# Focal Surfaces

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Related KMR-pages:

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Other related sources of information:

Evolutes at Wikipedia
Involutes at Wikipedia

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The interactive simulations on this page can be navigated with the Free Viewer
of the Graphing Calculator.

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Focal Surfaces in Two Dimensions

A two-dimensional focal surface is called an evolute.

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Ellipse – Evolute:

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EllipseEvolute – Involutes:

The interactive simulation that created this movie.
Drag the purple point at the bottom in order to change the involute.

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Ellipse – Parallel Curves:

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Focal Surfaces in Three Dimensions

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Hyperbolic Paraboloid – Focal Surface (patch 1):

The yellow normal to the green Hyperbolic Paraboloid is moving along the light-blue principal direction of the green Hyperbolic Paraboloid. At the same time, this yellow normal is rolling along a light-blue geodesic on the top focal sheet of this Hyperbolic Paraboloid.

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Hyperbolic Paraboloid – Focal Surface (patch 2):

The yellow normal to the green Hyperbolic Paraboloid is moving along the red principal direction of the green Hyperbolic Paraboloid. At the same time, this yellow normal is rolling along a red geodesic on the bottom focal sheet of this Hyperbolic Paraboloid.

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Hyperbolic Paraboloid – Focal Parallels:

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Hyperbolic Paraboloid – Focal Surface total 1:

The yellow normal to the green Hyperbolic Paraboloid is moving along the blue principal direction of the green Hyperbolic Paraboloid. At the same time, this yellow normal is rolling along a blue geodesic on the top focal sheet of this Hyperbolic Paraboloid.

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Hyperbolic Paraboloid – Focal Surface total 2:

The yellow normal to the green Hyperbolic Paraboloid is moving along the red principal direction of the green Hyperbolic Paraboloid. At the same time, this yellow normal is rolling along a red geodesic on the bottom focal sheet of this Hyperbolic Paraboloid.

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Elliptic Paraboloid – Focal surface patch:

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Elliptic Paraboloid – Focal Surface total:

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