2.1. CONIC SECTIONS119Here, we give a simplified and anachronistic version of the basic ideas inBook I, bowlderizing [25, pp. 355-9].Conical Surface:Start with a horizontal circleC; on the vertical linethrough the center ofC(theaxis6) pick a pointAdistinct from thecenter ofC. The union of the lines throughAintersectingC(thegenerators) is a surfaceKconsisting of two cones joined at theircommon vertex (Figure2.1). If we put the origin atA, the axisFigure 2.1: Conical SurfaceKcoincides with thez-axis, andKis the locus of the equation inrectangular coordinatesz2=m2(x2+y2)(2.1)wherem= cotαis the cotangent of the angleαbetween the axis and the generators.Horizontal Sections:A horizontal planeHnot containingAintersectsKin a circle centered on the axis. Theyz-plane intersectsHin a linewhich meets this circle at two points,BandC; clearly the segmentBCis a diameter of the circle. Given a point
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