Reuleaux Triangle
A curve of constant width constructed by drawing arcs from each polygon vertex of an equilateral triangle between the other two vertices. The Reuleaux triangle has the smallest area
for a given width of any curve of constant width.
Let the arc radius be 
. Since the area
of each meniscus-shaped portion of the Reuleaux triangle is a circular
segment with opening angle 
,
But the area of the central equilateral triangle with 
is
 |
(3)
|
so the total area is then
 |
(4)
|
Because it can be rotated inside a square, as illustrated
above, it is the basis for the Harry Watt square drill bit.
When rotated inside a square of side length 2 having corners at 
), the
envelope of the Reuleaux triangle is a region of the square with rounded corners.
At the corner 
, the envelope of the boundary
is given by the segment of the ellipse with parametric
equations
for ![beta in [pi/6,pi/3]](/images/equations/ReuleauxTriangle/Inline18.gif)
, extending a distance
from the corner (Gleißner
and Zeitler 2000). The ellipse has center 
, semimajor
axis 
, semiminor axis 
, and
is rotated by 
, which has the Cartesian equation
 |
(7)
|
The fractional area covered as the Reuleaux triangle rotates
is
 |
(8)
|
(OEIS A066666). Note that Gleißner and Zeitler (2000) fail to simplify their equivalent equation, and then proceed to assert
that (8) is erroneous.
The geometric centroid does not stay fixed as the triangle is rotated, nor does it move along
a circle. In fact, the path consists of a curve composed
of four arcs of an ellipse (Wagon 1991). For a bounding
square of side length 2, the ellipse in the lower-left quadrant has the parametric
equations
for ![beta in [pi/6,pi/3]](/images/equations/ReuleauxTriangle/Inline30.gif)
. The ellipse has center
, semimajor axis 
, semiminor
axis 
, and is rotated by 
, which has the Cartesian equation
 |
(11)
|
The area enclosed by the locus of the centroid is given by
 |
(12)
|
(Gleißner and Zeitler 2000; who again fail to simplify their expression). Note that the geometric centroid's path can be closely
approximated by a superellipse
 |
(13)
|
with 
and 
.
SEE ALSO: Curve of Constant Width,
Delta Curve,
Equilateral
Triangle,
Flower of Life,
Piecewise
Circular Curve,
Reuleaux Polygon,
Reuleaux
Tetrahedron,
Rotor,
Roulette,
Triquetra
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Math. Ann. 76, 504-513, 1915.
Bogomolny, A. "Shapes of Constant Width." http://www.cut-the-knot.org/do_you_know/cwidth.shtml.
Croft, H. T.; Falconer, K. J.; and Guy, R. K. Unsolved
Problems in Geometry. New York: Springer-Verlag, p. 8, 1991.
Dark, H. E. The Wankel Rotary Engine: Introduction and Guide. Bloomington, IN: Indiana University
Press, 1974.
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Constants. Cambridge, England: Cambridge University Press, 2003.
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Gardner, M. "Curves of Constant Width." Ch. 18 in The Unexpected Hanging and Other Mathematical Diversions. Chicago, IL: University
of Chicago Press, pp. 212-221, 1991.
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Mass." Result. Math. 37, 335-344, 2000.
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Raton, FL: CRC Press, pp. 176-177, 1997.
Kunkel, P. "Reuleaux Triangle." http://whistleralley.com/reuleaux/reuleaux.htm.
Math Forum. "Reuleaux Triangle, Reuleaux Drill." http://mathforum.org/~sarah/HTMLthreads/articletocs/reuleaux.triangle.html.
Peterson, I. "Ivar Peterson's MathLand: Rolling with Reuleaux." Oct. 21,
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Rademacher, H. and Toeplitz, O. The Enjoyment of Mathematics: Selections from Mathematics for the Amateur. Princeton,
NJ: Princeton University Press, 1957.
Reuleaux, F. The Kinematics of Machinery: Outlines of a Theory of Machines. London: Macmillan, 1876. Reprinted as The
Kinematics of Machinery. New York: Dover, 1963.
Sloane, N. J. A. Sequence A066666
in "The On-Line Encyclopedia of Integer Sequences."
Smith, S. "Drilling Square Holes." Math. Teacher 86, 579-583,
Oct. 1993.
Taimina, D. and Henderson, D. W. "Reuleaux Triangle." http://kmoddl.library.cornell.edu/math/2/.
Wagon, S. Mathematica
in Action. New York: W. H. Freeman, pp. 52-54 and 381-383, 1991.
Yaglom, I. M. and Boltyanskii, V. G. Convex
Figures. New York: Holt, Rinehart, & Winston, 1961.
CITE THIS AS:
Weisstein, Eric W. "Reuleaux Triangle."
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