デカルトの卵形線の短軸および卵形面 Minor Axis of the Oval of Descartes and Ovaliod
Descartes' oval is defined as mr<SUB>1</SUB> + nr<SUB>2</SUB> = kc by using bipolar coordinates. Where, if m=n, it is ellipse. According to this definition and a number of the properties, it can be said that the Descartes' oval is essential extension of ellipse.<BR>This time, the minor axis of oval that has the similar properties to those of the minor axis of ellipse is found. This minor axis is the segment connecting the middle point 0 of the major axis (the axis of symmetry) of oval and the point Np on the oval, which is at the shortest distance from the point 0. The length of this minor axis is expressed by α√1-e<SUB>L</SUB>e<SUB>R</SUB>, where α is a half of the length of the major axis, and e<SUB>L</SUB> and e<SUB>R</SUB> are left and right eccentricities, respectively. As for this minor axis, its proof and a number of the properties are discussed.<BR>Next, the method of defining ovaloid which is convex, closed curved surface in space by extending the oval on plane is found, therefore, it is reported. This ovaloid has, as the contours of the orthographic projection from three directions, circle, Descartes' oval and a fourth order curve like ellipse. Further, the parametric expression of this ovaloid is derived. In this way, the new properties of oval are able to be added, therefore, it is reported.
図学研究 (68), 3-8, 1995-06-01