Obtain a quadratic approximation to a 2D Mapping

#### Description:

This function returns the co-efficients of a quadratic fit to the supplied Mapping over the input area specified by LBND and UBND. The Mapping must have 2 inputs, but may have any number of outputs. The i’ th Mapping output is modelled as a quadratic function of the 2 inputs (x,y):

output_i = a_i_0 $+$ a_i_1$\ast$x $+$ a_i_2$\ast$y $+$ a_i_3$\ast$x$\ast$y $+$ a_i_4$\ast$x$\ast$x $+$ a_i_5$\ast$y$\ast$y

The FIT array is returned holding the values of the co-efficients a_0_0, a_0_1, etc.

#### Invocation

RESULT = AST_QUADAPPROX( THIS, LBND, UBND, NX, NY, FIT, RMS, STATUS )

#### Arguments

##### THIS = INTEGER (Given)
Pointer to the Mapping.
##### LBND( $\ast$ ) = DOUBLE PRECISION (Given)
An array containing the lower bounds of a box defined within the input coordinate system of the Mapping. The number of elements in this array should equal the value of the Mapping’ s Nin attribute. This box should specify the region over which the fit is to be performed.
##### UBND( $\ast$ ) = DOUBLE PRECISION (Given)
An array containing the upper bounds of the box specifying the region over which the fit is to be performed.
##### NX = INTEGER (Given)
The number of points to place along the first Mapping input. The first point is at LBND( 1 ) and the last is at UBND( 1 ). If a value less than three is supplied a value of three will be used.
##### NY = INTEGER (Given)
The number of points to place along the second Mapping input. The first point is at LBND( 2 ) and the last is at UBND( 2 ). If a value less than three is supplied a value of three will be used.
##### FIT( $\ast$ ) = DOUBLE PRECISION (Returned)
An array in which to return the co-efficients of the quadratic approximation to the specified transformation. This array should have at least " 6$\ast$Nout" , elements. The first 6 elements hold the fit to the first Mapping output. The next 6 elements hold the fit to the second Mapping output, etc. So if the Mapping has 2 inputs and 2 outputs the quadratic approximation to the forward transformation is:

X_out = fit(1) $+$ fit(2)$\ast$X_in $+$ fit(3)$\ast$Y_in $+$ fit(4)$\ast$X_in$\ast$Y_in $+$ fit(5)$\ast$X_in$\ast$X_in $+$ fit(6)$\ast$Y_in$\ast$Y_in Y_out = fit(7) $+$ fit(8)$\ast$X_in $+$ fit(9)$\ast$Y_in $+$ fit(10)$\ast$X_in$\ast$Y_in $+$ fit(11)$\ast$X_in$\ast$X_in $+$ fit(12)$\ast$Y_in$\ast$Y_in

##### RMS = DOUBLE PRECISION (Returned)
The RMS residual between the fit and the Mapping, summed over all Mapping outputs.
##### STATUS = INTEGER (Given and Returned)
The global status.

#### Returned Value

• This function fits the Mapping’ s forward transformation. To fit the inverse transformation, the Mapping should be inverted using AST_INVERT before invoking this function.