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Disk (mathematics)

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Disk (mathematics)

Disc with circumference (C) in black, diameter (D) in cyan, radius (R) in red, and centre (geometry) (O) in magenta.

In geometry, a disk (also spelled disc)[1] is the region in a plane bounded by a circle. A disk is said to be closed or open according to whether or not it contains the circle that constitutes its boundary.[2]

Contents

  • Formulas 1
  • Properties 2
  • See also 3
  • References 4

Formulas

In Cartesian coordinates, the open disk of center (a, b) and radius R is given by the formula[1]

D=\{(x, y)\in {\mathbb R^2}: (x-a)^2+(y-b)^2 < R^2\}

while the closed disk of the same center and radius is given by

\overline{ D }=\{(x, y)\in {\mathbb R^2}: (x-a)^2+(y-b)^2 \le R^2\}.

The area of a closed or open disk of radius R is πR2 (see area of a disk).[3]

Properties

The Euclidean disk has circular symmetry.

The open disk and the closed disk are not topologically equivalent (that is, they are not homeomorphic), as they have different topological properties from each other. For instance, every closed disk is compact whereas every open disk is not compact.[4] However from the viewpoint of algebraic topology they share many properties: both of them are contractible[5] and so are homotopy equivalent to a single point. This implies that their fundamental groups are trivial, and all homology groups are trivial except the 0th one, which is isomorphic to Z. The Euler characteristic of a point (and therefore also that of a closed or open disk) is 1.[6]

Every continuous map from the closed disk to itself has at least one fixed point (we don't require the map to be bijective or even surjective); this is the case n=2 of the Brouwer fixed point theorem.[7] The statement is false for the open disk:[8] consider for example the function

f(x,y)=\left(\frac{x+\sqrt{1-y^2}}{2},y\right)

which maps every point of the open unit disk to another point of the open unit disk slightly to the right of the given one.

See also

References

  1. ^ a b Clapham, Christopher; Nicholson, James (2014), The Concise Oxford Dictionary of Mathematics, Oxford University Press, p. 138,  .
  2. ^ Arnold, B. H. (2013), Intuitive Concepts in Elementary Topology, Dover Books on Mathematics, Courier Dover Publications, p. 58,  .
  3. ^ Rotman, Joseph J. (2013), Journey into Mathematics: An Introduction to Proofs, Dover Books on Mathematics, Courier Dover Publications, p. 44,  .
  4. ^ Maudlin, Tim (2014), New Foundations for Physical Geometry: The Theory of Linear Structures, Oxford University Press, p. 339,  .
  5. ^ Cohen, Daniel E. (1989), Combinatorial Group Theory: A Topological Approach, London Mathematical Society Student Texts 14, Cambridge University Press, p. 79,  .
  6. ^ In higher dimensions, the Euler characteristic of a closed ball remains equal to +1, but the Euler characteristic of an open ball is +1 for even-dimensional balls and −1 for odd-dimensional balls. See Klain, Daniel A.;  .
  7. ^ Arnold (2013), p. 132.
  8. ^ Arnold (2013), Ex. 1, p. 135.
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