A representation of digital hyperbolas y = 1/x α + β

Abstract

It is proved that digital hyperbola segments and their least squares hyperbola fits are in one-to-one correspondence. This enables a constant space representation of a digital hyperbola segment inscribed into the integer grid. Such a representation is (x 1 , n, a, b), where x 1 is the x-coordinate of the left endpoint of the digital hyperbola segment, n is the number of its integer points , while a and b are the coefficients of the least squares hyperbola fit Y = 1/x a + b of the given digital hyperbola segment. An O(n max{log n, log x 1 }) algorithm for obtaining a digital hyperbola segment from its least squares hyperbola fit is described.