Difference between revisions of "Surface of constant width"
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A surface of constant width (orbiform) is a convex form whose width, measured by the distance between two opposite parallel planes touching its boundary, is the same regardless of the direction of those two parallel planes. | A surface of constant width (orbiform) is a convex form whose width, measured by the distance between two opposite parallel planes touching its boundary, is the same regardless of the direction of those two parallel planes. | ||
==Reuleaux triangle== | ==Reuleaux triangle== | ||
| − | [[File:Reuleaux triangle.png| | + | [[File:Reuleaux triangle.png|350px|right]] |
The boundary of a Reuleaux triangle is a constant width curve based on an equilateral triangle. | The boundary of a Reuleaux triangle is a constant width curve based on an equilateral triangle. | ||
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Let <math>\overset{\Large\frown} {AC}</math> be the arc centered at <math>B</math> with radius <math>BB' = BA, \angle ABC = 60^\circ.</math> | Let <math>\overset{\Large\frown} {AC}</math> be the arc centered at <math>B</math> with radius <math>BB' = BA, \angle ABC = 60^\circ.</math> | ||
| − | Arcs <math>AB</math> and <math>BC</math> define similarly. | + | Arcs <math>\overset{\Large\frown} {AB}</math> and <math>\overset{\Large\frown} {BC}</math> define similarly. |
| − | All points on this arcs are equidistant from the opposite vertex. Distance is <math>|AB|.</math> | + | All points on this arcs are equidistant from the opposite vertex. |
| + | |||
| + | Distance is <math>|AB|.</math> | ||
Revision as of 07:23, 10 August 2024
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. A surface of constant width (orbiform) is a convex form whose width, measured by the distance between two opposite parallel planes touching its boundary, is the same regardless of the direction of those two parallel planes.
Reuleaux triangle
The boundary of a Reuleaux triangle is a constant width curve based on an equilateral triangle.
Let 
be equilateral triangle.
Let 
be the arc centered at 
with radius 
Arcs 
and 
define similarly.
All points on this arcs are equidistant from the opposite vertex.
Distance is 
