Algorithms for Triangulated Terrains
Summary (3 min read)
1 Introduction
- Geographic Information Systems are large software packages that store and operate on geographic data.
- Natural geographic data includes elevation above sea level, annual percipitation, soil type, and much more.
- GIS store the di erent t ypes of geographic data in di erent map layers, so there is a map layer with the major roads, one with the rivers, one with the current land use, and one with the elevation above sea level.
- This paper surveys the common models to store elevation data, in particular the triangulated irregular network.
- To analyse and express the e ciency of these algorithms we'll use big-Oh notation.
2 Digital elevation models
- In the computer the true geographic elevation function has to beapproximated by some nite representation of it.
- The standard choice is the so-called Delaunay triangulation that will be discussed later.
- One of the ways to obtain elevation data is by digitizing the contour lines on contour maps, so one may have to deal with the contour model nevertheless.
- In the edge-based structure for a TIN, any edge is an object that has a dual purpose.
- The triangle objects have references to the three edge objects that bound the triangle.
4 Visualization and traversal of a TIN
- The map can be enhanced by hill shading, a technique where an imaginary light source is placed in 3-dimensional space, and parts of the terrain that don't receive much light are shaded.
- The algorithms required for visualization are standard graph algorithms on the TIN structure in both cases.
4.1 Contour maps
- To determine all contour lines of, say, 1000 meters, on a TIN representing a terrain, observe that any triangle contains at most one line segment that is part of the contour lines of 1000 meters.
- In fact, the contour lines of 1000 meters are nothing else than the cross-section of the terrain as a 3-dimensional surface, and the horizontal plane z = 1000.
- Similarly, to compute hill shading for a TIN, one needs to determine the slope of each triangle and its aspect the compass direction to which the triangle is facing, in the xy-projection.
- This graph has a node for every triangle, and two nodes are connected by an arc if the corresponding triangles share an edge in the triangulation.
- Similarly, one can compute hill shading in linear time.
4.2 Perspective views
- The authors can produce such a view using the Painter's Algorithm, where all triangles are drawn from back to front, so that the ones more to the front erase the ones more to the back.
- Again, this directed graph is implicitly present in either of the two structures for storing a TIN.
- It is the same dual graph as the authors used for depth-rst search, but now the arcs have a direction.
- The direction of an arc can bedetermined by checking the coordinates of the endpoints of the dual edge of that arc, and the coordinates of the view point.
5 Construction of a TIN
- We'll now study algorithms for constructing a TIN from elevation data.
- Then the authors assume that the input is a large grid of regularly spaced data points, and the problem is to produce a TIN that approximates the grid to within a speci ed maximum error.
- The most popular triangulation of a set of points without doubt is the Delaunay triangulation.
- If P doesn't contain four points that are co-circular, then the de nition just given really de nes a unique triangulation.
- This property implies that the triangles generally will bewell-shaped, which is important for the interpolation function it de nes.
5.1 Delaunay triangulation on a point set
- The authors sketch one that is simple and requires.
- The expectation in the running time is only dependent on the random choices made by the algorithm and is valid for any set of points, independent of the distribution.
- One insertion of a point requires locating the triangle of the triangulation that contains the point, and then the actual insertion.
- We'll skip the location part and assume that the new point p i+1 falls inside some triangle t of triangulation T i .
- Any ip destroys one triangle incident to p i+1 and another triangle, and creates two p i+1 p i+1 Figure 9 : Flipping an edge because the empty circle property is violated.
5.2 Delaunay triangulation to approximate an elevation grid
- The algorithm we'll describe selects a subset of the grid points, such that the Delaunay triangulation of this subset is a TIN that approximates the elevation at all grid points to within a prespeci ed error .
- The right TIN shows the situation if the square grid point on the left is the one with maximum error.
- This update step of the Delaunay triangulation is the same as in the incremental construction algorithm.
- The lists of points of the destroyed triangles contain p and the points that must bedistributed among the new triangles, and stored in new lists.
- One can expect that k is usually constant, and after a couple of iterations of the algorithm, m will probably bemuch smaller than n.
6 Compression and decompression of a TIN
- Assuming that the Delaunay triangulation is used for the data points.the authors.
- We'll give a simple and e cient algorithm for compression and for decompression.
- A structure like the Delaunay triangulation can be compressed by omitting all structural information egdes and triangles, leaving only the vertices.
- On log n time to reconstruct the TIN, since it takes this much time to construct the Delaunay triangulation of n points from scratch.
- The algorithm to compute this particular order takes.
