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How to Move USER_SDO_GEOM_METADATA like a Grandpa

Sometimes moving a small amount of data in Oracle database requires more work than the actual task you need to complete. DBAs always have their tools (PL/SQL Developer Text Importer is my favorite one) to move a small amount of data between databases.
But when it comes to nested tables this might be a bit challenging. USER_SDO_GEOM_METADATA table (for those of you who are not familiar with it, it is the catalog information of spatial layers in an Oracle database) has no exception to that. Here how it is easy to handle it
The structure of USER_SDO_GEOM_METADATA table has the following structure

SQL> desc user_sdo_geom_metadata
 Name					   Null?    Type
 ----------------------------------------- -------- ----------------------------
 TABLE_NAME				   NOT NULL VARCHAR2(32)
 COLUMN_NAME				   NOT NULL VARCHAR2(1024)
 DIMINFO					    MDSYS.SDO_DIM_ARRAY
 SRID						    NUMBER

Obviously DIMINFO column is the problematic part since it is in SDO_DIM_ARRAY type. Now create a temporary table by unfolding this nested table at the source database site.

CREATE TABLE temp_metadata AS 
  SELECT t.table_name, 
         t.column_name, 
         d.*, 
         t.srid 
  FROM   user_sdo_geom_metadata t, 
         TABLE(t.diminfo) d; 

Once you are done check the structure of temp_metadata table which only contains primitive SQL types for the columns

SQL> desc temp_metadata
 Name					   Null?    Type
 ----------------------------------------- -------- ----------------------------
 TABLE_NAME					    VARCHAR2(32)
 COLUMN_NAME					    VARCHAR2(1024)
 SDO_DIMNAME					    VARCHAR2(64)
 SDO_LB 					    NUMBER
 SDO_UB 					    NUMBER
 SDO_TOLERANCE					    NUMBER
 SRID						    NUMBER

Choose your favorite tool to move this table to the target database since it contains just a few rows. Then as the final step import this data into USER_SDO_GEOM_METADATA table at the target database (Ensure that there is no other record with the same table_name.column_name key on the target USER_SDO_GEOM_METADATA table otherwise trigger on it will warn you)

insert into user_sdo_geom_metadata
SELECT m.table_name, m.column_name,
   CAST(MULTISET(SELECT sdo_dimname,sdo_lb,SDO_UB,sdo_tolerance
                 FROM temp_metadata n
                 WHERE n.table_name = m.table_name and n.column_name = m.column_name)
   AS SDO_DIM_ARRAY) as diminfo,
   m.srid
FROM (select distinct table_name,column_name,srid from temp_metadata) m;
commit;

You are done! Check everything works fine at the application site.

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Line of Sight (LoS) Analysis: Optimizing the Observers for Best Coverage (Part 4)

After my last post on multiple observers, I will discuss another topic related to multiple observers. The question is “What should be the optimal choice of N observers on terrain such that visible region (as many green points as possible by our convention) is maximized ?”.

We first define a pseudo code in order to find the optimal (Not guaranteed. Keep in mind that optimization problems are usually NP-complete by their nature) layout of N observers. For simplicity we will assume that all observers have the same height (7 units) which can be relaxed later.

We will implement a constructive way of finding optimal layout for N observers. Here is the pseudo code:

  1. Find the optimal layout for 1 observer and compute coverage ratio (best coverage for one observer)
  2. Add another random observer ((uniform(-8,8),uniform(-8,8))) and compute the coverage for those two observers (random one and the best observer from Step 1).
    1. If the new coverage is better than the coverage in Step 1, use this as the input of optimization solver
    2. Otherwise repeat Step 2 to find a better coverage.
  3. For number of observers greater than 2 apply the idea in Step 2 recursively.

There are some blur points in this pseudo code. We will define those before moving further with the implementation.

