Source code   By the author (Paul Bourke): C, Basic, FORTRAN-77 (the original) Java version: Render.java and Conrec.java by Bradley White. JavaScript by Jason Davies: conrec.js, and conrec_js.txt. Adapted to C++ by Nicholas Yue. cpp.c version by Bjorn Harpe. Ada version by Gautier de Montmollin. Visual Basic by James Craig. Visual Basic version by Michelle Smith. Delphi translation courtesy of Alexander Weidauer. Actionscript3 version: ConRec.as by Jonathan Vanderspek. C# : conrec.cs as part of OxyPlot. C# version for AutoCAD: autocad.zip by John Nichols. FORTRAN version: Aramini_c.txt by Francisco Frutos Alfaro. FORTRAN version: nncntr_c.txt by Michael Aramini. PHP version: conrec.php and test.php by Ashley Norris. GO version: conrec.go by Dan Kortschak. Converted to TypeScript by Erik Vullings

The famous BYTE magazine, in which the algorithm appeared. At least the algorithm survived.

Introduction

This article introduces a straightforward method of contouring some surface represented as a regular triangular mesh.
Contouring aids in visualizing three dimensional surfaces on a two dimensional medium (on paper or in this case a computer graphics screen). Two most common applications are displaying topological features of an area on a map or the air pressure on a weather map. In all cases some parameter is plotted as a function of two variables, the longitude and latitude or x and y axis. One problem with computer contouring is the process is usually CPU intensive and the algorithms often use advanced mathematical techniques making them susceptible to error.

To do contouring in software you need to describe the data surface and the contour levels you want to have drawn. The software given this information must call the algorithm that calculates the line segments that make up a contour curve and then plot these line segments on whatever graphics device is available.

CONREC satisfies the above description, it is relatively simple to implement, very reliable, and does not require sophisticated programming techniques or a high level of mathematics to understand how it works.

• The number of horizontal and vertical data points designated iub and jub.

• The number of contouring levels, nc.

• A one dimensional array z(0:nc-1) that saves as a list of the contour levels in increasing order. (The order of course can be relaxed if the program will sort the levels)

• A two dimensional array d(0:iub,0:jub) that contains the description of the data array to be contoured. Each element of the array is a sample of the surface being studied at a point (x,y)

• Two, one dimensional arrays x(0:iub) and y(0:jub) which contain the horizontal and vertical coordinates of each sample point. This allows for a rectangular grid of samples.

The contouring subroutine CONREC does not assume anything about the device that will be used to plot the contours. It instead expects a user written subroutine called VECOUT. CONREC calls VECOUT with the horizontal and vertical coordinates of the start and end coordinates of a line segment along with the contour level for that line segment. In the simplest case this is very similar the the usual LINE (x1,y1)-(x2,y2) command in BASIC. See the source code listing below.

As already mentioned the samples of the three dimensional surface are stored in a two dimensional real array. This rectangular grid is considered four points at a time, namely the rectangle d(i,j), d(i+1,j), d(i,j+1), and d(i+1,j+1). The centre of each rectangle is assigned a value corresponding to the average values of each of the four vertices. Each rectangle is in turn divided into four triangular regions by cutting along the diagonals. Each of these triangular planes may be bisected by horizontal contour plane. The intersection of these two planes is a straight line segment, part of the the contour curve at that contour height.

Depending on the value of a contour level with respect to the height at the vertices of a triangle, certain types of contour lines are drawn. The 10 possible cases which may occur are summarised below

• a) All the vertices lie below the contour level.
  b) Two vertices lie below and one on the contour level.
  c) Two vertices lie below and one above the contour level.
  d) One vertex lies below and two on the contour level.
  e) One vertex lies below, one on and one above the contour level.
  f) One vertex lies below and two above the contour level.
  g) Three vertices lie on the contour level.
  h) Two vertices lie on and one above the contour level.
  i) One vertex lies on and two above the contour level.
  j) All the vertices lie above the contour level.


•  

In cases a, b, i and j the two planes do not intersect, ie: no line need be drawn. For cases d and h the two planes intersect along an edge of the triangle and therefore line is drawn between the two vertices that lie on the contour level. Case e requires that a line be drawn from the vertex on the contour level to a point on the opposite edge. This point is determined by the intersection of the contour level with the straight line between the other two vertices. Cases c and f are the most common situations where the line is drawn from one edge to another edge of the triangle. The last possibility or case g above has no really satisfactory solution and fortunately will occur rarely with real arithmetic.

Example

As a simple example consider one triangle with vertices labelled m1,m2 and m3 with heights 0, 2 and 3 respectively

To calculate where a contour line at a height of 1 should be drawn, it can be seen that this is case f described earlier. Level 1 intersects line segment m1-m2 half the way along and it intersects line segment m1-m3 one third of the way along. A line segment is drawn between these two points. Each rectangular mesh cell is treated this way.

In summary, CONREC takes each rectangle of adjacent data points and splits it into 4 triangles after choosing the height at the centre of the rectangle. For each of the triangles the line segment resulting from the intersection with each contour plane. A routine is then called with the starting and stopping coordinates of the line segment.

An attempt is made at optimization by checking first to see if there are any contour levels within the present rectangle and second that there are some contour levels within the present triangle. The indices i and j are used to step through each rectangle in turn, k refers to each contour level and m to the four triangles in each rectangle.

Figure 5 Some of the notation used for identifying the rectangles and triangles in the subroutine.

Note that for large arrays the whole data array need not be stored in memory . Since the algorithm is a local one only requiring 4 points at a time, the data for each rectangle could be read from disk as required.

Contour map and the following function

Example 2

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Contour map and the following function

Example 3

This is a more "real life" example where a C version of CONREC has been used to contour a piece of landscape, the result is given below.

Images from the BYTE magazine version

 

On occasion users have reported gaps in their contour lines, this should of course never happen. There is however a pathological case that all local contouring algorithms suffer from (local meaning that they only use information in the immediate vicinity to determine the contour lines). The problem arises when all four vertices of a grid cell have the same value as the contour level under consideration. There are a number of strategies that can be employed to overcome this special event, the correct way is to consider a larger region in order to join up the contours on either side of the problem cell. CONREC doesn't do this and just leaves the cell without any contour lines thus resulting in a gap. This special case essentially never happens for real values data, it is most commonly associated with integer height datasets. The simplest solution is to offset the contour levels being drawn by a very small amount.