By Pascal Chossat
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Additional info for Methods in Equivariant Bifurcations and Dynamical Systems
Sample text
FIG. 6 37. Correspondence between harmonic conjugates. Given four harmonic points, A, B, C, D; if we fix A and C, then B and 38. Separation of harmonic conjugates 23 D vary together in a way that should be thoroughly understood. To get a clear conception of their relative motion we may fix the points L and M of the quadrangle K, L, M, N (Fig. 6). Then, as B describes the point-row AC, the point N describes the point-row AM perspective to it. Projecting N again from C, we get a pointrow K on AL perspective to the point-row N and thus projective to the point-row B.
On the fixed ray SD. , on the fixed line DS'. These last four harmonic points give four harmonic rays CA, CA1, CA2, CA3. Therefore the four points A which project to B in four harmonic rays also project to C in four harmonic rays. But C may be any point on the locus, and so we have the very important theorem, Four points which are on the locus, and which project to a fifth point of the locus in four harmonic rays, project to any point of the locus in four harmonic rays. 67. The theorem may also be stated thus: The locus of points from which, four given points are seen along four harmonic rays is a point-row of the second order through them.
Fundamental theorem. Postulate of continuity 33 48. We may also give an illustration of a case where two superposed projective point-rows have no self-corresponding points at all. Thus we may take two lines revolving about a fixed point S and always making the same angle a with each other (Fig. 10). They will cut out on any line u in the plane two point-rows which are easily seen to be projective. For, given any four rays SP which are harmonic, the four corresponding rays SP' must also be harmonic, since they make the same angles with each other.