File:How to make mathematical roller curve based on triangle.svg

Diagram showing how to construct a mathematical "roller" (i.e. constant-width curve) based on any triangle. In the diagram, A, B, C are the vertices of the triangle, while a, b, c are the lengths of the opposing sides. If x is defined as the sum of the lengths of the two longest sides of the triangle while y is an arbitrarily-chosen non-negative "padding factor", then the constant-width roller curve will be constructed from the following six circular arcs:

• Radius x+y-a centered at A
• Radius x+y-b-c centered at A
• Radius x+y-b centered at B
• Radius x+y-a-c centered at B
• Radius x+y-c centered at C
• Radius x+y-a-b centered at C

The arcs with two negative terms in the definition of the radius are on the same side of the triangle as the vertices at their centers, while the arcs with one negative term in the definition of the radius are on the opposite side of the triangle.

The overall diameter of constant width of the curve is calculated as the sum of the lengths of the two longest sides of the triangle (x) plus twice the padding factor (2y) minus the length of the shortest side of the triangle. The perimeter is pi times the diameter (as for a circle).

If the padding factor y is set at zero, then at least one of the arcs will be of zero radius, which means that there will be a point of tangent discontinuity where curves join at a vertex of the triangle. In the case of an isoceles triangle where the equal angles are less than 60°, y=0 means that there will two points of tangent discontinuity, while in the case of an equilateral triangle there will be three points of tangent discontinuity (i.e. a Reuleaux triangle).

In the diagram, there is a Pythagorean triangle with sides of relative length 3, 4, 5 (so x=9) while the padding factor is 1 unit (i.e. y=1). This means that there will be circular arcs of radius 5 and 3 centered on the right angle vertex (B in the diagram), circular arcs of radius 7 and 1 centered on the vertex opposite the shortest side of the triangle (C in the diagram), and circular arcs of radius 6 and 2 centered on the other vertex (A). The overall constant diameter is 8 units, and the perimeter (sum of the length of the six arcs) is 8π. This can be seen from the following calculation, proceeding counterclockwise from the top (where 36.869898°=atan(3/4), and angle is converted to arc length by multiplying it times radius, times the radian conversion factor π/180):

 Degrees:   Radius:   Arc Length:
  -----      ---      -----------
   90         5       2.5pi
36.869898     1       0.20483276pi
53.130102     6       1.77100341pi
   90         3       1.5pi
36.869898     7       1.43382935pi
53.130102     2       0.59033447pi
                        -------
Total perimeter:      8pi