Theory

The blue shape is a Reuleaux triangle.

Back to school for some principles. See https://en.wikipedia.org/wiki/Reuleaux_triangle for it. Reuleaux triangles are used for drilling square holes. In mechanical engineering, the priciples of square motion are possible - amongst others - by using Reuleaux triangles.

• Red and green: x- and y-axis.
• Magenta: equilateral triangle.
• Cyan: Construction circles.
• Blue: Reuleaux triangle

A Reuleaux triangle is characterized as a "curve of constant width". This means that when you place the triangle between two parallel lines, touching the triangle, the distance between the lines will not change when you rotate the triangle. About the points:

• D: The center of the magenta equilateral triangle.
• A, B, C: Corners. Distance AB = BC = AC = 1.
• E, F: The quadrants of one of the three equal cyan circles that can be used to create the triangle, Distance EF = 2. So AF = 1 too!
• F: Point between B and C on Reuleaux triangle.
• H: Point between B and C on equilateral triangle.

In the given situation we can say something about distances (in spreadsheet notation). Hi Pythagoras, AH^2+BH^2=BA^2 => AH^2+0.25=1 => AH=sqrt(0.75) => AH=sqrt(3)/2:

• AH = 3*DH =sqrt(3)/2.
• DH =sqrt(3)/6.
• DA = 2*DH =sqrt(3)/3.
• FH = 1-AH =1-sqrt(3)/2.

With a given base size of 1 we can say something about the inscribed and circumscribed circles of both triangles:

• Circumscribed radius = DA =sqrt(3)/3 =1/sqrt(3).
• Inscribed radius equilateral triangle = DH =sqrt(3)/6.
• Inscribed radius Reuleaux triangle = DF =1-sqrt(3)/3 =1-1/sqrt(3).

Additional information

Considering this picture where the triangle is locked inside a square frame en the triangle is rotated:

• From left to right the triangle is rotated in steps of 15 degrees.
• At number 1, 3 and 5 all corners touch the square, where at 2 and 4 one corner does not touch the square. The horizontal and vertical dimension of this no contact zone is typical FH.
• When turning clockwise, one rotation of the triangles results in three counter-clockwise rotations of the triangle centre point. The path of the centre point is close to but not circular.

What you get

The base output in a LISP supporting CAD system, is a closed 2D-polyline containing three arc segments.