dtm and tin

8/8/2019 DTM and TIN

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Source : Dr. Michela Bertolotto

TerrainTerrain modelingmodeling and TINand TIN


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Terrain dataTerrain data

Terrain dataTerrain data relates to the 3D configuration of the surface ofrelates to the 3D configuration of the surface of

the Earththe Earth

On the other hand,On the other hand, map datamap data refers to data located on therefers to data located on thesurface of the Earth (2D)surface of the Earth (2D)

The geometry of a terrain is modeled as a 2The geometry of a terrain is modeled as a 2 --dimensionaldimensional

surface, i.e., a surface in 3D space described by a bivariatesurface, i.e., a surface in 3D space described by a bivariatefunctionfunction


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Mathematical terrain modelsMathematical terrain models

AA topographic surfacetopographic surface oror terrainterrain can be mathematicallycan be mathematically

modeled by the image of a real bivariate functionmodeled by the image of a real bivariate function

z =z = JJ(x,y)(x,y)

defined over a domaindefined over a domain DD such that Dsuch that D 22

The pairThe pair TT=(=(DD,, JJ)) is called ais called a mathematical terrain modelmathematical terrain model

UnidimensionalUnidimensional

profile of aprofile of a

mathematicalmathematical

terrain modelterrain model

D

J


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Digital Terrain Models (DTM)Digital Terrain Models (DTM)

AA digitaldigital terrain modelterrain modelis a model providing ais a model providing a

representation of a terrain on the basis of a finite set ofrepresentation of a terrain on the basis of a finite set of

sampled datasampled data

Elevation dataElevation data refers to measures of elevation at a set ofrefers to measures of elevation at a set of

pointspoints VVof the domain plus possibly a setof the domain plus possibly a setEEof nonof non--

crossing line segments with endpoints incrossing line segments with endpoints in VV

D

J


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Elevation data acquisitionElevation data acquisition

Elevation data can be acquired through:Elevation data can be acquired through:

sampling technologiessampling technologies (by means of on(by means of on--site measurementssite measurements

or of remote sensing techniques)or of remote sensing techniques)

digitisationdigitisation of existing contour mapsof existing contour maps

Elevation data can be scattered (irregularly distributed)Elevation data can be scattered (irregularly distributed)

or form a regular gridor form a regular grid

The set of nonThe set of non--crossing lines can form a collection ofcrossing lines can form a collection ofpolygonal chainspolygonal chains


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ContoursContours

Given aGiven a terrain modelterrain model

TT= (= (DD,, JJ))

and a real valueand a real value v,v,the set ofthe set ofcontourscontours ofofTTat heightat height vvisis

{ (x,y){ (x,y)D,D,JJ(x,y) = v(x,y) = v}}

This is a set of simple lines (non selfThis is a set of simple lines (non self--intersecting)intersecting)

D

Planez = v


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DTMsDTMs

Digital terrain models represent an approximation ofDigital terrain models represent an approximation of

mathematical terrain modelsmathematical terrain models

Sampled modelSampled model Digital terrain modelDigital terrain model


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Sampled data distributionSampled data distribution

Sampled data can be scattered (irregularly distributed)Sampled data can be scattered (irregularly distributed)

or form a regular grid on the domainor form a regular grid on the domain

The distribution of the sampled data can depend on theThe distribution of the sampled data can depend on the

acquisition technique or on the specific applicationacquisition technique or on the specific application

Different distributions might be required by differentDifferent distributions might be required by different

configurations of the terrain reliefconfigurations of the terrain relief


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Sampled data distributionSampled data distribution

Sometimes it can be useful to haveSometimes it can be useful to have irregularly distributedirregularly distributedsets ofsets of

datadata

For example, only a few sampled points where the terrain isFor example, only a few sampled points where the terrain is

quite flat and more values where the surface presents specificquite flat and more values where the surface presents specific

features such as peaks etc.features such as peaks etc.


