It is well known that the line of intersection of an ellipsoid and a plane is an ellipse. In this note simple formulas for the semi-axes and the center of the ellipse are given, involving only the semi-axes of the ellipsoid, the componentes of the unit normal vector of the plane and the distance of the plane from the center of coordinates. This topic is relatively common to study, but, as indicated in [1], a closed form solution to the general problem is actually very difficult to derive. This is attemped here. As applications problems are treated, which were posed in the internet [1,2], pertaining to satellite orbits in space and to planning radio-therapy treatment of eyes
Many regulated health insurance markets include risk adjustment (aka risk equalization) to mitigate selection incentives for insurers. Empirical studies on the design and evaluation of risk‐adjustment algorithms typically focus on mandatory health insurance schemes. This paper considers risk adjustment in the context of voluntary health insurance, as found in Chile, Ireland, and Australia. In addition to the challenge of mitigating selection by insurers, regulators of these voluntary schemes have to deal with selection by consumers in and out of the market. A strategy for mitigating selection by consumers is to apply some form of risk rating. Our paper shows how risk adjustment and risk rating interact: (1) risk rating reduces the need for risk adjustment and (2) risk adjustment reduces premium variation across rating factors, thereby increasing incentives for consumers to select in and out of the market.
A contiguous derivation of radius and center of the insphere of a general tetrahedron is given. Therefore a linear system is derived. After a transformation of it the calculation of radius and center can be separated from each other. The remaining linear system for the center of the insphere can be solved after discovering the inverse of the corresponding coefficient matrix. This procedure can also be applied in the planar case to determine radius and center of the incircle of a triangle.
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