An Interesting Property of Hyperbolic Paraboloids
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1 Page v w Conider the generic hyperbolic paraboloid defined by the equation. u = where a and b are aumed a b poitive. For our purpoe u, v and w are a permutation of x, y, and z. A typical graph of uch a urface i hown in Figure. Figure t For > 0, the parametric pace curve, u( t = ( (, v t = t, w t = 0; a t acontruct a portion of the a parabola that i the interection of the u v plane (w = 0 with the hyperbolic paraboloid. The end point of thi parabolic arc have the u, v, w coordinate (, a,0 and (,,0 parametric pace curve, ( ( ( ( ( ( ( acoh t binh t a. Since =, the u t =, v t = acoh t, w t = binh t ; p t p contruct a portion of one branch of the hyperbola that i the interection of the hyperbolic paraboloid and the plane u =. Thi hyperbolic, a,0 and ha end point at, a coh ( p, b inh( p and arc i centered at the point (, a coh p, b inh p. Thee end point are labeled a and, repectively in Figure. Similarly, the parametric pace curve, ( ( ( ( ( u t, v t acoh t, w t binh t ; p t p a, coh, inh, a coh p, b inh p = = = contruct a portion of the econd branch of the ame hyperbola. Thi hyperbolic arc i centered at the point (,,0 labeled a 3 and 4 at ( a ( p b ( p and ( and ha end point, repectively. The 7//0
2 Page length of the hyperbolic arc to i identical to the length of the arc from 4 to 3 and i given by the following integral. p dv dw p p a + b b a dt a inh t b coh t dt e e dt p + = + = + + dt dt t t ( ( Figure t Along the negative u axi if > 0, the parametric pace curve, u( t = ( (, v t = 0, w t = t ; b t b b contruct a portion of the parabola that i the interection of the u w plane (v = 0 with the hyperbolic paraboloid. The end point of thi parabolic arc have the u, v, w coordinate (,0, b and (,0, b. The parametric pace curve, u( t =, v( t = ainh ( t, w( t = bcoh ( t ; p t p contruct a portion of one branch of the hyperbola that i the interection of the hyperbolic paraboloid and the plane centered at the point (,0, b (, a inh p, b coh p and ha end point at, ainh ( p, bcoh( p u =. Thi hyperbolic arc i and. Thee end point are labeled a 5 and 7, repectively in Figure. Similarly, the parametric pace curve, ( ( ( ( ( u t, v t ainh t, w t bcoh t ; p t p = = = contruct a portion of the econd branch of the ame hyperbola. Thi hyperbolic arc i centered at the point (,0, b and ha end 7//0
3 Page 3 point labeled a 6 and 8 at, ainh ( p, bcoh( p and, ainh ( p, bcoh( p, repectively. The length of the hyperbolic arc 7 to 5 i identical to the length of the arc from 8 to 6 and i given by the following integral. p dv dw p p a + b b a dt a coh t b inh t dt e e dt p + = + = + dt dt t t ( ( Now conider the line egment from to 5, to 6, 3 to 7, and 4 to 8. Under what condition do thee egment lie preciely on the urface of the hyperbolic paraboloid? The egment can be parameterized a follow: Segment Now all four parameterization have Parameterization u t = + t to 5 ( = coh( + inh coh( = inh + coh inh vt a p t a p a p wt b p t b p b p ut = + t to 6 ( ( = coh( + inh coh( ( = inh coh inh vt a p t a p a p wt b p t b p b p u t = + t 3 to 7 ( = coh ( inh coh ( = inh + coh inh vt a p t a p a p wt b p t b p b p u t = + t 4 to 8 ( = coh ( inh coh ( = inh coh inh vt a p t a p a p wt b p t b p b p ( coh ( p t coh ( p inh ( p coh ( p t inh( p coh( p a = + + wt ( inh ( p t inh ( p inh ( p coh ( p t coh( p inh( p = + b 7//0
4 Page 4 So that ( wt ( t inh ( p coh ( p coh ( p inh ( p = t p p p p inh coh ( coh inh( = + t + inh ( p p + t inh ( p coh ( p coh ( p inh ( p Along each egment if u( t t( match. Thu, ( ( wt = + i to equal + inh p p = inh coh( = coh inh ( p p p p The econd (t equation yield inh ( p coh ( p =± coh ( p inh ( p. If the + ign i taken, then = coh p inh p inh p coh p p p = e e which i impoible ince both and were aumed poitive. So, it mut be the cae that = + = + inh p coh p inh p coh p coh p inh p coh p inh p p p e = e ( p p = e The equation obtained from equating the t coefficient yield + inh p p = p ( p p p + ( e e = 0 Uing the quadratic formula, p p p p p p p p ( e e ± ( e e + 4 =, then the coefficient on t and t mut. 7//0
5 Page 5 p ( ( ( p p p p p p p e + e ± e + e + = ( p p ( p p ( p p e + e ± e + e p p p p e + e ± e + e = = ( p p = e or = e p p p p p p p p But a wa tated before, both and are aumed poitive, o ( p p ( p p = e = e, the ame condition required for the t term to match. Thu, the condition that ( p p = e i both neceary and ufficient for the four linear egment from to 5, to 6, 3 to 7, and 4 to 8 to lie preciely on the urface of the hyperbolic paraboloid. In particular, if = then p = p. Note: in mot cae the length of the hyperbolic arc at u = will not match the length of the hyperbolic arc at u = even if =. Only when a = b are thee arc of equal length. 7//0
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