Surface of constant width

A curve of constant width is a simple closed curve in the plane whose width (the distance between parallel supporting lines) is the same in all directions. A surface of constant width (orbiform) is a convex form whose width, measured by the distance between two opposite parallel planes touching its boundary, is the same regardless of the direction of those two parallel planes.

Reuleaux triangle

The boundary of a Reuleaux triangle is a constant width curve based on an equilateral triangle.

Let $\triangle ABC$ be equilateral triangle.

Let $\overset{\Large\frown} {AC}$ be the arc centered at $B$ with radius $BB' = BA, \angle ABC = 60^\circ.$

Arcs $\overset{\Large\frown} {AB}$ and $\overset{\Large\frown} {BC}$ define similarly.

All points on this arcs are equidistant from the opposite vertex.

Distance is $|AB|.$

vladimir.shelomovskii@gmail.com, vvsss

Reuleaux triangle analogue

Let $ABCD$ be the boundary of a Reuleaux triangle, $O$ be the centroid of $\triangle ABC.$

Let $\overset{\Large\frown} {FF_0}$ be the arc centered at $A$ with radius $AA', \angle FAF_0 = 60^\circ,$ points $F, A, C$ and $F_0, A, B$ are collinear.

Let $\overset{\Large\frown} {F_0F'}$ be the arc centered at $B$ with radius $|AA'| + |AB|,$ $\angle F'AF_0 = \angle ABC = 60^\circ,$ points $F', C, B$ are collinear.

Similarly define arcs from point $F'$ to $F.$

The width of this curve is $|AB| + 2|AA'|.$

vladimir.shelomovskii@gmail.com, vvsss

Rotation of the Reuleaux triangle

The left part of diagram shows the curve $ABC$ which is a Reuleaux triangle determined by the center $O$ and the vertex $A.$

The Reuleaux triangle $AB'C'$ is constructed by the rotation of the curve $ABC$ around axis $AO.$

The right part of the diagram shows the surface which arose as the result of the rotation $ABC.$

vladimir.shelomovskii@gmail.com, vvsss

Rotation of the Reuleaux triangle analogue

The left part of diagram shows the curve $G$ which is a Reuleaux triangle analogy determined by the center $O,$ the vertex $A$ and point $G$ on curve analogue.

Similarly, we use the Reuleaux triangle $AB'C'$ is constructed by the rotation of the curve $G$ around axis $AO$ and get the second position of the Reuleaux triangle analogue.

The middle part of the diagram shows part of the surface. We can see all four parts of this surface shown by different colors.

The right part of the diagram shows the surface which arose as the result of the rotation curve $G.$ It is impossible see the down side of the surface (blue) in this view.