We define the inner (+) and the outer ( - ) part of the Cartesian Oval as mr₁±nr₂=kc on the bipolar cooordinates. (if m=n, then it is ellipse and hyperbola. ), through these definition and some other properties, the Cartesian oval is the pure extention of conic section with the same composition.
Last time, we considerr on Minor axis of the inner part of the Oval. This time, we consider on axis of the outer part ( mr₁ - nr₂ =kc: k>m>n>o) . This asymmetry major axis (of the outer part) is a segment which connects the middle point 0 of symmetry - axis and the point F _p on the oval, which is at the longest distance from the point 0. Then, the point F _p on the outer part of the oval is equally distant from two pales (foci) . And the length of major axis is α√1+e_L⋅e_R ( α is a half of the length of the symmetry-axis), and e _L, e _R are left and right eccentricity of the Oval, respectively.Moreover, the line OF _p is the normal line at point F _p of the Oval.About this major axis, the proof and properties are the same as the minor axis. Next, we show that the perpendicular bisector of minor axis of inner part, or major axis of outer part pas through the third focus point.
When we consider on both parts of the Oval, it apears that Cardioid (r= a (1-cosθ ) ) is the special case of Cartesian Oval. In this case, e _L, e _R equals 1 and the length of the major axis is √2 α .That is to say, the radius of Circumcircle of Cardioid is √2 times a half length of symmetry - axis. Moreover, we have found the following Lemma which is universal on Euclidean metric.
[Lemma] Let the length of Minor axis of the inner part of the Oval be b_i, and let the half length of symmetry - axes of the inner, outer part be a_i, a_o, respectively.Let the length of the major axis of the outer part be b _o. Then we have
[b_i/a_i] ²+ [b_o/a_o] ²=2
These additional new properties of the oval of Descartes will be shown in this paper.

抄録全体を表示