Coverage

The very first thing to be defined is the coverage idea. As you will remember from second post, we have defined our 3D terrain by evaluating our height function over outer product of x & y values varying over [-8,8] with a step size of 0.1 units. We have 1681 different (x,y) tuples. Here is the definition of coverage based on our conventions:

  • Coverage Ratio is the ratio of points within LoS of a given observer/group of observers (at least one of the observers mark those set of points as green) to the total number of points (1681)

How to Find Optimal Coordinates of Observers ?

Optimality is a very common word used in place of many different concepts  in real life or engineering. Let me define it once more for our purpose:

  • Optimization is the process of searching for an N-dimensional vector using a technique to maximize/minimize a function of that N-dimensional vector.

Now let’s substitute three italic words of definition for our problem:

  • N-dimensional vector in our problem is the vector of first to components of observer dimensions. Such as, (x1,y1,x2,y2,…,xn,yn).
  • Technique to be used is the Nelder and Mead Technique (A version of it implemented in R).
  • Function to be maximized is the coverage function which we have defined for a given set of observers.

Implementation

Let’s start by defining the function to be optimized that is coverage of terrain for a given set of observers.


targetfunc<-function(observer){
m <- matrix(data=observer,ncol=2,byrow=TRUE)

# Compute merged status of all observers
mergedstatus <- rep("red",length(terrain$height))
for(oidx in seq(1:dim(m)[1])){
terrain$dist2observer <- distance(terrain, c(m[oidx,],7))

status <- LoS(terrain,c(m[oidx,],7),maxVisibleDistance)

mergedstatus <- updatestatus(mergedstatus,status)
}

sum(mergedstatus=="green")/1681
}

matrix routine allows us to create a table of two columns(first two dimensions of observers) and length(observer)/2 rows. We have used the technique discussed in part 3 to compute merged status of observers.  sum(mergedstatus==”green”) call is used to count number of green points on terrain with respect to observers.

Next is the computation of first input to be given to optimization solver. That’s because for any optimization technique starting point is critical. Without any formal definition we will use our pseudo code to choose a “good starting point/vector”.


n <- 2
baselineValue <- 0.541344
previousObserver <- c(1.15861411217711, 1.1499851362913)
observers <- c(previousObserver,runif(2,-8,8))
while(targetfunc(observers) <= baselineValue){
observers <- c(previousObserver,runif(2,-8,8))
}
print(observers)

Above code is an example to initialize observers vector for searching best 2 observer layout. It uses the best coverage ratio for 1 observer case (54.1344%) and adds a new random observer next to best observer found for single observer case.

Final point is the optimization solver which is very simple and totally handled by R


optim <- optim(observers, targetfunc,
control=list(fnscale=-1,trace=5,REPORT=1))

First parameter is the initial value for input vector (prepared by previous code piece). Second parameter is the name of the function to be maximized. optim function is implemented to solve minimization problems by default. Setting fnscale attribute of control parameter turns it to a maximization problem solver.

Now we can combine all to have our final script


library(rgl)
##################
# Functions
##################
# 3D Terrain Function
height <- function (point) {
sin(point$x)+0.125*point$y*sin(2*point$x)+sin(point$y)+0.125*point$x*sin(2*point$y)+3
}

# Linear Function
linear <- function (px, observer, target) {
v <- observer - target

y <- ((px - observer[1])/v[1])*v[2]+observer[2]
z <- ((px - observer[1])/v[1])*v[3]+observer[3]

data.frame(x=px,y=y, z=z)
}

# Linear Function
distance <- function (terrain, observer) {
sqrt((terrain$x-observer[1])^2+(terrain$y-observer[2])^2+(terrain$height-observer[3])^2)
}

LoS <- function(terrain, observer, maxVisibleDistance){
status = c()
for (i in seq(1:nrow(terrain))) {
if (observer[1] == terrain$x[i] && observer[2] == terrain$y[i]){
if(observer[3] >= terrain$height[i]){
if (terrain$dist2observer[i] > maxVisibleDistance){
status <- c(status,"yellow")
}else{
status <- c(status,"green")
}
}else{
status <- c(status,"red")
}
}else{
# All points on line
line <- linear(seq(from=min(observer[1],terrain$x[i]),
to=max(observer[1],terrain$x[i]),
by=0.1),
observer,
c(terrain$x[i],terrain$y[i],terrain$height[i]))
# Terrain Height
h <- height(line)

