A geometric look at corner cutting

G. de Rham (1947), Un peu de math ematiques a propos d'une courbe plane, Elemente der Mathematik, 2(4):89{104. G. de Rham (1956), Sur une courbe plane, Journal de Math ematiques pures et appliqu ees, 35:25{42. 27 Figure 28: Each point in the shaded area carries a continuum of lines such that the associated aane standard cut produces C 1-curves without line segments. Hence for each contact point (line) one can nd a continuum of contact lines (points) such that either the associated projective or aane standard cutting procedure produces a C 1-curve without line segments. Remark 13.2 Dual to Remark 10.2 one has that the aane standard cut produces parabolas if it lies on the parabola through a and b with tangents ac and bc. Addendum After writing the paper we became aware of unpublished notes by Bernd Mulansky with similar ideas as in the proof of Theorem 6.1. 26 Since A 0 and A 1 map lines onto lines, they also describe maps on {. These are the inverse dual maps B 0 = (A 0) ?1 and B 1 = (A 1) ?1 which map the ideal line onto itself and the lines (^ = points of {) ca, ab, bc onto da, ag, de and de, gb, be, respectively. Thus the envelope of the standard curve is obtained by successive applications of B 0 and B 1 to ca and bc. In other words, the polar scheme is given by the standard triangle ca, ab, bc in { , the contact line g and the contact point de while the ideal line of { assumes the role of m. Next we use the coordinates of Section 12 and the polarity with respect to the circle around m through a and b. Then the ideal line and the lines ca, ab, bc are mapped onto the points m, a, c, b. Further, the points on the cubic given in Remark 12.1 correspond to the tangents of the curve " x(t) y(t) # = 1 (t 2 ? t + 1) 2 " (1 ? t) 3 (1 + t) t 3 (2 ? t) # ; while the points in the interior of the shaded area shown in Figure 24 correspond to cuts which do not intersect this curve. This is illustrated in Figure 27: If the aane standard cut intersects the standard triangle only in the shaded …