Plane and solid analytic geometry; an elementary textbook . Ch. XV, § 102] HIGHER PLANE CURVES 197 Then FP = V(z - c)2 + ~y\ and FP = V(x + c)2 + = m2. Hence [0 - c)2 + f] [O + c)2 + y2] = m4, or (a? + y2 + c2)2 - 4 c2x2=m* is the equation of the Cassinian oval. The intercepts of the curve on the axes are ± Vc2 ± m2and ± Vm2 — c2. Hence if cra, the curve cuts the X-axis in four realpoints but does not cut the I^-axis. It must, therefore,consist of two distinct ovals, as shown in Fig. 100. If c = m, all the intercepts are zero and the curve. Fig. 100. goes through the origin. In thi
Plane and solid analytic geometry; an elementary textbook . Ch. XV, § 102] HIGHER PLANE CURVES 197 Then FP = V(z - c)2 + ~y\ and FP = V(x + c)2 + = m2. Hence [0 - c)2 + f] [O + c)2 + y2] = m4, or (a? + y2 + c2)2 - 4 c2x2=m* is the equation of the Cassinian oval. The intercepts of the curve on the axes are ± Vc2 ± m2and ± Vm2 — c2. Hence if cra, the curve cuts the X-axis in four realpoints but does not cut the I^-axis. It must, therefore,consist of two distinct ovals, as shown in Fig. 100. If c = m, all the intercepts are zero and the curve. Fig. 100. goes through the origin. In this case the equation re-duces to x2 + y2 = 2 c2(x> - y2), or in polar coordinates, p2=2c2 cos 2 6. This special form of the Cassinian oval is called thelemniscate. It has already been discussed on page 67. 198 ANALYTIC GEOMETRY [Ch. XV, § 103 103. The cissoid. — The cissoid may be defined asfollows : on any diameter OA of a circle, lay off equal dis-tances CM and CN on each side of the centre, and at thepoints M and N erect MK and NL perpendicular to the diameter. Draw OK and locus of the intersection ofOK with NL and OL with MKis the cissoid of Diodes. To obtain its rectangular equa-tion, let OA be the X-axis and 0the origin. Then 031= NA = x,and
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