e01eac creates a Thiessen triangulation with a given set of two-dimensional data points as nodes. This triangulation will be as equiangular as possible (Cline and Renka (1984)). See Renka and Cline (1984) for more detailed information on the algorithm, a development of that by Lawson (1977). The code is derived from Renka (1984).

The computed triangulation is returned in a form suitable for passing to e01ebc which, for a set of nodal function values, computes interpolated values at a set of points.

4 References

Cline A K and Renka R L (1984) A storage-efficient method for construction of a Thiessen triangulation Rocky Mountain J. Math. 14 119–139

Lawson C L (1977) Software for

C^{1}

surface interpolation Mathematical Software III (ed J R Rice) 161–194 Academic Press

Renka R L (1984) Algorithm 624: triangulation and interpolation of arbitrarily distributed points in the plane ACM Trans. Math. Software 10 440–442

Renka R L and Cline A K (1984) A triangle-based

C^{1}

interpolation method Rocky Mountain J. Math. 14 223–237

5 Arguments

1: $n$ – Integer Input: On entry: $n$ , the number of data points.

Constraint: $n \geq 3$ .
2: $x [n]$ – const double Input: On entry: the $x$ coordinates of the $n$ data points.
3: $y [n]$ – const double Input: On entry: the $y$ coordinates of the $n$ data points.
4: $triang [7 \times n]$ – Integer Output: On exit: a data structure defining the computed triangulation, in a form suitable for passing to e01ebc. Details of how the triangulation is encoded in triang are given in Section 9. These details are most likely to be of use when plotting the computed triangulation which is demonstrated in Section 10.
5: $fail$ – NagError * Input/Output: The NAG error argument (see Section 7 in the Introduction to the NAG Library CL Interface).

6 Error Indicators and Warnings

NE_ALL_DATA_COLLINEAR: On entry, all the $(x, y)$ pairs are collinear.
NE_ALLOC_FAIL: Dynamic memory allocation failed.
See Section 3.1.2 in the Introduction to the NAG Library CL Interface for further information.
NE_BAD_PARAM: On entry, argument $〈value〉$ had an illegal value.
NE_INT: On entry, $n = 〈value〉$ .
Constraint: $n \geq 3$ .
NE_INTERNAL_ERROR: An internal error has occurred in this function. Check the function call and any array sizes. If the call is correct then please contact NAG for assistance.
See Section 7.5 in the Introduction to the NAG Library CL Interface for further information.
NE_NO_LICENCE: Your licence key may have expired or may not have been installed correctly.
See Section 8 in the Introduction to the NAG Library CL Interface for further information.

7 Accuracy

Not applicable.

8 Parallelism and Performance

e01eac is not threaded in any implementation.

9 Further Comments

The time taken for a call of e01eac is approximately proportional to the number of data points,

n

. The function is more efficient if, before entry, the

(x, y)

pairs are arranged in x and y such that the

x

values are in ascending order.

The triangulation is encoded in triang as follows:

set $j_{0} = 0$ ; for each node, k=1,2,…,n, (using the ordering inferred from x and y)
- $i_{k} = j_{k - 1} + 1$
- $j_{k} = triang [6 \times n + k - 1]$
- $triang [j - 1]$ , for $j = i_{k}, \dots, j_{k}$ , contains the list of nodes to which node $k$ is connected. If $triang [j_{k} - 1] = 0$ then node $k$ is on the boundary of the mesh.

10 Example

In this example, e01eac creates a triangulation from a set of data points. e01ebc then evaluates the interpolant at a sample of points using this triangulation. Note that this example is not typical of a realistic problem: the number of data points would normally be larger, so that interpolants can be more accurately evaluated at the fine triangulated grid.

This example also demonstrates how to extract useful information from the data structure returned from e01eac. The provided function convex_hull returns, for the nodes on the boundary of the triangulation, the number of such nodes and the list of indices (ordered in anti-clockwise direction) for these nodes. The provided function triang2list returns: the number of triangles, the index of the vertices for each triangle, and whether the triangle has one or more edges on the boundary.