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Sampled data distribution (cont.d)Sampled data distribution (cont.d)

Regular samplingRegular samplingis good in areas where the terrainis good in areas where the terrain

elevation is more or less constantelevation is more or less constant


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DTMsDTMs

In general, a largerIn general, a larger

number of samplednumber of sampledpoints allows for apoints allows for a

better representation:better representation:


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Terrain modelsTerrain models

GlobalGlobalterrain models: defined by means of a singleterrain models: defined by means of a single

function interpolating all datafunction interpolating all data

L

ocalL

ocalterrain models: piecewise defined on a partitionterrain models: piecewise defined on a partitionof the domain into patches (regions)of the domain into patches (regions)

In other words, they represent the terrain by means of aIn other words, they represent the terrain by means of a

different function on each of the regionsdifferent function on each of the regions in which the domain isin which the domain is

subdividedsubdivided

In general it is very difficult to find a single functionIn general it is very difficult to find a single function

that interpolates all available data, so usually localthat interpolates all available data, so usually local

models are usedmodels are used


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Types of DTMsTypes of DTMs

Polyhedral terrain modelsPolyhedral terrain models

Gridded elevation modelsGridded elevation models

Contour mapsContour maps


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Polyhedral terrain models: definitionPolyhedral terrain models: definition

AApolyhedral terrain modelpolyhedral terrain modelfor a set of sampled pointsfor a set of sampled points VV

can be defined on the basis of:can be defined on the basis of:

1.1. aa partitionpartition of the domainof the domain DDinto polygonal regionsinto polygonal regionshaving their vertices at points inhaving their vertices at points in VV

2.2. aa functionfunctionffthat isthat is linearlinearover each region of theover each region of the

partition (i.e., the image ofpartition (i.e., the image offfover each polygonalover each polygonalregion is aregion is aplanar patchplanar patch this will guaranteethis will guarantee

continuity of the surface along the common edges)continuity of the surface along the common edges)

((ffis also called ais also called apiecewise linearpiecewise linearfunction). Imagefunction). Image

analysis techniques calledanalysis techniques calledfacet modelsfacet models have somehave some

similarity with these methods.similarity with these methods.


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Polyhedral terrain models: propertiesPolyhedral terrain models: properties

-- They can be used for any type of sampled pointsetThey can be used for any type of sampled pointset

(regularly and irregularly distributed)(regularly and irregularly distributed)

-- They can adapt to the irregularity of terrainsThey can adapt to the irregularity of terrains

-- They represent continuous surfacesThey represent continuous surfaces


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Triangulated Irregular NetworksTriangulated Irregular Networks

The most commonly used polyhedral terrain modelsThe most commonly used polyhedral terrain modelsareare Triangulated Irregular NetworksTriangulated Irregular Networks (TINs), where(TINs), where

each polygon of the domain partition is a triangleeach polygon of the domain partition is a triangle


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TINsTINs

Example of a TIN based on irregularly distributedExample of a TIN based on irregularly distributed

datadata


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TINs for regular dataTINs for regular data

Regular sampling is enough in areas where theRegular sampling is enough in areas where the

terrain elevation is more or less constantterrain elevation is more or less constant


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TINs: important propertiesTINs: important properties

They guarantee the existence of a planar patch forThey guarantee the existence of a planar patch for

each region (triangle) of the domain subdivisioneach region (triangle) of the domain subdivision

(three points define a plane): the resulting surface(three points define a plane): the resulting surface

interpolates all elevation datainterpolates all elevation data

The most commonly used triangulations areThe most commonly used triangulations are

Delaunay triangulationsDelaunay triangulations

Triangular Irregular Network (TIN), whichrepresents a surface as a set of non-overlapping

contiguous triangular facets, of irregular size and

shape.


