A conic of revolution (around the z axis) can be defined by the equation
s^2 – 2*R*z + (k+1)*z^2 = 0
where s^2=x^2+y^2, R is the vertex radius of curvature, and k is the conic constant: k<-1 for a hyperbola, k=-1 for a parabola, -1<k<0 for a tall ellipse, k=0 for a sphere, and k>0 for a short ellipse.
Write a function z=conic(s,R,k) to calculate height z as a function of radius s for given R and k. Choose the branch of the solution that gives z=s^2/(2*R)+... for small values of s. This defines a concave surface for R>0 and a convex surface for R<0.
The trick is to get full machine precision for all values of s and R. The test suite will require a relative error less than 4*eps, where eps is the machine precision.
Hint (added 2015/09/03): the straightforward solution is
z = (R-sqrt(R^2-(k+1)*s^2))/(k+1),
but this does not work if k=-1, gives the wrong branch of the solution if R<0, and is subject to severe roundoff error if s^2 is small compared to R^2. It is possible, however, to find a mathematically equivalent form of the solution that solves all three problems at once.
17469 Solvers
992 Solvers
Find common elements in matrix rows
844 Solvers
60 Solvers
Flip the vector from right to left
1005 Solvers
Problem Tags