(Solved): 27. Find the geodesic curves on the unit sphere, that is, the shortest curves connecting pairs of ...

27. Find the geodesic curves on the unit sphere, that is, the shortest curves connecting pairs of points on it. What changes if the radius of the sphere is \( \rho \) ? HINT: Look for a function \( \theta(\phi) \). At some opportune point, rewrite \( \sin \phi \sqrt{\sin ^{2} \phi-C_{1}^{2}}= \) \( \sin ^{2} \phi \sqrt{1-C_{1}^{2}\left(1+\cot ^{2} \phi\right)} \) (derive this!), and introduce a new variable \( u=\cot \phi \). The final result should yield \( z=a x+b y \) on the sphere, where \( a \) and \( b \) are constants.