6.1 Compression
- For simplicity w e'll assume that the point set lies in some rectangle of which the four vertices are also data points of the set, and that there are no other four co-circular points in P.
- The steps for one phase in the compression algorithm.
- 1.1 Find an independent set S. The algorithm starts with the Delaunay triangulation of a point set P.
- One can show that at least 5=121 of all non-corner vertices are chosen.
- So, an algorithm that chooses one vertex with degree at most 10 and throws away the neighbors chooses at least 1 of 11 vertices of degree 10, and in total 1=11 times 5m=11 vertices are chosen in the independent set S.
6.2 Decompression
- Decompression is done when a computer gets the data from background storage, or receives the data over a network.
- All steps of the algorithm basically are the reverse of some step of compression, and therefore the authors only sketch it brie y.
- During the traversal the authors need only test if the next triangle contains the rst point of the sequence.
- After all points between two end-of-phase markers are located in the triangles, the authors insert them in the Delaunay triangulation.
- Ok time for ipping if the next point has degree k.
7 Conclusions and further reading
- This paper surveyed a couple of geometric algorithms that can beused when working with digital elevation data.
- These algorithms were developed in the research areas of computational geometry and GIS.
- Both areas also have strong connections with computer graphics.
- Algorithms for the construction of TINs from digital elevation in another form has been studied extensively.
- Compression of digital elevation data hasn't been studied so much y et.
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Citations
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12 citations
Cites background or methods from "Algorithms for Triangulated Terrain..."
...A satellite is visible to the receiver if the line segment that joins them does not intercept any terrain features, such as mountains, or 3D objects, such as buildings ~Kreveld 1997!....
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...This method seamlessly joins both triangle sets, from TIN and objects, and maintains the Delaunay triangulation property ~Kreveld 1997! in TIN....
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11 citations
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Cites background from "Algorithms for Triangulated Terrain..."
...Most efficient spanner construction algorithms are geometric and they find practical applications in areas such as terrain construction [12,21], metric space searching [19], broadcasting in communication networks [11] and solving approximately geometric problems like traveling salesman problem [20]....
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References
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1,201 citations
"Algorithms for Triangulated Terrain..." refers background in this paper
...The two structures are simplified versions of data structures that can store arbitrary planar subdivsions such as the doublyconnected edge list, winged edge, or quad edge structure, commonly used in GIS, graphics, and computational geometry [6, 8, 14, 32]....
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1,163 citations
"Algorithms for Triangulated Terrain..." refers background in this paper
...A more complete description of randomized incremental construction of the Delaunay triangulation and its analysis can be found in [4, 6, 13]....
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854 citations
810 citations
"Algorithms for Triangulated Terrain..." refers methods in this paper
...The idea was introduced by Kirkpatrick [17], and used for e cient planar point location....
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Frequently Asked Questions (10)
Q2. How can the authors redistribute the points over the k triangles?
But redistribution of the points can also be done in O(k+m logm) time by sorting the m points by angle around p. Since all new triangles in the TIN are incident to p, the authors can distribute the m points over the k triangles by using the sorted order.
Q3. How many neighbors do the authors want in the chosen independent set?
we'd like all vertices in the chosen independent set to have constant degree in the graph, say, each chosen vertex has at most ten neighbors.
Q4. What are the types of man-made geographic data?
Borders of countries and provences, locations of roads and hospitals, and pollution of the lakes and rivers are types of man-made geographic data.
Q5. How does depth-rst search for a TIN map work?
Both the triangle-based and the edge-based TIN structures implicitly store this graph, and depth- rst search through all the triangles is easy if a mark bit is available in every object, to see if it has been visited before.
Q6. How can the authors produce a TIN from a view?
The authors can produce such a view using the Painter's Algorithm, where all triangles are drawn from back to front, so that the ones more to the front erase the ones more to the back.
Q7. What is the algorithm that we'll describe?
The algorithm we'll describe selects a subset of the grid points, such that the Delaunay triangulation of this subset is a TIN that approximates the elevation at all grid points to within a prespeci ed error .
Q8. What is the relation between the number of edges and triangles?
By Euler's relation for planar graphs, the number of edges and triangles is linear in the number of points, the vertices that determine the subdivision.
Q9. What is the common method of converting a TIN from digital elevation data?
This can be the triangulation between contour lines, grid to TIN conversion as in this paper, or producing a TIN from point data, with or without an interpolation method.
Q10. How can one compute contour lines of a given elevation in depth- rst?
So for a TIN with n vertices and, hence, O(n) edges and triangles, one can compute all contour lines of a given elevation in O(n) time by depth- rst search.