Open image in new window Fig. 1 Spherical orthotomic ruled surface.
In this paper, a method for determination of developable spherical orthotomic ruled surfaces is given by using dual vector calculus.
This solution includes an integral constant, thus we have infinitely many developable spherical orthotomic ruled surfaces such that each of them has a base curve (q(t)).
We show that dual vectorial expression of a developable spherical orthotomic ruled surface can be obtained from coordinates and the first derivatives of the base curve.
end{aligned}The graph of the developable spherical orthotomic ruled surface given by this equation for (C=0) in domain begin{aligned} D left{ begin{array}{c} -2le tle 2 -4le ule 4 end{array} right.
Moreover, note that (theta ^(t)), given by (4.6), has two values; by using the minus sign, then we obtain the reciprocal of the spherical orthotomic ruled surfaces obtained by using the plus sign for a given integral constant.
But in this case unknowns (Q_{1},Q_{2},) and (Q_{3}) are begin{aligned} Q_{1}&= theta ^sin varphi, nonumber Q_{2}&= theta ^cos varphi, Q_{3}&= -varphi ^. nonumber end{aligned} (4.7 Consider the same curve (q(t)=(t^{2},the2},2t+1),) the spherical orthotomic ruled surface is obtained as begin{aligned} m u,t)=left( t^{2}+frac{ut}{sqrt{2}},t^{2}-frac{ut}{sqrt{2}},2t+1right).
Then we can find a developable spherical orthotomic ruled surface such that, its base curve is the curve (q(t)) and by (4.3), we have begin{aligned} tan varphi =frac{Q_{1}}{Q_{2}},quad text theta ^=pm sqrt{ Q_{1}^{2}+Q_{2}^{2}}quad text { and }quad varphi ^=-Q_{3}.
end{aligned} (4.4 If this spherical orthotomic ruled surface is developable, then (Delta =0) and by (4.4), we get begin{aligned} frac{dvarphi ^{dt}frac{dvarphi }{dt}sin ^{2}theta +theta ^left( frac{dvarphi }{dt}right) ^{2}sin theta cos theta + frac{dtheta ^{dt}frac{dtheta }{dt}=0.
Secondly, a new cubature Kalman filtering (CKF) algorithm is developed on the basis of the spherical-radial cubature rule for approximating such nonlinear integrals.
The meta-cognitive component use the instantaneous error of the sample and spherical potential of the rule antecedents to select the best training strategy for the current sample.
