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HOROPTER CURVE

Curve studied by Helmholtz, Ludwig and Schur
in 1902.
From the Greek horos "limit" and opter
"observer".
Other name: cubic circle. See on wikipedia a more general definition of this curve in the study of binocular vision. |

System of Cartesian equations: .
Cartesian parametrization: , or, with t : = tan (t / 2): .
Rational 3D cubic. |

The horopter curve is the intersection between the cylinder of revolution with axis x = a, z = 0 and passing by O and the two rectangular hyperbolic paraboloids with of equation and .
We take out from this intersection the line Oy in the first case, and the line x= 2a, y=0 in the second case, this line being common to both the quadrics.
The horopter curve is obviously also the intersection between the two aforementioned hyperbolic paraboloids. |

The Cartesian parametrization shows that the horopter curve is a cylindrical tangent wave; when the cylinder on which it is traced is developed, we get a tangentoid.

The projections on the planes *xOy*, *xOz* and
*yOz* are the witch of Agnesi , the circle and the anguinea , respectively.

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© Robert FERRÉOL 2018