Create a ChebyMap
A ChebyMap is a form of Mapping which performs a Chebyshev polynomial transformation. Each output coordinate is a linear combination of Chebyshev polynomials of the first kind, of order zero up to a specified maximum order, evaluated at the input coordinates. The coefficients to be used in the linear combination are specified separately for each output coordinate.
For a 1-dimensional ChebyMap, the forward transformation is defined as follows:
f(x) = c0.T0(x’
) $+$
c1.T1(x’
) $+$
c2.T2(x’
) $+$
...
where:
Tn(x’
) is the nth Chebyshev polynomial of the first kind:
T0(x’
) = 1
T1(x’
) = x’
Tn$+$1(x’
) =
2.x’
.Tn(x’
) $+$
Tn-1(x’
)
x’
is the inpux axis value, x, offset and scaled to the range [-1, 1] as x ranges over
a specified bounding box, given when the ChebyMap is created. The input positions,
x, supplied to the forward transformation must fall within the bounding box
- bad axis values (AST__BAD) are generated for points outside the bounding
box.
For an N-dimensional ChebyMap, the forward transformation is a generalisation of
the above form. Each output axis value is the sum of "
ncoeff"
terms, where
each term is the product of a single coefficient value and N factors of the
form Tn(x’
_i), where "
x’
_i"
is the normalised value of the i’
th input axis
value.
The forward and inverse transformations are defined independantly by separate sets of coefficients, supplied when the ChebyMap is created. If no coefficients are supplied to define the inverse transformation, the astPolyTran method of the parent PolyMap class can instead be used to create an inverse transformation. The inverse transformation so generated will be a Chebyshev polynomial with coefficients chosen to minimise the residuals left by a round trip (forward transformation followed by inverse transformation).
"
ncoeff_f$\ast $( 2
$+$ nin )"
elements.
Each group of "
2 $+$
nin"
adjacent elements describe a single coefficient of the forward transformation.
Within each such group, the first element is the coefficient value; the next element is
the integer index of the ChebyMap output which uses the coefficient within its
defining expression (the first output has index 1); the remaining elements of the
group give the integer powers to use with each input coordinate value (powers
must not be negative, and floating point values are rounded to the nearest
integer). If "
ncoeff_f"
is zero, a NULL pointer may be supplied for "
coeff_f"
.
For instance, if the ChebyMap has 3 inputs and 2 outputs, each group consisting of 5
elements, A groups such as "
(1.2, 2.0, 1.0, 3.0, 0.0)"
describes a coefficient with
value 1.2 which is used within the definition of output 2. The output value is
incremented by the product of the coefficient value, the value of the Chebyshev
polynomial of power 1 evaluated at input coordinate 1, and the value of the
Chebyshev polynomial of power 3 evaluated at input coordinate 2. Input coordinate
3 is not used since its power is specified as zero. As another example, the
group "
(-1.0, 1.0, 0.0, 0.0, 0.0 )"
adds a constant value -1.0 onto output
1 (it is a constant value since the power for every input axis is given as
zero).
Each final output coordinate value is the sum of the "
ncoeff_f"
terms described by the
"
ncoeff_f"
groups within the supplied array.
"
ncoeff_i$\ast $( 2
$+$ nout )"
elements.
Each group of "
2 $+$
nout"
adjacent elements describe a single coefficient of the inverse transformation,
using the same schame as "
coeff_f"
, except that "
inputs"
and "
outputs"
are
transposed. If "
ncoeff_i"
is zero, a NULL pointer may be supplied for "
coeff_i"
. "
nin"
elements. "
nin"
elements. "
nout"
elements. "
nout"
elements. "
printf"
format specifiers identified by "
%"
symbols in the
normal way. "
options"
string contains "
%"
format specifiers,
then an optional list of additional arguments may follow it in order to supply
values to be substituted for these specifiers. The rules for supplying these are
identical to those for the astSet function (and for the C "
printf"
function). A null Object pointer (AST__NULL) will be returned if this function is invoked with the AST error status set, or if it should fail for any reason.