Figures

Chapter 5

Fig. 5-1 (above, left). Beginning the construction of the 90°-intercept transform of an ellipse about an interior point using four pairs of radius vectors.

Fig. 5-2 (above, right). Construction of conic sections in the non-incident polar-linear system about the point pole, pv, and line pole, pu for e = 1, e < 1, and e > 1.

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Fig. 5-3 (above, left). Two constructions of parabolas about a line pole, pv, and a circle pole, pu. There are two parabolas for each construction because there are both greatest and least distances to the circle pole from points on the locus.

Fig. 5-4 (above, right). The most symmetrical bipolar loci are the midline between the two poles and pairs of circles about the two poles.

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Fig. 5-5 (above, left). Parabola inversion quartics. The initial parabola (half of which is shown at left center) is inverted about points on the three lines shown (excluding the vertex, itself, which would yield the cissoid of Diocles, a cubic) to yield fourth-degree (quartic) curves.

Fig. 5-6 (above, right), The equilateral strophoid self inverts about the loop vertex, that is, distances from the loop vertex to the loop are the inverse of those from the loop vertex to the (exterior) arms along the same radius vector, one of which is shown.

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Fig. 5-7, Deriving the Devil's curve (three thick curvilinear segments) as the 90°-intercept transform of the centered circle about the centered equilateral hyperbola (sample intercepts X₁, X_1',Y₁) and, also, of the equilateral hyperbola about the circle (sample intercepts X₂ and Y₂). As far as is known, these identical 90°-reciprocal intercept transforms are unique.

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Fig. 5-8, Inversion quartics of an initial (or basis) ellipse about interior points on the lines of symmetry, and on the lines X = 3a/4 and Y = a/4, where a = the semi major axis. As for the parabola, the inverse loci about these poles are quartics.

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