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# ftkurvpd

Calculates an interpolatory spline for closed parametric curves; it also calculates first and second derivatives of the interpolatory spline.

## Prototype

```	procedure ftkurvpd (
xi  [*] : numeric,
yi  [*] : numeric,
t   [*] : numeric,
xo  [*] : float,    ; or double
yo  [*] : float,    ; or double
xd  [*] : float,    ; or double
yd  [*] : float,    ; or double
xdd [*] : float,    ; or double
ydd [*] : float     ; or double
)
```

## Arguments

xi

A 1D array of any size (npts) containing the abscissae for the input function.

yi

A 1D array containing the npts functional values of the input function - yi(k) is the functional value at xi(k) for k=0,npts-1.

t

Contains a 1D array of mpts values for the parameter mapping onto the interpolated curve.

xo

(output)
A 1D array containing the X values for the interpolated points. t(k) maps to (xo(k),yo(k)) for k=0,mpts-1.

yo

(output)
A 1D array containing the Y values for the interpolated points. t(k) maps to (xo(k),yo(k)) for k=0,mpts-1.

xd

(output)
A 1D array containing the first derivatives of the X component with respect to t.

yd

(output)
A 1D array containing the first derivatives of the Y component with respect to t.

xdd

(output)
A 1D array containing the second derivatives of the X component with respect to t.

ydd

(output)
A 1D array containing the second derivatives of the Y component with respect to t.

## Description

ftkurvpd is in the Fitgrid package - a package containing 1D and 2D interpolators using cubic splines under tension.

There are some parameters that can alter the behavior of ftkurvpd. These parameters all have reasonable default values. However, users may change any of these parameters by invoking ftsetp prior to calling ftkurvpd. ftkurvpd is called after all of the desired values for control parameters have been set.

The only control parameter that applies to ftkurvpd is: sig.

Given a sequence of distinct input points ( (x(0),y(0)), ... , (x(n-1),y(n-1))), the interpolated curve is parameterized by mapping points in the interval [0.,1.] onto the interpolated curve. The resulting curve has a parametric representation both of whose components are splines under tension and functions of the polygonal arc length. The value 0. is mapped onto (x(0),y(0)) and the value 1. is mapped onto (x(0),y(0)) as well (completing the closed curve).

The value for the parameter sig specifies the tension factor. Values near zero result in a cubic spline; large values (e.g. 50) result in nearly a polygonal line. A typical value is 1. (the default).

You can extrapolate values with ftkurvpd (that is, calculate interpolated values for abscissae outside of the domain of the input), but these values are, in general, unreliable.

## Examples

```begin
x = (/  3.0,  4.0,  9.0, 16.0, 21.0, 27.0,    \
34.0, 36.0, 34.0, 26.0, 18.0         /)
y = (/  2.4,  9.6, 14.4, 12.0,  9.6,  8.4,    \
13.2, 21.6, 30.0, 37.2, 38.4         /)

mpts = 201
u = fspan(0.,1.,mpts)

xo  = new( (/ mpts /), float)
yo  = new( (/ mpts /), float)
xd  = new( (/ mpts /), float)
yd  = new( (/ mpts /), float)
xdd = new( (/ mpts /), float)
ydd = new( (/ mpts /), float)
ftkurvpd(x, y, u, xo, yo, xd, yd, xdd, ydd)
end
```

## Errors

= 1 if the number of input points is less than 2.
= 2 if adjacent coordinate pairs coincide.