Radial Transformations

These one-dimensional grids can be transformed to other kinds of intervals using the radial transform module.

Radial Transformation

Transform	Domain	Co-Domain	\(r(x)\)
BeckeRTransform	\([-1,1]\)	\([r_{min},\infty)\)	\(r_i = R \frac{1 + x_i}{1 - x_i} + r_{min}\)
LinearFiniteRTransform	\([-1,1]\)	\([r_{min},r_{max}]\)	\(r_i = \frac{r_{max} - r_{min}}{2} (1 + x_i) + r_{min}.\)
InverseRTransform	Co-Domain	Domain	\(x_i(r_i)\)
IdentityRTransform	Domain	Domain	\(x_i\)
LinearInfiniteRTransform	\([0,\infty)\)	\([r_{min},r_{max})\)	\(r_i = \frac{(r_{max} - r_{min})}{N - 1} x_i + r_{min},\)
ExpRTransform	\([0,\infty)\)	\([r_{min},r_{max}]\)	\(r = r_{min} e^{x \log\bigg(\frac{r_{max}}{r_{min} / (N - 1)} \bigg)},\)
PowerRTransform	\([0,\infty)\)	\([r_{min},r_{max}]\)	\(r = r_{min} (x + 1)^{\frac{\log(r_{max}) - \log(r_{min})}{N}},\)
HyperbolicRTransform	\([0,\infty)\)	\([0,\infty)\)	\(r_i = \frac{a x_i}{(1 - bx_i)},\)
MultiExpRTransform	\([-1,1]\)	\([r_{min},\infty)\)	\(r_i = -R \log \left( \frac{x_i + 1}{2} \right) + r_{min}\)
KnowlesRTransform	\([-1,1]\)	\([r_{min},\infty)\)	\(r_i = r_{min} - R \log \left( 1 - 2^{-k} (x_i + 1)^k \right)\)
HandyRTransform	\([-1,1]\)	\([r_{min},\infty)\)	\(r_i = R \left( \frac{1+x_i}{1-x_i} \right)^m + r_{min}\)
HandyModRTransform	\([-1,1]\)	\([r_{min},r_{max}]\)	\(r_i = \frac{(1+x_i)^m (r_{max} - r_{min})} { 2^m (1 - 2^m + r_{max} - r_{min}) - (1 + x_i)^m (r_{max} - r_{min} - 2^m )} + r_{min}\)