Some two-dimensional loci connected with cross ratios

Abstract

THnORUM I. If the points z1, Z2, Z3 vary independently and have circular regions as their respective loci, then the locus of the point Z4 defined by the real constant cross ratio X=(Zl, Z2, Z3, Z4) is also a circular region. It is the purpose of the present paper to consider generalizations of and other results related to Theorem I, primarily the determination of the locus of the point Z4 defined as in Theorem I, when zi, Z2, Z3 vary independently so as to have certain prescribed loci. Thus one may raise the following question: If two variable points lie respectively on two fixed circles, where does the mid-point of their segment lie? The answer is contained in Theorem VI. Or again, if the loci of the points zi, Z2, Z3 are regions each bounded by a number of circles, what can be said of the locus of z4? The answer is given by Theorem VII. All the results proved concerning such loci as these can be interpreted in terms of the roots of the jacobian of two binary forms. Such an interpretation is given in ?12. The chief method used in the present paper to determine the locus of Z4 in any given case is that suggested in I (pp. 102, 103, footnote), namely the determination of the locus of z4 when z1 and Z2 are held fast and Z3 varies over its locus; the determination of the locus of this locus of z4 when z1 is kept fixed but Z2 varies over its locus; and finally the determination of the locus of this new locus of Z4 when z1 varies over its prescribed locus. In the present paper this method is carried through geometrically. The same method has been used by Professor A. B. Coblet to determine analytically the locus of Z4 in Theorem I,