# LoS Analysis
aboveTerrain <- round((line$z-h),2) >= 0.00

visible <- !is.element(FALSE,aboveTerrain)
if (visible){
# Second Rule
if(terrain$dist2observer[i] <= maxVisibleDistance){
status <- c(status,"green")
}else{
status <- c(status,"yellow")
}
}else{
status <- c(status,"red")
}
}
}

status
}

updatestatus <- function(status1,status2){
mergedstatus<-c()

for(i in seq(length(status1))){
if (status1[i] == "green" || status2[i] == "green"){
mergedstatus <- c(mergedstatus,"green")
}else if (status1[i] == "yellow" || status2[i] == "yellow"){
mergedstatus <- c(mergedstatus,"yellow")
}
else{
mergedstatus <- c(mergedstatus,"red")
}
}

mergedstatus

}

##################
# Input
##################

# Max visible distance
maxVisibleDistance = 8

# Generate points with a step size of 0.1
x <- seq(from=-8,to=8,by=0.4)

xygrid <- expand.grid(x=x,
y=x)

terrain <- data.frame(xygrid,
height=height(xygrid)
)

targetfunc<-function(observer){
#print(observer)
m <- matrix(data=observer,ncol=2,byrow=TRUE)

# Compute merged status of all observers
mergedstatus <- rep("red",length(terrain$height))
for(oidx in seq(1:dim(m)[1])){
terrain$dist2observer <- distance(terrain, c(m[oidx,],7))

status <- LoS(terrain,c(m[oidx,],7),maxVisibleDistance)

mergedstatus <- updatestatus(mergedstatus,status)
}

sum(mergedstatus=="green")/1681
}

n <- 3
baselineValue <- 0.541344
previousObserver <- c(-1.32661956044593, 2.18870625357827)

# List of observers (x1,y1,z1,x2,y2,z2)
observers <- c(previousObserver,runif(2,-8,8))
while(targetfunc(observers) <= baselineValue){
observers <- c(previousObserver,runif(2,-8,8))
}
print(observers)

optim <- optim(observers, targetfunc,
control=list(fnscale=-1,trace=5,REPORT=1))

Results

Here is the coverage ratio for different number of observers after optimization

You can test covering more than 98% of whole terrain is not trivial by using only 6 random observers but requires “careful” choice of their layout.

Finally let’s check step by step improvement in coverage as we add more optimal observers.

Single Observer

Two Observers

Three Observers

Four Observers

Five Observers

Six Observers

Line of Sight (LoS) Analysis: Multiple Observers (Part 3)

In this part of my LoS Analysis series, I will try to extend 3D LoS analysis for multiple observers. Assume that you drop multiple observers into a terrain with the aim of covering it perfectly (100% green).

We will reuse R codes used in Part 2. However we need to add a simple code piece to be used to merge Line of Sight results of multiple observers. If a point on terrain is visible by any of the observers that means point is visible, if the point is visible but far from all observers that means point is out of LoS due to distance (marked with yellow), for all other conditions point on terrain is red. updatestatus function is implemented for this purpose.

library(rgl)
##################
# Functions
##################
# 3D Terrain Function
height <- function (point) {
sin(point$x)+0.125*point$y*sin(2*point$x)+sin(point$y)+0.125*point$x*sin(2*point$y)+3
}

# Linear Function
linear <- function (px, observer, target) {
v <- observer - target

y <- ((px - observer[1])/v[1])*v[2]+observer[2]
z <- ((px - observer[1])/v[1])*v[3]+observer[3]

data.frame(x=px,y=y, z=z)
}

# Linear Function
distance <- function (terrain, observer) {
sqrt((terrain$x-observer[1])^2+(terrain$y-observer[2])^2+(terrain$height-observer[3])^2)
}

