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 has the least area of any curve of given constant width.
Read more about this topic: Curve Of Constant Width
Famous quotes containing the word properties:
“A drop of water has the properties of the sea, but cannot exhibit a storm. There is beauty of a concert, as well as of a flute; strength of a host, as well as of a hero.”
—Ralph Waldo Emerson (1803–1882)
“The reason why men enter into society, is the preservation of their property; and the end why they choose and authorize a legislative, is, that there may be laws made, and rules set, as guards and fences to the properties of all the members of the society: to limit the power, and moderate the dominion, of every part and member of the society.”
—John Locke (1632–1704)