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Every [[regular polygon]] with an odd number of sides gives rise to a curve of constant width, a [[Reuleaux polygon]], formed from circular arcs centered at its vertices that pass through the two vertices farthest from the center. For instance, this construction generates a Reuleaux triangle from an equilateral triangle. Some irregular polygons also generate Reuleaux polygons.{{r|bs|cr}} The Reuleaux polygons are a special case of a more general construction, called by [[Martin Gardner]] the "crossed-lines method", in which any [[arrangement of lines]] in the plane (no two parallel), sorted into cyclic order by their slopes, are connected by a smooth curve formed from circular arcs between pairs of consecutive lines in the sorted order, centered at the crossing of these two lines. The radius of the first arc must be chosen large enough to cause all successive arcs to end on the correct side of the next crossing point; however, all sufficiently-large radii work. For two lines, this forms a circle; for three lines on the sides of an equilateral triangle, with the minimum possible radius, it forms a Reuleaux triangle, and for the lines of a regular [[star polygon]] it can form a Reuleaux polygon.{{r|gardner|bs}}
[[Leonhard Euler]] constructed curves of constant width as the [[involute]]s of curves with an odd number of [[Cusp (singularity)|cusp singularities]], having only one [[tangent line]] in each direction (that is, [[Hedgehog (geometry)|projective hedgehogs]]). If the starting curve is smooth (except at the cusps), the resulting curve of constant width will also be smooth.{{r|euler|robertson}} An example of a starting curve with the correct properties for this construction is the [[deltoid curve]], and involutes of the deltoid form smooth curves of constant width, not
Another construction chooses half of the curve of constant width, meeting certain conditions, and then completes it to a full curve. The construction begins with a convex curved arc, connecting a pair of closest points on two parallel lines whose separation is the intended width <math>w</math> of the curve. The arc must have the property (required of a curve of constant width) that each of its supporting lines is tangent to a circle of radius <math>w</math> containing the entire arc; intuitively, this prevents its [[curvature]] from being smaller than that of a circle of radius <math>w</math> at any point. As long as it meets this condition, it can be used in the construction. The next step is to intersect an infinite family of circular disks of radius <math>w</math>, both the ones tangent to the supporting lines and additional disks centered at each point of the arc. This intersection forms a body of constant width, with the given arc as part of its boundary. This construction is universal: all curves of constant width may be constructed in this way.{{r|rt}}
Given any two bodies of constant width, their [[Minkowski sum]] forms another body of constant width.{{r|mmo}}
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