Results 1 to 2 of 2

Math Help - How to prove that a point in Euclidian space lies OUTSIDE a given triangular pyramid?

  1. #1
    Junior Member
    Joined
    Nov 2009
    Posts
    60

    How to prove that a point in Euclidian space lies OUTSIDE a given triangular pyramid?

    So if given a particular point, say (0,3,4)

    how can you prove it lies OUTSIDE or INSIDE a pyramid with 4 vertices, say A(0,2,0), B(2,0,0), C(0,2,2), D(2,2,2)? (i just made these numbers up.


    help much appreciated!!!
    Follow Math Help Forum on Facebook and Google+

  2. #2
    Super Member
    Joined
    Mar 2008
    Posts
    934
    Thanks
    33
    Awards
    1
    Quote Originally Posted by Yehia View Post
    So if given a particular point, say (0,3,4)

    how can you prove it lies OUTSIDE or INSIDE a pyramid with 4 vertices, say A(0,2,0), B(2,0,0), C(0,2,2), D(2,2,2)? (i just made these numbers up.


    help much appreciated!!!
    Here is one way that will work-- doubtless there are others.

    1. Find the equation of the plane through ABC in the form f(x,y,z) = ax + by + cz = 0. I'm guessing you know how to compute a, b, and c. If not, let us know.

    2. Compute f(D). In your example, that is 2a +2b + 2c.

    3. Compute f(P) where P is your point of interest. In your example, that is 0a + 3b + 4c.

    4. If the numbers computed in steps 2 and 3 have the same sign, then P and D lie on the same side of the plane ABC, so P may be inside the tetrahedron. If they don't have the same sign, P is definitely outside the tetrahedron.

    5. If P passes the test (f(P) and f(D) have the same sign), repeat steps 1-4 for the other three faces of the tetrahedron: ABD, ACD, and BCD. If P passes the test for all the faces, it is inside the tetrahedron. That's because the tetrahedron is convex and is the intersection of the four half-spaces formed by the bounding planes.
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. triangular pyramid
    Posted in the Pre-Calculus Forum
    Replies: 1
    Last Post: June 10th 2011, 09:51 AM
  2. Replies: 3
    Last Post: February 24th 2011, 02:57 PM
  3. Replies: 0
    Last Post: February 21st 2011, 07:47 AM
  4. Area of a triangular pyramid
    Posted in the Geometry Forum
    Replies: 2
    Last Post: April 18th 2010, 12:58 PM
  5. [SOLVED] Geometry HELP!!! Triangular Pyramid
    Posted in the Geometry Forum
    Replies: 1
    Last Post: February 5th 2008, 11:39 PM

Search Tags


/mathhelpforum @mathhelpforum