A stadium is a two-dimensional

geometric shape A shape or figure is a graphical representation of an object or its external boundary, outline, or external surface, as opposed to other properties such as color, texture, or material type. A plane shape or plane figure is constrained to lie on ...

constructed of a

rectangle In Euclidean plane geometry, a rectangle is a quadrilateral with four right angles. It can also be defined as: an equiangular quadrilateral, since equiangular means that all of its angles are equal (360°/4 = 90°); or a parallelogram containi ...

with

semicircle In mathematics (and more specifically geometry), a semicircle is a one-dimensional locus of points that forms half of a circle. The full arc of a semicircle always measures 180° (equivalently, radians, or a half-turn). It has only one line of ...

s at a pair of opposite sides. The same shape is known also as a pill shape, discorectangle, squectangle, obround, or sausage body. The shape is based on a

stadium A stadium ( : stadiums or stadia) is a place or venue for (mostly) outdoor sports, concerts, or other events and consists of a field or stage either partly or completely surrounded by a tiered structure designed to allow spectators to stand o ...

, a place used for

athletics Athletics may refer to: Sports * Sport of athletics, a collection of sporting events that involve competitive running, jumping, throwing, and walking ** Track and field, a sub-category of the above sport * Athletics (physical culture), competiti ...

and

horse racing Horse racing is an equestrian performance sport, typically involving two or more horses ridden by jockeys (or sometimes driven without riders) over a set distance for competition. It is one of the most ancient of all sports, as its basic p ...

tracks. A stadium may be constructed as the

Minkowski sum In geometry, the Minkowski sum (also known as dilation) of two sets of position vectors ''A'' and ''B'' in Euclidean space is formed by adding each vector in ''A'' to each vector in ''B'', i.e., the set : A + B = \. Analogously, the Minkowski ...

of a

disk Disk or disc may refer to: * Disk (mathematics), a geometric shape * Disk storage Music * Disc (band), an American experimental music band * ''Disk'' (album), a 1995 EP by Moby Other uses * Disk (functional analysis), a subset of a vector sp ...

and a

line segment In geometry, a line segment is a part of a straight line that is bounded by two distinct end points, and contains every point on the line that is between its endpoints. The length of a line segment is given by the Euclidean distance between ...

. Alternatively, it is the

neighborhood A neighbourhood (British English, Irish English, Australian English and Canadian English) or neighborhood (American English; see spelling differences) is a geographically localised community within a larger city, town, suburb or rural area, ...

of points within a given distance from a line segment. A stadium is a type of

oval An oval () is a closed curve in a plane which resembles the outline of an egg. The term is not very specific, but in some areas (projective geometry, technical drawing, etc.) it is given a more precise definition, which may include either one or ...

. However, unlike some other ovals such as the

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 ...

s, it is not an

algebraic curve In mathematics, an affine algebraic plane curve is the zero set of a polynomial in two variables. A projective algebraic plane curve is the zero set in a projective plane of a homogeneous polynomial in three variables. An affine algebraic plane c ...

because different parts of its boundary are defined by different equations.

Formulas

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 a stadium is calculated by the formula

P = 2 (\pi r+a)

where ''a'' is the length of the straight sides and ''r'' is the radius of the semicircles. With the same parameters, the

area Area is the quantity that expresses the extent of a region on the plane or on a curved surface. The area of a plane region or ''plane area'' refers to the area of a shape A shape or figure is a graphics, graphical representation of an obje ...

of the stadium is

A = \pi r^2 + 2ra = r(\pi r + 2a)

Bunimovich stadium

When this shape is used in the study of

dynamical billiards A dynamical billiard is a dynamical system in which a particle alternates between free motion (typically as a straight line) and specular reflections from a boundary. When the particle hits the boundary it reflects from it without loss of speed ( ...

, it is called the

Bunimovich stadium A dynamical billiard is a dynamical system in which a particle alternates between free motion (typically as a straight line) and specular reflections from a boundary. When the particle hits the boundary it reflects from it without loss of speed ...

Leonid Bunimovich Leonid Abramowich Bunimovich (born August 1, 1947) is a Soviet and American mathematician, who made fundamental contributions to the theory of Dynamical Systems, Statistical Physics and various applications. Bunimovich received his bachelor's degr ...

used this shape to show that it is possible for billiard tracks to exhibit

chaotic behavior Chaos theory is an interdisciplinary area of scientific study and branch of mathematics focused on underlying patterns and deterministic laws of dynamical systems that are highly sensitive to initial conditions, and were once thought to have c ...

(positive

Lyapunov exponent In mathematics, the Lyapunov exponent or Lyapunov characteristic exponent of a dynamical system is a quantity that characterizes the rate of separation of infinitesimally close trajectories. Quantitatively, two trajectories in phase space with ini ...

and exponential divergence of paths) even within a convex billiard table.

Related shapes

A '' capsule'' is produced by revolving a stadium around the

line of symmetry In mathematics, reflection symmetry, line symmetry, mirror symmetry, or mirror-image symmetry is symmetry with respect to a reflection. That is, a figure which does not change upon undergoing a reflection has reflectional symmetry. In 2D ther ...

that bisects the semicircles.

References

External links

*{{MathWorld, title=Stadium, urlname=Stadium Piecewise-circular curves