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 ...

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 contain ...

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 o ...

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), competi ...

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 pr ...

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 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 i ...

. 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 ...

. However, unlike some other ovals such as the

ellipse In mathematics, an ellipse is a plane curve surrounding two 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 type of ellipse in ...

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 ...

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 pr ...

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 or planar lamina, while '' surface area'' refers to the area of an op ...

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. 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 that bisects the semicircles.

References

External links

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