Compiled with assistance from Thomas K. Poiker, Simon Fraser
University
NOTES
UNIT 39 - THE TIN MODEL
Compiled with assistance from Thomas K. Poiker, Simon Fraser
University
A. INTRODUCTION
- the Triangulated Irregular Network model is a significant
alternative to the regular raster of a DEM, and has been
adopted in numerous GISs and automated mapping and
contouring packages
- the TIN model was developed in the early 1970's as a
simple way to build a surface from a set of irregularly
spaced points
- several prototype systems were developed in the 1970's
- commercial systems using TIN began to appear in the
1980's as contouring packages, some embedded in GISs
The TIN model
- irregularly spaced sample points can be adapted to the
terrain, with more points in areas of rough terrain and
fewer in smooth terrain
- an irregularly spaced sample is therefore more
efficient at representing a surface
- in a TIN model, the sample points are connected by lines
to form triangles
- within each triangle the surface is usually
represented by a plane
- by using triangles we ensure that each piece of the
mosaic surface will fit with its neighboring pieces - the
surface will be continuous - as each triangle's surface
would be defined by the elevations of the three corner
points
- it might make sense to use more complex polygons as
mosaic tiles in some cases, but they can always be broken
down into triangles
- for example, if a plateau is eroded by gullies, the
remaining plateau would be a flat (planar) area
bounded by an irregular, many-sided polygon. In the
TIN model it would be represented by a number of
triangles, each at the same elevation
diagram
- for vector GISs, TINs can be seen as polygons having
attributes of slope, aspect and area, with three vertices
having elevation attributes and three edges with slope
and direction attributes
- the TIN model is attractive because of its simplicity and
economy
- in addition, certain types of terrain are very
effectively divided into triangles with plane facets
- this is particularly true with fluvially-eroded
landscapes
- however, other landscapes, such as glaciated ones,
are not well represented by flat triangles
- triangles work best in areas with sharp breaks in
slope, where TIN edges can be aligned with breaks,
e.g. along ridges or channels
Creating TINs
- despite its simplicity, creating a TIN model requires
many choices:
- how to pick sample points
- in many cases these must be selected from some
existing, dense DEM or digitized contours
- normally, a TIN of 100 points will do as well
as a DEM of several hundred at representing a
surface
- how to connect points into triangles
- how to model the surface within each triangle
- this is almost always resolved by using a plane
surface
- however, if the surface is contoured, the
contours will be straight and parallel within
each triangle, but will kink sharply at
triangle edges
diagram
- consequently, some implementations of TIN
represent the surface in each triangle using a
mathematical function chosen to ensure that
slope changes continuously, not abruptly, at
the edges of the triangle
B. HOW TO PICK POINTS
- given a dense DEM or set of digitized contours, how
should points be selected so that the surface is
accurately represented?
- consider the following 3 methods for selecting from
a DEM
- all of them try to select points at significant
breaks of the surface
- such breaks are common on terrain, absent on
smooth mathematical surfaces
1. Fowler and Little algorithm
- this approach is based on the concept of surface-specific
points which play a specific role in the surface
- e.g. represent features such as peaks and pits
Procedure
- first examine the surface using a 3x3 window, looking at
a small array of 9 points at each step
- next the surface is examined using a 2x2 window
- except at the edges, every point appears in four
positions of the window
- a point is a potential ridge point if it is never
lowest in any position of the window
- a point is a potential channel point if it is never
highest in any position of the window
- then starting at a pass, search through adjacent ridge
points until a peak is reached
- similarly, search from the pass through adjacent
channel points until a pit is reached
Finishing the TIN
- the result of this process is a connected set of peaks,
pits, passes, ridge lines and channel lines
- Fowler and Little recommend that the number of
points in each ridge and channel line be reduced by
thinning using a standard thinning algorithm
- it may be desirable to add additional points from
the DEM which are not on ridges or channels if we
can significantly reduce any substantial differences
from the real surface by doing so
- triangles are built between all selected points
- the resulting surface will differ from the original DEM,
perhaps substantially in some areas
Comments
- the Fowler and Little algorithm is complex
- performs better on some types of landscape than
others, particularly where there are sharp breaks of
slope along ridges, and where channels are sharply
incised
- it may require substantial "fine tuning" to work
