The " Kissing " Circle A.K.A the Osculating Circle

Question

Find the equation of the osculating circle of the curve resulting as the intersection of the plane that the vector valued function r(t) = < cost , sint, sint > lies and a cylinder

Answer 1

x = cos(t), y = sin(t), z = sin(t). If the cylinder is lined up with the z-axis, then we have x²+y²=r², where r is the radius, then substituting, we'd have cos²(t)+sin²(t) = r² -> r=1. If the cylinder is lined up with the y-axis, then we have x²+z²=r², where r is the radius, then substituting, we'd have cos²(t)+sin²(t) = r² -> r=1. Finally, if the cylinder is lined up with the y-axis, then we have y²+z²=r², where r is the radius, then substituting, we'd have sin²(t)+sin²(t) = r² -> r²=2sin²(t).