Posts tagged 'bilinear'

F.2 Bilinear Transformations

The canonical bilinear transformation B(x) = (1 + x)/(1 - x) and its inverse B^{-1}(y) = (y - 1)/(y + 1) link the internal division ratio kappa = x/r of a segment AB to the Apollonian distance ratio k = (1 + kappa)/(1 - kappa).

F.27 Harmonic Mean

If C and D are harmonic conjugates with respect to segment AB (so |AC|/|CB| = |AD|/|BD|), then the length |CD| is the harmonic mean of |BD| and |AD|: |CD| = 2 |BD| |AD| / (|BD| + |AD|) = r (gamma + 1/gamma).