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Why Delaunay TriangulationsWhy Delaunay Triangulations

They generate the most equiangular triangles in theThey generate the most equiangular triangles in the

domain subdivision (thus minimising numericaldomain subdivision (thus minimising numerical

problems: e.g.,problems: e.g.,point locationpoint location))

Their Dual is a Voronoi diagram. Therefore, someTheir Dual is a Voronoi diagram. Therefore, some

proximity queries can be solved efficientlyproximity queries can be solved efficiently


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Delaunay TriangulationsDelaunay Triangulations

Intuitively: given a set V of points, among all the triangulationsIntuitively: given a set V of points, among all the triangulations

that can be generated with the points of V, the Delaunaythat can be generated with the points of V, the Delaunay

triangulation is the one in which triangles are as muchtriangulation is the one in which triangles are as much

equiangular as possibleequiangular as possible

In other words, Delaunay triangulations tend to avoid long andIn other words, Delaunay triangulations tend to avoid long and

thin triangles: important for numerical problemsthin triangles: important for numerical problems

t P

DoesDoes P lie inside t or on its boundary?lie inside t or on its boundary?


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Voronoi DiagramsVoronoi Diagrams

Given a set V of points in the plane, theGiven a set V of points in the plane, the VoronoiVoronoi Diagram for V is theDiagram for V is the

partition of the plane into polygons such that each polygon contains onepartition of the plane into polygons such that each polygon contains one

pointpointpp of V and is composed of all points in the plane that are closer toof V and is composed of all points in the plane that are closer topp

than to any other point of Vthan to any other point of V


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Voronoi Diagrams (cont.d)Voronoi Diagrams (cont.d)

Property: the straightProperty: the straight--lineline dualdual of theof the VoronoiVoronoi diagram of V is adiagram of V is a

Delaunay triangulation of VDelaunay triangulation of V

Dual:Dual: obtained by replacing each polygon with a point and each pointobtained by replacing each polygon with a point and each point

with a polygon.Connect all pairs of points contained inwith a polygon. Connect all pairs of points contained in VoronoiVoronoi cells thatcells thatshare an edgeshare an edge


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Voronoi Diagrams (cont.d)Voronoi Diagrams (cont.d)

VoronoiVoronoi diagrams are used as underlying structures to solvediagrams are used as underlying structures to solveproximityproximity

problems (queries):problems (queries):

Nearest neighbour (what is the point of V nearest to P?)Nearest neighbour (what is the point of V nearest to P?)

KK--nearest neighbours (what are the k points of V nearest to P?)nearest neighbours (what are the k points of V nearest to P?)

Etc.Etc.

P


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Why Delaunay Triangulations (cont.d)Why Delaunay Triangulations (cont.d)

It has been proven that they generate the bestIt has been proven that they generate the bestsurface approximation (in terms of roughness).surface approximation (in terms of roughness).

There are several efficient algorithms to calculateThere are several efficient algorithms to calculate

themthem


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Gridded modelsGridded models

AAGridded Elevation ModelGridded Elevation Modelis defined on the basis of ais defined on the basis of a

domain partition into regular polygonsdomain partition into regular polygons


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RSGsRSGs

The most commonly used gridded elevation models areThe most commonly used gridded elevation models are

Regular Square Grids (RSGs)Regular Square Grids (RSGs)where each polygon in thewhere each polygon in the

domain partition is a squaredomain partition is a square

The function defined on each square can be a bilinearThe function defined on each square can be a bilinear

function interpolating all four elevation pointsfunction interpolating all four elevation points

corresponding to the vertices of the squarecorresponding to the vertices of the square


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RSG: an exampleRSG: an example


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RSG (cont.d)RSG (cont.d)

Alternatively, a constant function can be associatedAlternatively, a constant function can be associated

with each square (i.e., a constant elevation value). Thiswith each square (i.e., a constant elevation value). This

is called ais called a stepped modelstepped model(it presents discontinuity steps(it presents discontinuity steps

along the edges of the squares)along the edges of the squares)

D

UnidimensionalUnidimensional

profile of aprofile of a steppedstepped

modelmodel


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TINs & RSGsTINs & RSGs

Both models support automated terrain analysisBoth models support automated terrain analysisoperationsoperations

RSGs are based on regular data distributionRSGs are based on regular data distribution

