Question

5. A torus is a tube that's been wrapped around into a circle. It's basically a donut. Its parametrization is given by x(0,0) =(R+r cos ) cose y(0 ) =(R+r cosp) sino z(0.) =r sin where R is the distance from the center of the circle made by the torus to the center of the tube, ris the radius of the tube, and 0 and p are the same as in spherical coordinates. Find an equation for the tangent plane at (xo, Yo,2o) at 0(0,0).

Answer 1

so, the equation of the tangent plane at the given point will be,

The parametric equation of the torus given as - $69,4) - (2(8,), y(0.4), 7(0,4)). . where, a(8,4)= (R+ reos 8) caso Y 10,4) - (R+7 €456) Siro 0,4€ (0,27? 2 (0,4) = Y Sine - and tangent blank of $ at P., Tp Q is spanned by 5000$? so, Tp is perpendicular to the veckor 04 x 24 News, 24 = f(r +965 ) Sine, (R+Ycos() ws B, o) and de - (-7 sin puse, -vsin plino, o losop). - I Como) = (0,.R+,0), 2 - ateloro) . li ş il .). Los 60,0) De Ico) loor. del (0,0) = (0,0,0). S ohlecox bello Er(R 48) ? = ((R+r), 0, 0) - now, of Co-ordinate of p is (%o, Yo, 2o) then, (x0, 40, 20) = 0(0,0). . 30, . (20, Yo, Zo) - (R+r,0,0). ..Tps is a plane in 12% passed through (Xo, Yo, 2o) and perpendicador to ñ - agen bola .com, -fr(R+r), 0, 0) - so, equation of tps will be (R+). (x – K.) + 0.(4-40) +0.(2-2) -0. CS Scannes virs R+Y)(x- (R+r)) = 0.ie x = R+r. CamSlannel

x = r + R, y,z in |R.