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I need to deal with comparing directions of normals and in order to do this I need to know whether normals are always treated as if the surface that they come from was the origin or whether normals can in some cases be defined with respect to the "global" origin?

Visually,

enter image description here

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    The normal is given as vector notation (meaning that it technically extends from the origin) but you consider it translated to the point on the object to which it is perpendicular. Hence the green is representative. – 2017-02-11
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    depends what you are doing. The ambiguity you mention led (in differential geometry), eventually, to the tangent bundle. Look up the Frenet Serret frame – 2017-02-11
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    @lulu That the vector starts from the point of the surface where it's. I.e. that it's "as if" the surface point was the origin. – 2017-02-11
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    The red arrow is the sum of the position vector of a point on the surface and the normal vector at that point. I suppose that vector might be useful for _something,_ but I don't know what. – 2017-02-11

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