
Application
Girth is sometimes used by postal services and delivery companies as a basis for pricing. For example,
Canada Post
Canada Post Corporation (french: Société canadienne des postes), trading as Canada Post (french: Postes Canada), is a Crown corporation that functions as the primary postal operator in Canada. Originally known as Royal Mail Canada (the opera ...
requires that an item's length plus girth not exceed a maximum allowed value.
For a rectangular box, the girth is 2 * (height + width), i.e. the perimeter of a projection or cross section perpendicular to its length.
References
{{reflist
Euclidean solid geometry