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Math Help - intersection of two curves on unit sphere

  1. #1
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    intersection of two curves on unit sphere

    Hi everyone,

    I am working on automatic 3D Mesh morphing.

    I have two sets of two points each. Say, A1 and A2; and B1 and B2 all lying on the same unit sphere (centered at origin). I want to find whether the shortest curve (also lying on the sphere) joining A1 and A2 intersects the curve joining B1 and B2...and also wish to know the location of the intersection point.

    Note that the curves do not extend beyond A1,A2 and B1,B2.

    Thanks a lot in advance.
    Thanks a lot in advance.
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  2. #2
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    Quote Originally Posted by zeeshanzia84 View Post
    Hi everyone,

    I am working on automatic 3D Mesh morphing.

    I have two sets of two points each. Say, A1 and A2; and B1 and B2 all lying on the same unit sphere (centered at origin). I want to find whether the shortest curve (also lying on the sphere) joining A1 and A2 intersects the curve joining B1 and B2...and also wish to know the location of the intersection point.

    Note that the curves do not extend beyond A1,A2 and B1,B2.

    Thanks a lot in advance.
    Thanks a lot in advance.
    I cannot answer your question because I don't know how to find the equation of a spherical curve bounded by two points on the sphere.

    I know that the shortest curve lies on the great circle passing through the two given points. I now also that if curve A1A2 and curve B1B2 intersect each other, then there is one point on the sphere that is common to both spherical curves. So if you know the equations of both curves, just solve them simultaneously to find if they have this common point, the intersection point.

    If there are more than one intersection points, then the two curves lie on the same great circle and they are touching/overlapping each other on those many intersection points.
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  3. #3
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    Quote Originally Posted by zeeshanzia84 View Post
    Hi everyone,

    I am working on automatic 3D Mesh morphing.

    I have two sets of two points each. Say, A1 and A2; and B1 and B2 all lying on the same unit sphere (centered at origin). I want to find whether the shortest curve (also lying on the sphere) joining A1 and A2 intersects the curve joining B1 and B2...and also wish to know the location of the intersection point.

    Note that the curves do not extend beyond A1,A2 and B1,B2.

    Thanks a lot in advance.
    Thanks a lot in advance.
    I would have thought that it will depend on the coordinates of the two points ..... Some simple experiments with an orange should demonstrate this ......
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