TINs can be based both on regular and irregularTINs can be based both on regular and irregular

data distributiondata distribution

Irregular data distribution allows to adapt to theIrregular data distribution allows to adapt to thevariability of the terrain relief: more appropriatevariability of the terrain relief: more appropriate

and flexible representation of the topographicand flexible representation of the topographic

surfacesurface


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Calculations from a Gridded

DEM


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Slope and Aspecta b c

d p e

f g h

Consider a 3x3 neighbourhood at every pixel for computation

Slope = Rate of change of terrain property at a given point

Aspect = Direction of Change

Both are expressed as angles


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Slope and Aspect

Slope = S: tan S =

1

22 2

( ) ( )z z

x y

x x

x x

Where z is the altitude and x and y are the coordinate

axes

Aspect: tan A = /

/

z y

z x

x x

x x

Let dxx = (a + 2d + f) (c + 2e + h)

dyy = (a + 2b + c) (f + 2g + h)


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Slope

z

x

x !x

dxx/(8*dx) where dx = X-resolution

z

y

x

xdyy/(8*dy) where dy = Y-resolution

Slope

S = arctan

(To convert S into degrees, multiply by 57.29578)

1

22 2

( ) ( )z z

x y

x x

x x


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Aspect

a) If and are positive,

Aspect = 90o + 57.3*tan-1[abs()]

b) If > 0 and < 0,

Aspect = 90 - 57.3*tan-1[abs()]

z

x

x

x

z

y

x

x

z

x

x

x

z

y

x

x

Aspect: tan A =/

/

z y

z x

x x

x x


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Aspect

c) If < 0 and > 0,

Aspect = 270 - 57.3*tan-1[abs( )]

d) If < 0 and < 0,

Aspect = 180 - 57.3*tan-1[abs( )]

In all cases, if Aspect < 0, Aspect = Aspect + 360

/

/

z y

z x

x x

x x

z

x

x

x

z

y

x

x

z

x

x

x

z

y

x

x

/

/z yz x

x x

x x


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Digital Contour MapsDigital Contour Maps

Given a sequence {Given a sequence { vv00, ,v, ,vnn } of real values, a} of real values, a digitaldigital

contour mapcontour map of a mathematical terrain model (of a mathematical terrain model (DD,, JJ) is) is

an approximation of the set of contour linesan approximation of the set of contour lines

{ ({ (x,yx,y))D,D,JJ(x,y) = v(x,y) = vii}} i = 0, , ni = 0, , n

A set of contourA set of contour

lineslines


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Digital Contour MapsDigital Contour Maps

A line interpolating points of a contour can be obtainedA line interpolating points of a contour can be obtained

in different waysin different ways

ExamplesExamples:: polygonal chainspolygonal chains, or lines described by, or lines described by

higher order equationshigher order equations


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Digital Contour Maps: propertiesDigital Contour Maps: properties

They are easily drawn on paperThey are easily drawn on paper

They are very intuitive for humansThey are very intuitive for humans

They are not good for complex automated terrainThey are not good for complex automated terrain

analysisanalysis


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DTMs: accuracyDTMs: accuracy

In general, the more data is available, the better theIn general, the more data is available, the better the

representationrepresentation

Using very large datasets requires large storage spaceUsing very large datasets requires large storage spaceand processing timeand processing time

Use of generalisation techniques: select a subset of dataUse of generalisation techniques: select a subset of data

that still maintains acceptable accuracy in thethat still maintains acceptable accuracy in therepresentationrepresentation


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DTMs: accuracy (cont.d)DTMs: accuracy (cont.d)

AnAn approximate terrainapproximate terrain model is a model that uses amodel is a model that uses a

subset of the data availablesubset of the data available

The approximationThe approximation errorerror EEis calculated with respect tois calculated with respect toa reference model built using the whole dataseta reference model built using the whole dataset

For exampleFor example: the error could be the maximum: the error could be the maximum

difference between the elevation value at a point anddifference between the elevation value at a point andthe interpolated value in the approximated modelthe interpolated value in the approximated model


