In geometry, a curve of constant width is a convex planar shape whose width, defined as the distance between two opposite parallel lines touching its boundary, is the same regardless of the direction of those two parallel lines. One defines the width of the curve in a given direction to be the perpendicular distance between the parallels perpendicular to that direction.
More generally, any compact convex planar body D has one pair of parallel supporting lines in a given direction. A supporting line is a line that has at least one point in common with the boundary of D but no points in common with the interior of D. One defines the width of the body as before. If the width of D is the same in all directions, then one says that the body is of constant width and calls its boundary a curve of constant width, and the planar body itself is referred to as an orbiform.
If you have a circle, then the width is constant, i.e. the diameter. If you have a square then the width varies between the side length and the length of the diagonal. If the width is constant in all directions then do you have a circle? The surprising answer is that there are many non-circular shapes of constant width. A nontrivial example is the Reuleaux triangle. To construct this, take an equilateral triangle ABC and draw the arc BC on the circle centered at A, the arc CA on the circle centered at B, and the arc AB on the circle centered at C. The resulting figure is of constant width.
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Properties
Curves of constant width can be rotated between parallel line segments. To see this, simply note that one can rotate parallel line segments (supporting lines) around curves of constant width by definition. Consequently, a curve of constant width can be rotated in a square.
A basic result on curves of constant width is Barbier's theorem, which asserts that the perimeter of any curve of constant width is equal to the width (diameter) multiplied by π. A simple example of this would be a circle with width (diameter) d having a perimeter of πd.
By the isoperimetric inequality and Barbier's theorem, the circle has the maximum area of any curve of given constant width. The Blaschke-Lebesgue theorem says that the Reuleaux triangle had the least area of any curve of given constant width.
Because of the constant diameter, these answer the question/brain teaser:
- "What shape can you make a manhole cover so that it cannot fall down through the hole?"
In practice, manhole covers are not made in such shapes, for lack of compelling reason. Circles are easier to machine, and need not be rotated to a proper alignment to seal the hole.
Generalizations
Δ curves, or curves which can be rotated in the equilateral triangle, have many similar properties to curves of constant width. The generalization of the definition of bodies of constant width to convex bodies in R³ and their boundaries leads to the concept of surface of constant width (in the case of a Reuleaux triangle, this does not lead to a Reuleaux tetrahedron, but to Meissner bodies). There is also the concept of space curves of constant width, which are space curves whose normal planes intersect them at exactly two points and are normal to them at both points.
Examples
Famous examples of a curve of constant width are the British 20p and 50p coins. Their heptagonal shape with curved sides means that the currency detector in an automated coin machine will always measure the correct diameter through the middle, no matter which angle it takes its measurement from.
There exists a polynomial f(x,y) of degree 8, whose zero-set (i.e., set of points in R2 for which f(x,y) = 0) is a curve of constant width.[1]
External links
- How round is your circle? contains a chapter on this topic.
- Star Construction of Shapes of Constant Width at cut-the-knot
- Weisstein, Eric W. "Curve of Constant Width." From MathWorld--A Wolfram Web Resource.
- Multi-angle-wheel bicycle appears in Qingdao - bicycle with wheels using this property.
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