By Luigi. BIANCHI
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Additional resources for Congruenze di sfere di Ribaucour e superficie di Peterson
19 hands, stop the time at 9:40 and start to slowly turn it around. You 17 may do this on a piece of paper where you have already marked the positions of 12, 3, 6, and 9 as if on a compass (N, E, S, W); you may of course mark the other hours in between as well. 19 demonstrates, your turning watch’s hands will approximately show 2:00 or 4:10, right? Could you tell how much you should turn your watch in order to ‘attain’ these times? ), while a clockwise turn by 195 0 will ‘change the time’ from 9:40 to approximately 4:10 (in fact 4:12′ :44 ″ , but you don’t have to worry about that right now).
21. How do we find the image P′ of any given point P under this rotation? We simply draw KP, measure it either with a ruler or with a compass, then ‘build’ a 70 0 angle ‘to the left hand’ of KP with the help of a protractor, and finally pick a point P ′ on the angle’s ‘new’ leg so that |KP ′ | = |KP|. That’s all! Fig. 4 It’s an isometry! 2 and prove that every rotation is indeed an isometry. We return to our watch example and prove that |LS| = |L′ S ′ |, which says that the distance between the two images L′ , S′ is equal to the distance between the two original points L, S; the general case is proven in exactly the same way.
25 Similarly, y′ = |OF| = |OB| + |BF| = |OB| + |HP′| = |OB| + |KP′ |sinθ ′ = = |OB| + |KP|sinθ cosφ + |KP|cosθ sinφ = |OB| + |GP|cosφ + |GK|sinφ = = |OB| + (|OD| − |OB|)cosφ + (|OA| − |OC|)sinφ = = b + (y − b)cosφ + (a − x)sinφ = = (sinφ )a + (1− cos φ )b − (sinφ )x + (cosφ )y, as claimed. 26 below: now θ ′ = θ − φ , hence cosθ′ = cosθ cos φ + sinθsinφ and sinθ′ = sinθcosφ − cosθsinφ; |GK| = |OA| − |OC| and |GP| = = |OD| − |OB| remain valid. You should be able to fill in the details and derive the claimed formulas for x ′ and y′ .