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DTMs: accuracy (cont.d)DTMs: accuracy (cont.d)

Example:Example:

Model builtModel built

using the wholeusing the whole

data set:data set:

reference modelreference model

Model builtModel built

using a subset ofusing a subset of

the data:the data:

approximatedapproximated

modelmodel


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DTMs: approximation errorDTMs: approximation error

Maximum difference: approximationMaximum difference: approximation errorerror

Calculate all differences betweenCalculate all differences between

elevation data and interpolatedelevation data and interpolated

datadata


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DTMs: accuracy (DTMs: accuracy (cont.dcont.d))

TheThe accuracyaccuracy of the model is inversely proportional toof the model is inversely proportional to

the errorthe errorEEassociated with the model and is defined as:associated with the model and is defined as:

Some applications might require to build anSome applications might require to build an

approximate model with accuracy (error) within aapproximate model with accuracy (error) within a

given thresholdgiven threshold

E11


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How to pick points

Given a set of points with known values or given a set of

digitized contours, how should points be selected so thatthe surface is accurately represented?


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VIP Algorithm

Each point has 8 neighbors, forming 4 diametricallyopposite pairs, i.e. up and down, right and left, upperleft and lower right, and upper right and lower left

For each point, examine each of these pairs ofneighbors in turn

Connect the two neighbors by a straight line, andcompute the perpendicular distance of the central

point from this line


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VIP Algorithm

Average the four distances to obtain a measure of

"significance" for the point

Delete points from the DEM in order of increasingsignificance, 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


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Comments

Due to its local nature, this method is best when the

proportion of points deleted is low.

Due to its emphasis on straight lines, and the TIN's useof planes, it is less satisfactory on curved surfaces.


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Given a set of points, calculate a triangulationGiven a set of points, calculate a triangulation

Particular properties, e.g., equiParticular properties, e.g., equi--angularity (Delaunayangularity (Delaunaytriangulation)triangulation)

TriangulationCalculationTriangulationCalculation


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Given a set of points, calculate a triangulationGiven a set of points, calculate a triangulation

Particular properties, e.g., equiangularity (DelaunayParticular properties, e.g., equiangularity (Delaunaytriangulation)triangulation)

TriangulationCalculationTriangulationCalculation


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Given a setGiven a set VVof points, calculate aof points, calculate a DelaunayDelaunaytriangulationtriangulation with vertices at points ofwith vertices at points ofVV

Watsons algorithm is one of the soWatsons algorithm is one of the so--calledcalled onon--lineline

methodsmethods: based on the modification of an existing: based on the modification of an existingDelaunay triangulation when a new point is insertedDelaunay triangulation when a new point is inserted

In onIn on--line methodsline methods, the first step consists of building a, the first step consists of building aDelaunay triangulation of the domain (containing allDelaunay triangulation of the domain (containing all

data points)data points)

Then all points ofThen all points ofVVare added incrementallyare added incrementally

Watsons algorithm (1981)Watsons algorithm (1981)


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InWatsons algorithm, the initial triangulation of theIn Watsons algorithm, the initial triangulation of the

domain is built by considering a fictitious triangledomain is built by considering a fictitious triangle

containing all points ofcontaining all points ofVVin its interiorin its interior

Watson: initial stepWatson: initial step

Dataset VDataset V


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After building the initial triangle, all points ofAfter building the initial triangle, all points ofVVareare

added one at a timeadded one at a time

Finally the initial triangle and all edges incident at itsFinally the initial triangle and all edges incident at its

vertices are deletedvertices are deleted

The main step in this algorithm is the insertion of aThe main step in this algorithm is the insertion of a

new point in the current Delaunay triangulationnew point in the current Delaunay triangulation

