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Math Help - how to define the orientation of a surface

  1. #1
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    how to define the orientation of a surface

    Hi;

    Normals n are imaginary lines perpendicular to each point on a surface. If u and v gives the orientation of the surface at a given point, then the triplet (u,v,n) matches the orientation of R^3 according to the right-hand rule.

    My question is given a point on the surface, how can one determine the direction of u and v? If we determine the direction of u and v then the direction of the normal vector will be determined by using the right-hand rule.

    I don't know how to determine the direction of u and v at a given point on the surfcae.

    Every guidance is highly appreciated.

    Thank you in advance
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  2. #2
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    Re: how to define the orientation of a surface

    Hey student2011.

    Orientation of a surface (in 3D) is determined by the tangent vectors and the Jacobian. If the Jacobian is positive you have a right-handed orientation. If it is negative, you have a left handed orientation. If it is zero, then you need to evaluate what is going on.

    You can't determine orientation from a point alone, but I'm guessing you want to do what I said above. Calculate the Jacobian and see if you get a positive or negative answer.
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  3. #3
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    Re: how to define the orientation of a surface

    Thank you so much chiro for your very helpful reply.

    I know that for a convex polygon, a surface normal can be calculated as the vector cross product of two non parallel edges of the polygon. The direction of the normal vector is then given by the right-hand rule.

    If we want to apply this for example to a plane oriented clockwise. Then the magnitude of the cross product between any two non-parallel edges is same. But the problem is with the direction. Take for example a point on the plane at the lower-left corner, note that the orientation normal to this point is pointing away from us, unlike the direction of the orientation normal to the upper-right point on the plane which is pointing towards us. This is a contradiction, since the plane is oriented. So we must define the orientation normal to the palne in a unique way either its away or towards us.


    May be I miss understood the idea, how can I define the direction of the orientation normal to an orientable surface?

    Thank you in advance
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  4. #4
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    Re: how to define the orientation of a surface

    In 3D space, you can check orientability by putting in the vectors in the order of tangent, bi-normal, and normal and if the Jacobian is positive then you have positive orientability. If its negative you have a left handed system and if its zero, then you have zero vector or linear dependence.

    A simple example for a plane that spans the z = 0 surface has <1,0,0> , <0,1,0> and <0,0,1> as tangent, binormal and normal vectors respectively (that have been normalized). If I throw these in a matrix I get

    [1 0 0]
    [0 1 0]
    [0 0 1]

    The determinant of this is +1 indicating a right hand orientation. If for some reason the triple was [0 1 0] [1 0 0] [0 0 1] then the determinant would be -1 indicating a left hand orientation.

    If you want to define a plane that is oriented clock-wise, the determinant of your tangent, bi-normal and normal must satisfy the positive determinant property.
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