Curve of constant width

A constant width is a curve of constant width whose closed line always touches all four sides within a suitable square in any position.

Or detailed words, the width of a curve is defined as the distance between two parallel lines touching the cam on opposite sides. These lines are called supporting lines. Curves of constant width are those curves in which for the distance of this line always yields the same value, regardless of where the figure attack the line.

The simplest non-trivial equal thickness of all is the Reuleaux triangle. It is also the same thickness with the smallest area, whereas the circle is the one with the largest, in between there are infinitely many others. So you can save material: A circle with the same diameter has more volume a larger area, a cylindrical roller.

Well-known examples of a uniform thickness, the British 20 - and 50 - pence coins. Your heptagonal shape with curved sides means that material is saved, however, always identify machines in the determination of the diameter of the correct value.

After the set of barber applies to the scope of each orbiform the width b:

That and the scale of equal thickness and circle the same diameter is the same.

However, a constant diameter must not be made of circular arcs or be somehow symmetrical. There are an unlimited number of identical thicknesses. Together you is the convex shape.

A drill bit with the cross section of a Reuleaux triangle can be used for drilling " squared " holes. Invented this drill almost four square holes has generated, the British engineer Harry James Watt in 1914 (U.S. Patent 1,241,175 and following).

The three-dimensional generalization

A spatial constant width is a convex body of constant width: a body without indentations, which always affects all six sides within a suitable die in every position: the orientation in which such a body is clamped between two parallel plates, always the two plates are exactly the same far apart.

A simple non-trivial spatial constant diameter is the body of revolution formed by rotating a Reuleaux triangle about one of its axes of symmetry. But all the other around a symmetry axis rotated Reuleaux polygons are bodies of constant width. Thus there are infinitely many different spatial thickness equal the same constant width.

Contrary to the intuitive assumption that the Reuleaux tetrahedron is also of constant width, it is in this body to no equal thickness. However, can be on the basis of equal thickness spatial construct that are not rotating body, the two Meissner body.

Equal thickness in the production

Especially during the rolling of cylindrical or cylindrical -like shapes such as screws or plugs, it is due to tolerances the rule rather than the exception that instead of cylinders equal thickness arise. By measuring between two parallel surfaces is to determine on the basis of the above definition, no difference. However, problems occur when mold closing is needed. Can be detected a Reuleaux triangle, if instead of a vernier calliper a downward pointing equilateral triangle is used, the upper side is movable and the constant diameter is then rotated.