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ORTHOPTIC SURFACE

From the Greek orthos "right" and optikos
"relating to sight". |

The *orthoptic* surface of a surface is the locus
of the points through which pass 3 planes tangent to the surface and perpendicular
2 by 2.

Examples:

- the orthoptic surface of a sphere
with radius *R* is a concentric sphere with radius .

- more generally, the orthoptic surface
of a centered quadric is a sphere
called *orthoptic sphere* or *Monge sphere *; for example, for
the ellipsoid of half-axes *a, b, c,* the Monge sphere has same center
that the ellipsoid and radius .

- The orthoptic surface of the elliptic
paraboloid :
is a plane called Monge plane, of equation .

A similar notion, bearing the same name, is that of *orthoptic
surface* of a *subset* of the space: locus of the vertices of the
right trihedra that circumscribe X (i.e. "containing" X, and the three
faces of which meet X).

Example: the orthoptic of a circle with radius *R*
is a sphere with the same center and radius .

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© Robert FERRÉOL 2022