One way to get a curve on a sphere of radius 1 is to take an arbitrary curve Y(t) in three-space not passing through the origin and define X(t) = unit(Y(t)). The curve X(t) then lies on the unit sphere and the normal to that sphere is N(t). We then get X(t)*X(t) = 1 so 2X'(t)*X(t) = 0 and 2X''(t)*X(t) + 2X'(t)*X'(t) but X''(t) = s''(t)T(t) + s'(t)2(kg(t)U(t) + kN(t)N(t)) so X''(t)*N(t) = s'(t)2kN(t) + s'(t)2 = 0 so kN(t) = -1 for all t. Moreover N'(t) = X'(t) = s'(t)T(t) so N'(t)*U(t) = 0 = ?g(t)s'(t) and ?g(t) = 0.
- A sphere is specified in the control window, and its radius may
be changed by changing the variable R.
- A curve is displayed on the circle, and the curve is the
projection of the curve x(t) onto the sphere. This projected
curve is specified in the control window.
- A point along this image curve is a white sphere specified by
changing the value of t0.
- The tangent vector at this point T is displayed in red. The
normal vector N is displayed in blue. The U vector is displayed in magenta.
- Readouts in the control window display the values of geodesic
curvature, geodesic torsion, and normal curvature at the white point
specified by t0.
- Readouts in the control window display the total geodesic
curvature, geodesic torsion, and normal curvature along the whole image
- The normal curvature along the whole curve is displayed in pink
in the "Kn(t)" window.
- The geodesic curvature along the whole curve is displayed in
orange in the "Kg(t)" window.
- The geodesic torsion along the whole curve is displayed in cyan
in the "Tg(t)" window.