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Intersection of Three Planes

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Agenda

  • How Can Three Planes Intersect?
  • Practice

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1.

How Can Three Planes Intersect?

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Intersecting Three Planes

  • Look at this GeoGebra Applet to determine all the ways that three planes intersect
  • When solving for solutions, we are always looking at the normals of the planes to see how the intersections occur or do not occur

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Intersection of Three Planes

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Case 1: Intersection at a Point

  • No collinear normals
  • Solve system of equations from three planes using scalar form
  • Result is single, unique point

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Case 2a, 2b: Intersection In Line

  • No collinear normals, or two collinear normals
  • Solve system of equations from three planes using scalar form
  • Result is single line with one free parameter
    • And a line 0=0

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Case 3: All planes Coincident

  • All normals collinear
  • All scalar forms are multiples of each other
  • Infinite number of solutions

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Non-Intersection of Three Planes

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Case 1: Three Distinct Intersection Lines in Pairs

  • No collinear normals
  • In the solution of the three scalar forms, get result 0=some non-zero constant

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Case 2: Two Distinct Intersection Lines

  • Two collinear normals
  • In the solution of the three scalar forms, get result 0=some non-zero constant

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Case 3, 4: No Intersection, Planes Parallel

  • All normals are collinear
  • Either two or zero planes are multiples of each other

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2.

Practice

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Practice

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Credits

Special thanks to all the people who made and released these awesome resources for free:

  • Presentation template by SlidesCarnival

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