#### 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∗x + a_i_2∗y + a_i_3∗x∗y + a_i_4∗x∗x + a_i_5∗y∗y

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

#### Synopsis

int QuadApprox( AstMapping ∗this, const double lbnd[2], const double ubnd[2], int nx, int ny, double ∗fit, double ∗rms )

#### Parameters:

##### this

Pointer to the Mapping.

##### lbnd

Pointer to an array of doubles 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

Pointer to an array of doubles containing the upper bounds of the box specifying the region over which the fit is to be performed.

##### nx

The number of points to place along the first Mapping input. The first point is at `"` lbnd[0]`"` and the last is at `"` ubnd[0]`"` . If a value less than three is supplied a value of three will be used.

##### ny

The number of points to place along the second 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.

##### fit

Pointer to an array of doubles in which to return the co-efficients of the quadratic approximation to the specified transformation. This array should have at least `"` 6∗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[0] + fit[1]∗X_in + fit[2]∗Y_in + fit[3]∗X_in∗Y_in + fit[4]∗X_in∗X_in + fit[5]∗Y_in∗Y_in Y_out = fit[6] + fit[7]∗X_in + fit[8]∗Y_in + fit[9]∗X_in∗Y_in + fit[10]∗X_in∗X_in + fit[11]∗Y_in∗Y_in

##### rms

Pointer to a double in which to return the RMS residual between the fit and the Mapping, summed over all Mapping outputs.

#### Returned Value

##### astQuadApprox()

If a quadratic approximation was created, a non-zero value is returned. Otherwise zero is returned and the fit co-efficients are set to AST__BAD.

#### Notes:

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

- A value of zero will be returned if this function is invoked with the global error status set, or if it should fail for any reason.

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