Watson: processWatson: process


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We call theWe call the influence polygoninfluence polygon RRPP of a pointof a point PP in ain a

triagulationtriagulation TTthe union of all triangles ofthe union of all triangles ofTTwhosewhose

circumscribing circle containcircumscribing circle contain PP

After insertingAfter inserting PP inin TT, we update, we update TTby deleting allby deleting all

edges internal toedges internal to RRPP and by joiningand by joining PPwith all thewith all thevertices ofvertices ofRR

PP

Watson: insertion of a new pointWatson: insertion of a new point

P

RP


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We call theWe call the influence polygoninfluence polygon RRPP of a pointof a point PP in ain a

triagulationtriagulation TTthe union of all triangles ofthe union of all triangles ofTTwhosewhose

circumscribing circle containscircumscribing circle contains PP

After insertingAfter inserting PP inin TT, we update, we update TTby deleting allby deleting all

edges internal toedges internal to RRPP and by joiningand by joining PPwith all thewith all thevertices ofvertices ofRR

PP

Watson: insertion of a new pointWatson: insertion of a new point

P

RP


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Points inPoints in VVare added one at a time in the currentare added one at a time in the current

Delaunay triangulationDelaunay triangulation

Watson: insertion stepWatson: insertion step


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First vertexFirst vertex PP11: the fictitious triangle is its influence: the fictitious triangle is its influencepolygonpolygon RRP1P1; join; join PP11 with its three verticeswith its three vertices

Watson: insertion step (cont.d)Watson: insertion step (cont.d)


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Inserting the second vertexInserting the second vertexPP

22: calculation of: calculation ofRRP1P1



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Inserting the second vertexInserting the second vertexPP

22: updating the current: updating the currenttriangulationtriangulation



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Inserting new verticesInserting new vertices



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Delete the fictitious triangle and all edges incident atDelete the fictitious triangle and all edges incident at

its verticesits vertices

Watson: final stepWatson: final step


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Delete the fictitious triangle and all edges incident atDelete the fictitious triangle and all edges incident at

its verticesits vertices



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GivenGiven VV, Watsons algorithm calculates the Delaunay, Watsons algorithm calculates the Delaunay

triangulation with vertices at points of oftriangulation with vertices at points of ofVV



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The time complexity ofWatsons algorithm is O(The time complexity ofWatsons algorithm is O(nn22))

wherewhere nn is the number of input points (worst case)is the number of input points (worst case)

This depends on the fact that the insertion of a newThis depends on the fact that the insertion of a new

point requires changing all triangles of the influencepoint requires changing all triangles of the influence

polygon. In the worst case, the influence polygonpolygon. In the worst case, the influence polygon

includes all triangles of the current triangulationincludes all triangles of the current triangulation

NOTE: remember that the number of triangles in aNOTE: remember that the number of triangles in a

triangulation is O(triangulation is O(vv), with v the number of vertices in), with v the number of vertices inthe triangulationthe triangulation

Watson: complexityWatson: complexity


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Although the worst case time complexity for WatsonsAlthough the worst case time complexity for Watsonsalgorithm is O(algorithm is O(nn22), it has been shown that in the), it has been shown that in the

average case the number of triangles in the influenceaverage case the number of triangles in the influence

polygon is equal to 6polygon is equal to 6

Therefore, the average case time complexity is linearTherefore, the average case time complexity is linear

in the number of input pointsin the number of input points

Watson: complexity (cont.d)Watson: complexity (cont.d)


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Storage of Triangle Data


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Storage of Triangle Data


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The structure of a TIN. The TIN is a topological data model. The data are stored in

a set of tables that retain the coordinate values as well as the spatial relations of

the facets.

TIN S


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TIN Storage

Performance of any structure based operation, e.g.,

algorithm development depends on the way the dataare stored.

In TIN based DTM, data can be stored as

Triangle-based (for slope, aspect, volume

computations) Node-based

Side-based (for contouring or any traversal procedure)

Combination of the above (for visibility analysis)

TIN structure is a case of establishing a good andefficient triangle topology

TIN A li ti


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TIN Applications

Slope and aspect Contouring

Finding drainage networks

Creating cross-sections

Visualization of a 3-dimensional surface

dtm and tin

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