This page is a sub-page of our page on Substitution in Integrals
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The sub-pages of this page are:
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Related KMR-pages:
• Partial Differential Equations
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Other relevant sources of information:
• Separation of variables in Differential Equations
• Orthogonal coordinates
• Curvilinear coordinates
• Confocal conics sections
• Elliptic coordinate system
• Parabolic coordinates
• Elliptic cylindrical coordinates
• Ellipsoidal coordinates
• Confocal quadrics
• Paraboloidal coordinates
• Confocal Ellipsoidal coordinates
• Confocal Paraboloidal coordinates
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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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The interactive simulation that created this movie
The interactive simulation that created this movie
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The interactive simulation that created this movie
The interactive simulation that created this movie
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The interactive simulation that created this movie
The interactive simulation that created the thee movies above.
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The interactive simulation that created this movie.
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The interactive simulation that created this movie.
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The interactive simulation that created this movie
The interactive simulation that created this movie
The interactive simulation that created this movie
The interactive simulation that created this movie
The interactive simulation that created this movie
Parabolic cylindrical coordinates
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Confocal paraboloids acting as mirrors to a planar wavefront
Confocal paraboloids reflecting a planar wavefront into parallel Dupin cyclides
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Reflection of a planar wavefront in a saddle surface
turns the wavefront into a Dupin cyclide:
Confocal-paraboloids-planar-wavefront-into-cyclide-reflection.gcf
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A set of (horizontally positioned) confocal paraboloids give the same (Dupine cyclide) reflection of a planar wave front that is incident along the vertical axis:
The interactive simulation that created this movie.
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The same variation of the confocal paraboloids mirrors is shown here without the reflected wavefront (which is the same for all the paraboloids of the confocal family:
Confocal Paraboloids – reflect vertical rays into Dupin cyclide.gcf
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