Editing Projection-slice theorem (section)

== The FHA cycle ==

If the two-dimensional function ''f''('''r''') is circularly symmetric, it may be represented as ''f''(''r''), where ''r''&nbsp;=&nbsp;|'''r'''|. In this case the projection onto any projection line
will be the [[Abel transform]] of ''f''(''r''). The two-dimensional [[Fourier transform]]
of ''f''('''r''') will be a circularly symmetric function given by the zeroth-order [[Hankel transform]] of ''f''(''r''), which will therefore also represent any slice through the origin. The projection-slice theorem then states that the Fourier transform of the projection equals the slice or

: <math>F_1 A_1 = H,</math>

where ''A''<sub>1</sub> represents the Abel-transform operator, projecting a two-dimensional circularly symmetric function onto a one-dimensional line, ''F''<sub>1</sub> represents the 1-D Fourier-transform
operator, and ''H'' represents the zeroth-order Hankel-transform operator.