Figures

Appendix I

Fig. A-1 (above, right). Biovular Cassinians ("b" biovals) self-invert at 0° and 180°. But the monovular members ("b" monovals) self-invert at 90° [e.g., the outer monoval in b (labeled "0.5") inverts to the second-most inner monoval in "a" (labeled "0.5")]. The equilateral lemniscate (labeled in b) is the known curve of demarcation between Cassinian ovals and biovals ("b"). The equilateral hyperbola (labeled in "a") is the newly discovered curve of demarcation of inverse Cassinian ovals and biovals ("a"); discussed in text.

Fig. A-2 (above, left). The eccentricity of conic sections can be illustrated as a 'visible' property by erecting a line through a focus at a right angle to the axis (F), known as a latus rectum. The magnitude of the slope of a tangent-line to the curve where the latus rectum intersects it is the conic's eccentricity.

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Fig. A-3 (above, left). The most symmetrical curves in the previously largely unknown non-incident (point-pole not on circle-pole) polar-circular coordinate system. At a1, for point-poles outside the circle, and u = v, a two arm hyperbola; one arm for the least-distance from the circle construction, the other for the greatest distance. The foci are at the point-pole and center of the circle. At a2, for the point pole within the circle-pole, and u = v, an ellipse. Only a least distance construction exists. Again, the point-pole and circle-center become foci. For both a1 and a2, the circle-pole is a directrix. At b1, for the construction rule u - v = Cd, a 2-arm hyperbola with arms of different sizes, one for a greatest-distance, the other for a least-distance construction. At b2, for the construction rule u + v = Cd, two ellipses of different sizes, for greatest- and least-distance constructions. In all cases, the point-poles and circle-centers become foci of all curves.

Fig. A-4 (above, right). Incident polar-circular conics. At a, for u - v = d/2, a 1-arm hyperbola for the greatest distance construction. At b, for u + v = d/2, composite joined segments of an hyperbola and an ellipse, for least-distance constructions. At c, for v - u = d/2, composite joined, supplemental segments of the same ellipse and hyperbola as in b, both for least-distance constructions. At d, for u - v = d, a line (hyperbola for e = infinity), for greatest distances. At e, for u + v = d, a composite joined line and ellipse segments, for least-distance constructions. At f, for v - u = d, the supplemental joined line and ellipse segments of illustration (e). Point-poles and circle-centers become foci of resulting partial conics.

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Fig. A-5 (above, left). Linear-circular parabolas, u ± v = ±d (with d = 2R for all constructions). At a, for ugreatest - v = d, a conventional parabola. At b, for uleast - v = d, joined open parabolic segments from two conventional parabolas. At c, for uleast + v = d, joined, closed, finite segments from three conventional parabolas. At d, for v - uleast = d, joined open parabolic segments (one finite, the others infinite) from two conventional parabolas. These four constructions (Figs. AI. 5a-d) complement one another in including all parts of all four participating conventional parabolas.

Fig. A-6 (above, right). Linear-circular ellipses, u =ev =v/2. At a, for d = 5R/2, greatest- and least-distance constructions exist, with the circle-center being a focus of both. At b, with d = 2R, a least-distance construction exists but the other construction for the small inner ellipse collapses to a point locus at the center of the circle. At c, a least-distance constructions yield both ellipses. Similarly at d, where the inner ellipse becomes tangent to both the outer ellipse and the line pole. In all cases both the circle- and line-poles are directrices, since there is no constant term in the construction rules.

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Fig. A-7. Symmetry of rectangular circles. A series from the least to the most symmetrical circles in the rectangular coordinate system is illustrated. At the top, the general circle about a point in the plane, with five terms and four parameters (R,h,k,2), i.e., (5,4). The next-most symmetrical circle is the general bisector (5,3). Next-most is the general tangent circle (5,3). Next comes the tangent bisector circle (5,2). There is some arbitrariness in selecting the most symmetrical circle with four terms. In Fig. AI-7, the sequence general circle including the origin (4,3), .bisector circle (4,2), and general axial circle (4,3) have been selected.). Next-most symmetrical is the axial circle including the origin (3,2) and the centered circle (2,1). Most symmetrical is the centered point circle (2,0).

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Fig. A-8. The most symmetrical bipolar loci and their corresponding linear construction rules. The e = _ and e = 2 limacons (d1 and d2, respectively) are obtained by inverting the e = _ ellipse and e = 2 hyperbola (c1 and c2, respectively) about their traditional foci.

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