Body of constant brightness
geometric measure
In convex geometry, a body of constant brightness is a three-dimensional convex set all of whose two-dimensional projections have equal area. A sphere is a body of constant brightness, but others exist. Bodies of constant brightness are a generalization of curves of constant width, but are not the same as another generalization, the surfaces of constant width.
The name comes from interpreting the body as a shining body with isotropic luminance, then a photo (with focus at infinity) of the body taken from any angle would have the same total light energy hitting the photo.
Properties
A body has constant brightness if and only if the reciprocal Gaussian curvatures at pairs of opposite points of tangency of parallel supporting planes have almost-everywhere-equal sums.
According to an analogue of Barbier's theorem, all bodies of constant brightness that have the same projected area
A
{\displaystyle A}
as each other also have the same surface area,
A
/
π
{\displaystyle \textstyle {\sqrt {A/\pi }}}
. This can be proved by the Crofton formula.
Example
The first known body of constant brightness that is not a sphere was constructed by Wilhelm Blaschke in 1915. Its boundary is a surface of revolution of a curved triangle (but not the Reuleaux triangle). It is smooth except on a circle and at one isolated point where it is crossed by the axis of revolution.
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This entry incorporates text from “Body of constant brightness” on English Wikipedia. Contributors are listed in the page history. Text is available under the Creative Commons Attribution-ShareAlike 4.0 License. Selected authority identifiers and statements are retrieved from Wikidata under CC0; their references and qualifiers remain part of the verification path.