. Plane and solid analytic geometry . Fia. 1 62 ^ c2 a2 + -=1, ^+y-=i. a^ The parts of these ellipses which lie in the quarter-planesbounding the first octant connect the points (a, 0, 0), (0, b, 0),(0, 0, c), as shown. The section of (1) by a plane z = k parallel to the {x, y)-plane has the equations : 548 QUADRIC SURFACES 549 If Ti^ < c2, these equations represent, in the plane z = k, anellipse whose center is on the axis of z and whose axes lie inthe {z, x)- and (y, 2)-planes and have the lengths 2a \/i-! Wi-J. As k increases from 0 toward c as its limit, this ellipse, risingfrom the sec
. Plane and solid analytic geometry . Fia. 1 62 ^ c2 a2 + -=1, ^+y-=i. a^ The parts of these ellipses which lie in the quarter-planesbounding the first octant connect the points (a, 0, 0), (0, b, 0),(0, 0, c), as shown. The section of (1) by a plane z = k parallel to the {x, y)-plane has the equations : 548 QUADRIC SURFACES 549 If Ti^ < c2, these equations represent, in the plane z = k, anellipse whose center is on the axis of z and whose axes lie inthe {z, x)- and (y, 2)-planes and have the lengths 2a \/i-! Wi-J. As k increases from 0 toward c as its limit, this ellipse, risingfrom the section by the (x, 2/)-plane, grows continuouslysmaller and shrinks finally to a point, — the point (0, 0, c).Similarly, if k decreases from 0 toward — c as its limit. The surface generated by thechanging ellipse is the 2 (or Fig. 6 of Ch. XXII *)shows it in its entirety. The sur-face is evidently symmetric in theorigin, 0, and in the coordinateaxes and coordinate planes. O iscalled the center of the ellipsoid; Fig. 2 the coordinate axes, the axes of the ellipsoid ; and the coordinate planes, the principal planes of theellipsoid. The sections by the principal planes are known asthe principal sections. The dimensions of the ellipsoid, measured along the axes,are 2a, 2b, 2c. These numbers, in the order of their magni-tude, are known as the major axis, mean axis, and minor axisof the ellipsoid. Here, and throughout the chapter, we speak of a (plane)section of a surface only when the plane in ques
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