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Caption: Mathematical Models Catalog of a Collection of Models of Ruled Surfaces
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20 the square of the distance from the axis of any point at a height z is therefore z2 x2 + y2 = r2 cos2 0 t- . g ? 2 siii2 0, - > and this is the equation of the surface. N o w with reference to the string in the the tangent plane ; before deformation, its distance from the axis will be r sec <p; after deformation, the position of its upper end will be x = r sec (p cos (<p + 6), y = r sec ^ sin (<p -{- 0), 2 = c, and of its lower end x = r sec p cos (^ — 0), ?/ = r sec ^ sin (^ — o)3 z ~ ~ c. Hence its equations will be x — r sec <p cos (p + 0) r sec f {cos ( p + 0) — cos (^ — o)} < __ 3/ cos <p — t sec ^ sin (<p + 0) __z — c "~ r sec 5 {sin (/> + 0) — sin (p -—6)) ~~ 3 <• or, # cos p — r cos ( p + 0) < r sin <£ sin 0 __ 3/ cos p — r sin (<-/> -f 0) __ £ — c r cos ^ sin 5 c We easily find from these 1 1 ^ z . -gin 0 — tan 0 cos ( 9 2c r e t z — 2c r - — cos 0 — tan < sin 0 p r c and we can eliminate tan <p by simple division. This gives us the equation of the surface, since < is a parameter which p indicates only the position of the line in the surface, a hyperbolic paraboloid. It will be an easy and useful exercise for the student to find the other system of generating lines of the paraboloid, and to show that they are horizontal. T h e leading properties connected with rectilinear generation are as follows :— A tangent plane to the hyperboloicl cuts the surface in two right lines, which intersect at the point of contact.
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