Curves and surfaces of constant width

#constant width#orbiform#surface of revolution#lambda-convexity#Meissner

Date of design: 1911 Designer: Prof. Dr. Ernst Meissner in ZürichTitle in the catalog: (de) Modell Nr. I represents an algebraic surfaces of revolution of constant width b = 12 cm.Size: 12x12x12 cm Price in the catalog: 40 marks (for models No. 1-3, measuring apparatus and cloth cylinders)

Surface (body) of constant width is the surface such that the distance between any two parallel tangent (supporting) planes is constant.

The model is a body with constant width $b = 12$ cm., and is obtained by revolution of a profile curve given by the parametric equation

$x(t) = p(t) \cdot \cos t - p'(t) \cdot \sin t,$

$y(t) = p(t) \cdot \sin t + p'(t) \cdot \cos t,$

where $p(t) = \frac{b}{2} \left(1 + \frac{1}{8} \cos (3t)\right)$ and $p'$ is the derivative of $p$ w.r.t. $t$.

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Illustration from the catalog: