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Caption: Mathematical Models Catalog of a Collection of Models of Ruled Surfaces
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29 Ruled Surface with director Plane, Let the equations of the line be A.x + B y + G z = 7 c ax + % = 1 (1.) ' (2.) in which «, £, and /?,, are arbitrary constants, Differentiating implicitly A + Bf + Cft + t?)=° (3-) a# «-Mg = 0. (4.) Proceeding to a second differentiation, in which,- by virtue dy of the last equation, w e m a y consider-^ as constant, w e dx obtain dx2 "*" ~ tfa% WW ' r/ya \dxj ' V ; Then, if we take (2) as the director plane, that is to say, if w e make the director plane parallel to the axis of z, w e must substitute in (5) the value of -~ obtained, from (4), and we get aar ax ay ay* or in the usual notation b2r — 2abs + a2t^ 0. (6.) If, however, we take (1) as the director plane, we must determine y- from (3), which gives dy __ __ Gp ~}~ A. <S ~~ ~~ G q -f B and the differential equation becomes (C? •+ B)2r - 2 (C? •+ B) (Gp + A) 5 + (Cp 4 A)2 * = 0. " (7.) which reduces to the same Form ns (6) if C ~= 0. \a# dx dxj x '
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