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Math Help - Test for 2 pentagrams being diametric that share an outer end point.

  1. #1
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    Test for 2 pentagrams being diametric that share an outer end point.

    Hi,

    I'm having trouble finding a good way to test for 2 pentagrams being diametric.

    The pentagrams share two lines and an endpoint, so it looks like they're attached.

    I want to test if they're diametric or if they overlap.

    I thought it would be easy. I get the coordinates of the common endpoint, I take the coordinates of the center of each pentagon and make sure one is above it, one below it and makes sure the same is true for left and right. Good? No, sorry. Two stars can be diametric and have both centers of their pentagons on one side of the common endpoint. That's because the star arms can be long and skewed, which puts the center of the pentagon at different places.

    Anybody have any clue how else I could test this??


    Any help is appreciated. Thanks.
    Last edited by Stockgoblin; September 20th 2013 at 01:14 PM.
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  2. #2
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    Re: Test for 2 pentagrams being diametric that share an outer end point.

    Could you define your terms? I Googled "diametric pentagons" and "diametric polygons" -- nothing. For that matter, what is a pentagram? From your reference to skewed arms, I assume you don't mean a regular pentagram. Now pentagon: surely you don't mean any polygon with 5 vertices? I suspect your term pentagon refers to a convex polygon with 5 vertices??
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  3. #3
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    Re: Test for 2 pentagrams being diametric that share an outer end point.

    sorry, a pentagram is a star made of five lines, the inner points make a pentagon. When 2 of them share an endpoint they either overlap or they don't (diametric).
    Last edited by Stockgoblin; September 20th 2013 at 05:56 PM.
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  4. #4
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    Re: Test for 2 pentagrams being diametric that share an outer end point.

    Hi again,
    Look at the two attachments; I won't try and prove anything unless this is what you want.

    Test for 2 pentagrams being diametric that share an outer end point.-mhfgeometry40.png

    Test for 2 pentagrams being diametric that share an outer end point.-mhfgeometry40a.png

    I think it is impossible that P5 and P6 are on "opposite" rays.
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  5. #5
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    Re: Test for 2 pentagrams being diametric that share an outer end point.

    wow, that's great and exactly what I'm dealing with. Thanks. Yes, I think I see what you're saying. So for a star to be "diametric" or "opposite" to another, the outer points on the rays that make up the common endpoint must be in reverse order to the comparison star or an overlapping star that shares the common endpoint.

    I think I can work with that.

    P.s. - I don't need a mathematical proof as such. Just a test where I take 2 stars and see if they're like your first diagram, second or there's even a third scenario. Here's some sample data of it, 2 stars sharing two lines and an end point but laying side on side: check it out:

    slope: Y-intercept: Outer X: Outer Y:

    star1:
    1.98,36.58,6.819672131, 50.08295082
    0.76,44.9, -38.34375, 15.75875
    0.44,32.63, 1328, 616.95
    0.43,45.91, -30.16358325, 32.9396592
    -9.84,-263.87, -25.41878173, -13.74918782

    star2:
    1.98, 36.58, 89.76923077, 214.3230769
    1.85, 48.25, -5.830188679, 37.46415094
    1.32, 45.16, 50.42857143, 111.7257143
    1.11, 55.75, -31, 21.34
    0.76, 44.90, 6.819672131, 50.08295082
    Last edited by Stockgoblin; September 22nd 2013 at 10:35 AM.
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  6. #6
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    Re: Test for 2 pentagrams being diametric that share an outer end point.

    Scenario one - "opposite" data - compared to star1 above
    slope:, Y-intercept
    1.30, 84.75
    0.76, 44.90
    0.44, 32.63
    0.38, 22.29
    -0.82,-25.79

    Scenario 2: "overlap" data - compared to star1 above
    slope:, Y-intercept
    1.98, 36.58
    1.85, 48.25
    -0.82, -25.79
    -1.40 ,-24.57
    -9.84 ,-263.87
    Last edited by Stockgoblin; September 22nd 2013 at 11:39 AM.
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  7. #7
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    Re: Test for 2 pentagrams being diametric that share an outer end point.

    I organize my data like above, with the slopes in order of largest to smallest and the X & Y coordinates of the outer points beside it.

    I have noticed that for scenario 1 "opposite" and scenario 2 "overlap", matching slopes between those stars were in the same position in the list of slopes whereas in scenario 3, the slopes involved were in a different ranked position.

    To distinguish between scenario 1 & 2, I'm thinking I'll have to involve the coordinate pairs though.

    any ideas?
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