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
(5)
(6)
for ,
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
(9)
(10)
for .
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
