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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. 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. The ellipse has center , semimajor axis , semiminor axis , and is rotated by , which has the Cartesian equation.
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 For a bounding square of side length 2, the ellipse in the lower-left quadrant has the parametric equations. Note that the geometric centroid 's path can be closely approximated by a superellipse.
Unsolved Problems in Geometry. The Wankel Rotary Engine: Indiana University Press, Cambridge University Press, University of Chicago Press, pp.
The Enjoyment of Mathematics: Selections from Mathematics for the Amateur. Princeton University Press, The Kinematics of Machinery: Outlines of a Theory of Machines. Reprinted as The Kinematics of Machinery. Teacher 86 , , Oct. Explore thousands of free applications across science, mathematics, engineering, technology, business, art, finance, social sciences, and more. Walk through homework problems step-by-step from beginning to end.
Hints help you try the next step on your own. Unlimited random practice problems and answers with built-in Step-by-step solutions. Practice online or make a printable study sheet. Collection of teaching and learning tools built by Wolfram education experts: Mon May 7 Contact the MathWorld Team. Drilling a Square Hole Stan Wagon. Drilling a Hexagonal Hole Barry Cox.