ISPRS, Vol.34, Part 2W2, “Dynamic and Multi-Dimensional GIS”, Bangkok, May 23-25, 2001 
4. Discussions and Future Works 
4.1 Effectiveness of the break-line for simple shapes 
In our methods, we only add a break-line from the centroid of a 
flat triangle area. This method aims at keeping the smoothness 
of the TIN at all points of the area. Fig. 7 (particularly at the 
middle lower part) shows an example of contour lines generated 
without new break-lines for simple shapes. The result at the far 
end from the non-flat TIN is smooth, but not so around the 
non-flat area. (The overlapped contours are caused by invisible 
shapes, which are not catered for in this result) 
Fig.7 Contour line generated without new break-lines for simple 
shapes 
Fig.8 shows the corresponding result where break-line from the 
centroid to the starting point of the non-flat TIN. It is obvious that 
the smoothness is maintained in all the area. 
Fig.8 Contour line generated new break-lines for simple shapes 
4.2 Exceptions and solutions 
The proposed method will not be valid when the contour line is 
broken, as shown in Fig.9 (a). In this case, the flat-TIN area 
cannot be traced properly and will consequently remain as it is. 
This problem can be solved by adding a line (shown in thick 
dashed line) as shown in Fig.9 (b), and then regrouping the 
contour lines connected to both end as the same contour line. 
(b) solution 
Fig. 9 The exceptional case and its solution 
The contour lines connecting the broken ones can mostly be 
added automatically by considering the height value, uniqueness 
of the neighboring broken lines or the degree of connectivity. 
Here, the degree of connectivity of two adjacent broken contour 
lines can be determined by their relative angle, direction. One 
example of automatic connection can be found in Fig.6, where 
the broken contour lines are automatically connected, as a result, 
the flat triangles are correctly processed. 
Even if the broken contour lines are connected improperly, the 
generated TIN will be the same as when they are not added, 
which means the same flat area will remain. If they needed to be 
corrected, such kind of area can be automatically identified from 
the location of broken contour lines and the flat TINs. 
4.3 Future works 
In general, the methods proposed can successfully solve the 
problem of flat triangles that will occur when using contour lines 
as constrains. 
There are other ways of positioning the new break-lines, which 
are not discussed in detail here. For example, similar to the 
suggestion by Peng [2] for determining the position of newlly 
added random points, the new break-lines can be drawn by 
connecting the centroid of flat triangles. There are also other 
ways for determining the break-lines for simple shapes, which 
still cause unnatural interpolation result in some cases. More 
experiments will be performed to compare the effectiveness of 
these options. 
We will further apply the results obtained with the 
above-mentioned methods to analysis of topography. In this case, 
we will face the problem of continuous interpolation of contour 
lines. The contour lines shown in this paper are interpolated in 
linear method. As a result, the density (relative distance between 
new contour lines) is not continuous between different original 
contours. The ideal distance between interpolated contour lines 
should be in proportion to the gradient of adjacent height values. 
References: 
1. Hiroshi Akima: A Method of Bivariate Interpolation and Smooth 
Surface Fitting for Irregularly Distributed Data Points. ACM 
Transactions on Mathematical Software, Vol.4, No.2, pp.148-159, 
1978 
2. Jonathan Richard Shewchuk: Triangle: Engineering a 2D 
Quality Mesh Generator and Delaunay Triangulator. First 
Workshop on Applied Computational Geometry (Philadelphia, 
Pennsylvania), pp. 124-133, ACM, May 1996. 
3. Wanning Peng, Morakot Pilouk, Klaus Tempfli: Generalizing 
Relief Representation Using Digitized Contours, International 
Archives of Photogrammetry and Remote Sensing. Vol. XXXI, 
Part B4. Vienna 1996 
4. Jonathan Richard Shewchuk: A Two-Dimensional Quality 
Mesh Generator and Delaunay Triangulator. 
http://www.cs.cmu.edu/~quake/triangle.html 
(a) An example of broken contour line