This page is a sub-page of our page on Geometric Optics.
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Related KMR-pages:
• Conics
• Optical properties of Conics
• Evolutes and Involutes
• Parallel Curves
• Caustics
• Quadrics
• Optical properties of Quadrics
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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:
The interactive simulation that created this movie.
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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:
The interactive simulation that created this movie.
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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.
The interactive simulation that created this movie.
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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.
The interactive simulation that created this movie.
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Hyperbolic Paraboloid – Focal Parallels:
Interactive simulation of Hyperbolic Paraboloid – Focal Parallels 1 (rotating camera).
Interactive simulation of Hyperbolic Paraboloid – Focal Parallels 1 (rainbow focal sheets).
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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.
The interactive simulation that created this movie.
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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.
The interactive simulation that created this movie.
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Elliptic Paraboloid – Focal surface patch:
The interactive simulation that created this movie.
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Elliptic Paraboloid – Focal Surface total:
The interactive simulation that created this movie.
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