well
2. VIP (Very Important Points) Algorithm
- unlike the previous algorithm which tries to identify the
major features of the terrain, VIP works by examining the
surface locally using a window
- this is a simplification of the technique used in ESRI's
ARC/INFO
Procedure
- each point has 8 neighbors, forming 4 diametrically
opposite pairs, i.e. up and down, right and left, upper
left and lower right, and upper right and lower left
- for each point, examine each of these pairs of neighbors
in turn
- connect the two neighbors by a straight line, and
compute the perpendicular distance of the central
point from this line
diagram
- average the four distances to obtain a measure of
"significance" for the point
- delete points from the DEM in order of increasing
significance, deleting the least significant first
- this continues until one of two conditions is met:
- the number of points reaches a predetermined
limit
- the significance reaches a predetermined limit
Comments
- because of its local nature, this method is best when the
proportion of points deleted is low
- because of its emphasis on straight lines, and the TIN's
use of planes, it is less satisfactory on curved surfaces
diagram
3. Drop heuristic
- this method treats the problem as one of optimization
- given a dense DEM, find the best subset of a
predetermined number of points such that when the
points are connected by triangles filled with
planes, the TIN gives the best possible
representation of the surface
Procedure
- start with the full DEM
- examine each point in turn
- temporarily drop the point and modify the
surrounding triangles accordingly
diagram
- find the triangle containing the dropped point
- measure the difference between the elevation of the
point, and the elevation of the new surface at the
point
- restore the dropped point, storing the calculated
elevation difference
- continue the process dropping each point in turn
- when all the points have been dropped, remove the point
which produced the least difference and start the process
again
Comments
- the TIN will likely be more accurate if the differences
are measured not only for the point being dropped, but
for all previously dropped points lying within the
modified triangles as well, but this would be time-
consuming
- rather than select points from the DEM, the best solution
(in the sense of producing the best possible TIN for a
given number of points) may be to locate TIN points at
locations and elevations not in the original raster
- these points may be chosen from air photographs or
ground surveys
C. HOW TO TRIANGULATE A TIN
- having selected a set of TIN points, these will become
the vertices of the triangle network
- there are several ways to connect vertices into
triangles
- "fat" triangles with angles close to 60 degrees are
preferred since this ensures that any point on the
surface is as close as possible to a vertex
- this is important because the surface representation
is likely most accurate at the vertices
- consider the following two methods for building the
triangles
- in practice almost all systems use the second
1. Distance ordering
Procedure
- compute the distance between all pairs of points, and
sort from lowest to highest
- connect the closest pair of points
- connect the next closest pair if the resulting line
does not cross earlier lines
- repeat until no further lines can be selected
- the points will now be connected with triangles
- this tends to produce many skinny triangles instead
of the preferred "fat" triangles
2. Delaunay triangulation
- by definition, 3 points form a Delaunay triangle if and
only if the circle which passes through them contains no
other point
diagram
- another way to define the Delaunay triangulation is as
follows:
- partition the map by assigning all locations to the
nearest vertex
- the boundaries created in this process form a set of
polygons called Thiessen polygons or Voronoi or
Dirichlet regions
overhead - Delaunay triangles from Thiessen polygons
- two vertices are connected in the Delaunay
triangulation if their Thiessen polygons share an
edge
- this method produces the preferred fat triangles
- the boundary edges on the Delaunay network form the
Convex Hull, which is the smallest polygon to contain all
of the vertices
Procedure
- there are several techniques for building the triangles:
1. since the convex hull will always be part of the
Delaunay network
- start with these edges and work inwards until
the network is complete
2. connect the closest pair which by definition must
be a Delaunay edge
- search for a third point such that no other
point falls in the circle through them
- continue working outward from these edges for
the next closest point
Problems
- Delaunay triangles are not hierarchical
- they cannot be aggregated to form bigger triangles
- if they are divided into smaller triangles, the
results tend to be poorly shaped (not "fat")
D. ALTERNATIVE METHODS OF CREATING TINS
Break lines
- methods presented above concentrate on finding TIN
vertices, then connecting them with triangles
- a major advantage of TINs is their ability to capture
