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  2. Curve of constant width - Wikipedia

    en.wikipedia.org/wiki/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 an orbiform, the name given to these shapes by Leonhard Euler. [1]

  3. Reuleaux triangle - Wikipedia

    en.wikipedia.org/wiki/Reuleaux_triangle

    The angles made by each pair of arcs at the corners of a Reuleaux triangle are all equal to 120°. This is the sharpest possible angle at any vertex of any curve of constant width. [9] Additionally, among the curves of constant width, the Reuleaux triangle is the one with both the largest and the smallest inscribed equilateral triangles. [15]

  4. Surface of constant width - Wikipedia

    en.wikipedia.org/wiki/Surface_of_constant_width

    One defines the width of the surface in a given direction to be the perpendicular distance between the parallels perpendicular to that direction. Thus, a surface of constant width is the three-dimensional analogue of a curve of constant width, a two-dimensional shape with a constant distance between pairs of parallel tangent lines.

  5. Reuleaux polygon - Wikipedia

    en.wikipedia.org/wiki/Reuleaux_polygon

    Gambian dalasi coin, a Reuleaux heptagon. In geometry, a Reuleaux polygon is a curve of constant width made up of circular arcs of constant radius. [1] These shapes are named after their prototypical example, the Reuleaux triangle, which in turn is named after 19th-century German engineer Franz Reuleaux. [2]

  6. Barbier's theorem - Wikipedia

    en.wikipedia.org/wiki/Barbier's_theorem

    These Reuleaux polygons have constant width, and all have the same width; therefore by Barbier's theorem they also have equal perimeters. In geometry, Barbier's theorem states that every curve of constant width has perimeter π times its width, regardless of its precise shape. [1] This theorem was first published by Joseph-Émile Barbier in ...

  7. Hedgehog (geometry) - Wikipedia

    en.wikipedia.org/wiki/Hedgehog_(geometry)

    It can be used to characterize curves of constant width: a convex hedgehog has constant width if and only if its support function is formed by adding / to the support function of a projective hedgehog. That is, the curves of constant width are exactly the convex hedgehogs formed as sums of projective hedgehogs and circles.

  8. Girth (geometry) - Wikipedia

    en.wikipedia.org/wiki/Girth_(geometry)

    More generally, if S is a surface of constant width w, then every projection of S is a curve of constant width, with the same width w. All curves of constant width have the same perimeter, the same value πw as the circumference of a circle with that width (this is Barbier's theorem). Therefore, every surface of constant width is also a surface ...

  9. Blaschke–Lebesgue theorem - Wikipedia

    en.wikipedia.org/wiki/Blaschke–Lebesgue_theorem

    The same theorem is also true in the hyperbolic plane. [11] For any convex distance function on the plane (a distance defined as the norm of the vector difference of points, for any norm), an analogous theorem holds true, according to which the minimum-area curve of constant width is an intersection of three metric disks, each centered on a boundary point of the other two.