Question

Question 2. In this problem we consider the following two surfaces (i) The circular cylinder of radius 1/2 and axis of rotation the z-axis (ii) The sphere of radius 1 and centre (-1/2,0,0) These two surfaces intersect in a curve called Viviani's curve. (a) (7 marks) Show that for all real t, (t) = (cos2t -2 curve. (In fact y(t) is a parametrisation of Viviani's curve, but you need not prove this.) (b) (6 marks) Show that y(t) has speed V1 cos2 t. (c) (7 marks) Find the curvature K of the curve y(t) in terms of t sint cos t sin t) lies on Viviani's

Answer 1

whose axis is z-axts Equation f ctrcular cylinder of radius -(i) Tha sphere sradi centerad at (0,0) 1. 1 The intersecton of (1) and 2) in Viviani's CNrVe Cot-Snt (os t, sint) = ((t), (),( t)= Cot- = (sa) Given Y whore 2 ,_(t) = sint Cost = 1.2snt (ost Sin 2t . sint NGw Theretire Y () en on ) 2 ONd 4Catsins £(-25)sw 1 thenatore ) .. lies on 2) Viviani's aurve Hence y(t) uen Knoo that speed {^H}'+ } vLt]}* + 5x,'(1)- V v') 2si2t) We Sin 2t (t Cost Sint Tuane ure speed of (= N Esn 2+)+ /os 2t) 4eoa t)i Sinzt Los2t Zst

qet (t)'), (E), )) (-Sinet, Cos 2t, Cost) From secton (b) we Tharotora t)-2 Cos 2t, - 2sin2t, -simt) Now = - -Sin 2t+ oit+ Cost =N I+t (,x (, -2Cos2E and - Sin2t Cost 2Sin2t Sint (2 Sinat (st- Cos2t Sint) i- (sin zismt + 200324) j4 (2 Sint20sžt) 2 sinakest- s t sint)î - (snztsint+ 2n2t)f+ 22 ojar V')x 1 = 2sin2tlst- (os2t sint)(Sn2t sut +2%t'+ 2 Sn 2t Ca3t-Ca2t int (Sin zt Sint +2sit 4 Heuce curvadur k - IG7, (2sM 2t lst-lus2tsint)" Swet snt +22t) i 4