Πέμπτη, 9 Ιουνίου 2016

INVERSION - Angle

The angle between two smooth curves at a point of intersection is usually defined as the angle between the tangents to the curves at that point. Since the tangent itself is the limit of a set of chords, it will be sufficient for us to show that QPR (at left) is preserved under the inversion transformation.
Suppose that under the inversion, points P, Q, R map to points P', Q', R' as at right.
Then points P, P', Q', Q are concyclic (property of inverse points), so OPQ = OQ'P'. Similarly, points P, P', R', R are concyclic so OPR = OR'P'.
Now 
QPR = OPR – OPQ = OR'P' – OQ'P' = Q'P'R'. 
This shows that the angle between curves is preserved. Some care is required when the point of contact of the two curves is O itself. 
Here, for example, two curves touching at O invert into two parallel lines, and two circle intersecting at O invert into two straight lines not through O. In this last case, the angle is of course preserved.

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