breaks of slope, if edges can be aligned with known
ridges or channels
- this requires a different approach, where "breaklines"
are incorporated into the triangle network as edges after
the points have been triangulated
- the result is generally non-Delaunay, i.e. an edge
need not be an edge in the Delaunay network of the
vertices
- this approach is now incorporated into some TIN software,
e.g. the ARC/INFO TIN module
TINs from contours
- contours are a common source of digital elevation data
- rather than convert from contours to a grid (DEM) and
then to a TIN, it is more direct to obtain the TIN from
contours directly
- a TIN can be created by selecting points from the
digitized contour lines
- selection may create a triangle with three vertices on
the same contour (at the same elevation)
- such a "flat triangle" has no defined aspect, causes
problems in modeling runoff
- several ways of avoiding this problem have been
devised
E. STORING TINS
- there are basically two ways of storing triangulated
networks:
1. Triangle by triangle
2. Points and their neighbors
overhead - Storing TINs
1. Triangle by triangle
- in this case, a record usually contains:
- a reference number for the triangle
- the x,y,z-coordinates of the three vertices
- the reference numbers of the three neighboring
triangles
- since a vertex participates in, on the average, six
triangles, repetition of coordinates can be avoided by
creating a separate vertex file and referencing them in
the triangle files
2. Points and their neighbors
- the alternative is to store for every vertex:
- an identification number
- the xyz coordinates
- references (pointers) to the neighboring vertices in
clockwise or counter-clockwise order
- this structure was the original TIN structure (Peucker et
al, 1978)
Comparison of the two structures
- both structures are necessary, depending on the purpose
- slope analysis needs the first
- contouring and other traversing procedures work best
with the second
- as long as one can be extracted from the other in close
to linear time (i.e., without an exhaustive search per
point), either will do
- the second generally needs less storage space
- however, the savings within different TIN structures
is minor compared to the reduction of points from
the regular grid to the triangular network
F. ALGORITHMS ON TINS
Slope and aspect
- compared to the DEM, it is simple to find slope and
aspect at some location using a TIN - we simply find the
slope and aspect attributes of the containing triangle
Contouring
Finding Drainage Networks
REFERENCES
Chen, Z., and J.A. Guevara, 1987. "Systematic selection of
very important points (VIP) from digital terrain models
for construction triangular irregular networks,"
Proceedings, AutoCarto 8, ASPRS/ACSM, Falls Church, VA,
pp. 50-56. A description of ESRI's VIP approach to
constructing a TIN.
Fowler, R.J., and J.J. Little, 1979. "Automatic extraction of
irregular network digital terrain models," Computer
Graphics 13:199-207.
Heller, M., 1986. "Triangulation and Interpolation of
Surfaces," in R. Sieber and K. Brassel (eds), A Selected
Bibliography on Spatial Data Handling: Data Structures,
Generalization and Three-Dimensional Mapping, Geo-
Processing Series, vol 6, Department of Geography,
University of Zurich, pp 36 - 45. A good overview with
literature, mainly on triangulation.
Mark, D. M., 1975. "Computer Analysis of Topography: A
Comparison of Terrain Storage Methods," Geografisker
Annaler 57A:179-188. A quantitative comparison of
regular grids and triangulated networks.
Mark, D.M., 1979. "Phenomenon-Based Data-Structuring and
Digital Terrain Modelling," Geo-Processing 1:27-36. A
very interesting conceptual article proposing a
phenomenon-based approach to data structuring. Such an
approach has to involve expert knowledge of the
phenomenon.
Peucker, T.K., R.J. Fowler, J.J. Little and D.M. Mark, 1978.
"The Triangulated Irregular Network," Proceedings,
American Society of Photogrammetry: Digital Terrain
Models (DTM) Symposium, St. Louis, Missouri, May 9-11,
1978, pp 516-540. The basic description of the original
TIN project.
DISCUSSION AND EXAM QUESTIONS
1. Argue the differences between the regular grid and the
triangular net approaches. Apply the argument to the
computation of slope, contouring and visibility.
2. Mark's article in 1979 argued that the TIN model was more
appropriate to the nature of certain geographical phenomena.
Do you agree? For what types of landforms is TIN most and
least appropriate?
3. Discuss the various methods proposed for selecting TIN
vertices from a DEM, and their relative strengths and
weaknesses.
4. Describe how information on directions of flow can be
obtained from a TIN, and the nature of the extracted stream
network. How does this compare to networks derived from
DEMs?
Back to Geography 370 Home Page
Back to Geography 470 Home Page
Back
to GIS & Cartography Course Information Home Page
Please send comments regarding content to: Brian
Klinkenberg
Please send comments regarding web-site problems to: The
Techmaster
Last Updated: August 30, 1997.