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Math Help - Sphere surface area

  1. #1
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    Sphere surface area

    I was wondering if someone could help explain the following to me.



    dw = sin(\theta) d\theta d\phi

    If only angles are given, how can you compute the surface area? And is it just trigonometry that is being used to computer the edge lengths?
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  2. #2
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    Grandad's Avatar
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    Hello floater
    Quote Originally Posted by floater View Post
    I was wondering if someone could help explain the following to me.



    dw = sin(\theta) d\theta d\phi

    If only angles are given, how can you compute the surface area? And is it just trigonometry that is being used to computer the edge lengths?
    The sphere is assumed to have a radius of 1 unit. \theta and \phi are the 'latitude' and 'longitude' of any point on its surface.

    \theta is the angle measured upwards from the horizontal ( x- z) plane through the centre of the sphere - the point's 'latitude', as it were.


    \phi is the 'longitude' of the point. This is the angle, measured anticlockwise around the 'equator', between two planes: the vertical plane containing the
    great circle* through the point and the y- z plane.

    A neighbouring point (\theta+d\theta, \phi+d\phi) is chosen, defining an area of the sphere's surface which, for small increases in \theta and \phi, is approximately rectangular.


    Simple trigonometry will show that the radius of the 'circle of latitude' is \sin\theta. So an angle of d\phi at its centre is subtended by an arc of length \sin\theta d\phi on its circumference.


    The angle d\theta is subtended at the centre of the great circle by an arc of length d\theta, since this circle has unit radius.


    Hence the dimensions of the approximate rectangle are \sin\theta d\phi \times d\theta. Its area is therefore given by
    d\omega = \sin\theta d\theta d\phi
    Grandad

    * A great circle drawn on the surface of a sphere is one that has its centre at the centre of the sphere; its radius is equal to the radius of the sphere.
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  3. #3
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    Thankyou Grandad. ( that sounds so wrong )
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