A stadium is a two-dimensional

geometric shape A shape is a graphical representation of an object's form or its external boundary, outline, or external surface. It is distinct from other object properties, such as color, texture, or material type. In geometry, ''shape'' excludes informat ...

constructed of a

rectangle In Euclidean geometry, Euclidean plane geometry, a rectangle is a Rectilinear polygon, rectilinear convex polygon or a quadrilateral with four right angles. It can also be defined as: an equiangular quadrilateral, since equiangular means that a ...

with

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

s at a pair of opposite sides. The same shape is known also as a pill shape, discorectangle, 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 completely or partially surrounded by a tiered structure designed to allow spectators to stand or sit ...

, 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), competitio ...

and

horse racing Horse racing is an equestrian performance activity, 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 bas ...

tracks. A stadium may be constructed as the

Minkowski sum In geometry, the Minkowski sum 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'': A + B = \ The Minkowski difference (also ''Minkowski subtraction'', ''Minkowsk ...

of a disk and a

line segment In geometry, a line segment is a part of a line (mathematics), straight line that is bounded by two distinct endpoints (its extreme points), and contains every Point (geometry), point on the line that is between its endpoints. It is a special c ...

. Alternatively, it is the

neighborhood A neighbourhood (Commonwealth English) or neighborhood (American English) is a geographically localized community within a larger town, city, suburb or rural area, sometimes consisting of a single street and the buildings lining it. Neigh ...

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 of mathematics (projective geometry, technical drawing, etc.), it is given a more precise definition, which may inc ...

. 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 cu ...

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

Formulas

The

perimeter A perimeter is the length of a closed boundary that encompasses, surrounds, or outlines either a two-dimensional shape or a one-dimensional line. The perimeter of a circle or an ellipse is called its circumference. Calculating the perimet ...

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 measure of a region's size on a surface. The area of a plane region or ''plane area'' refers to the area of a shape or planar lamina, while '' surface area'' refers to the area of an open surface or the boundary of a three-di ...

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 Elastic collision, witho ...

, it is called the Bunimovich stadium. Leonid Bunimovich used this shape to show that it is possible for billiard tracks to exhibit chaotic behavior (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 trajectory, trajectories. Quantitatively, two trajectories in phase sp ...

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 two-di ...

that bisects the semicircles.

References

External links

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