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Caption: Mathematical Models Catalog of a Collection of Models of Ruled Surfaces
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19 A n d since (a, /3, y) is an arbitrary point on the line, the equation of the surface m a y be obtained by writing x9 y, z, instead of a, $y y, in this equation, that is ayz -f hzx -f cxy + ctbc = 0, The asymptotic cone is ayz + bzx + cxy = 0, which may also be written as a b c - + - + - = 0. x y z These are the forms of equation which, arise most directly from the rectilinear generation of the surface. Another form of the equation is X* a2 2 z* "c2 ~ 1, the asymptotic cone them becc>ming X2 a* z2 0. c2 ~ Let us try to follow analytically the deformation indicated in model N o . 16, starting, for simplicity, from the right cylinder. Take for two of the axes the centre line of the cylinder and the line from its middle point to the middle point of its line of contact with the tangent plane. Instead of supposing that one ring only turns, it will be easier to suppose that w e turn them both equally, one backwards and the other forwards, through an angle = 0. Suppose, also, that for the tangent plane w e take a string whose distance from the point of contact is tan < . p Before deformation the equation of the cylinder will be , 2 + y2 = r2, and its tangent plane y = r. ' ? T h e ends of the line of contact will be, at the top x =: r cos 0, y = r sin 0, z = c, and at the bottom x = r cos 0, y = — r sin 0, z = — c. The equations of the line of contact will therefore be II Z x — v cos 0 = 0, . -= rsm 9 c
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