Thread: Proof using frenet equations

(R-Rc).(R-Rc)=a^2 where R=R(s) lies on the surface of a sphere

prove R(s)=Rc-pn -1/t*(dp/ds)b where p = curvature t = torsion n is the unit normal and b is the unit binormal

getting to the proof involves differentiating the original equation and then using the Frenet equations to simplify

Questions? (Rc) and (a) would be considered constants?

if so the
first derivative is 2*(dR/ds.(R-Rc))
sencond derviative is 2*(d^2R/ds^2.(R-Rc)+dR/ds.dR/ds)
third derivative is 2*(d^3R/ds^3.(R-Rc)+3*d^2R/ds^2.dR/ds)

I have tried using the Frenet equations along with R'(s) = v where v is unit tangent and R''(s)=dv/ds

any suggestions on where to go from here?

Lots of derviative taking and using the fact that unit vector cross itself = 0 and unit vector dot itself =1