LoS <- function(terrain, observer, maxVisibleDistance){
status = c()
for (i in seq(1:nrow(terrain))) {
if (observer[1] == terrain$x[i] && observer[2] == terrain$y[i]){
if(observer[3] >= terrain$height[i]){
if (terrain$dist2observer[i] > maxVisibleDistance){
status <- c(status,"yellow")
}else{
status <- c(status,"green")
}
}else{
status <- c(status,"red")
}
}else{
# All points on line
line <- linear(seq(from=min(observer[1],terrain$x[i]),
to=max(observer[1],terrain$x[i]),
by=0.1),
observer,
c(terrain$x[i],terrain$y[i],terrain$height[i]))
# Terrain Height
h <- height(line)

# LoS Analysis
aboveTerrain <- round((line$z-h),2) >= 0.00

visible <- !is.element(FALSE,aboveTerrain)
if (visible){
# Second Rule
if(terrain$dist2observer[i] <= maxVisibleDistance){
status <- c(status,"green")
}else{
status <- c(status,"yellow")
}
}else{
status <- c(status,"red")
}
}
}

status
}

updatestatus <- function(status1,status2){
mergedstatus<-c()

for(i in seq(length(status1))){
if (status1[i] == "green" || status2[i] == "green"){
mergedstatus <- c(mergedstatus,"green")
}else if (status1[i] == "yellow" || status2[i] == "yellow"){
mergedstatus <- c(mergedstatus,"yellow")
}
else{
mergedstatus <- c(mergedstatus,"red")
}
}

mergedstatus

}

##################
# Input
##################
# Observer location
#observers<-c(0,0, 6,1,1,6)

# Max visible distance
maxVisibleDistance = 8

# Generate points with a step size of 0.1
x <- seq(from=-8,to=8,by=0.4)

xygrid <- expand.grid(x=x,
y=x)

terrain <- data.frame(xygrid,
height=height(xygrid)
)

# List of observers (x1,y1,z1,x2,y2,z2)
observers <- c(runif(2,-8,8),6,runif(2,-8,8),6,
runif(2,-8,8),6,runif(2,-8,8),6,
runif(2,-8,8),6,runif(2,-8,8),6,
runif(2,-8,8),6,runif(2,-8,8),6)
m <- matrix(data=observers,ncol=3,byrow=TRUE)

# Compute merged status of all observers
mergedstatus <- rep("red",length(terrain$height))
for(oidx in seq(1:dim(m)[1])){
terrain$dist2observer <- distance(terrain, m[oidx,])

status <- LoS(terrain,m[oidx,],maxVisibleDistance)

mergedstatus <- updatestatus(mergedstatus,status)
}

# Set merged status as the ultimate status
terrain <- data.frame(terrain,status = mergedstatus)

rgl.open()
rgl.surface(x, x,
matrix(data=terrain$height,nrow=length(x),ncol=length(x)),
col=matrix(data=mergedstatus,nrow=length(x),ncol=length(x))
)
bg3d("gray")

# Mark all observers
for(oidx in seq(1:dim(m)[1])){
spheres3d(c(m[oidx,1]),
c(m[oidx,3]),
c(m[oidx,2]),
radius=0.25,
color="white"
)
}

rgl.viewpoint(-60,30)

A Few Examples

Here are a few examples. All those observers are uniformly distributed over terrain using runif function

Trivial Case: Single Observer

Trivial Case with One Observer

Two Observers

Two Random Observers

Four Observers

Four Random Observers

Eight Observers

Eight Random Observers

Line of Sight (LoS) Analysis: 3D Terrain Analysis (Part 2)

In my previous post on LoS Analysis, I have tried to explain briefly the basics of LoS in two dimensional space. Obviously real life problems are based on three dimensional terrains although basic concepts are all the same. In this second part I will try to adapt the same techniques with a few modifications for three dimensional terrains.

3D Terrain Visualization with R

One of the first differences in 3D LoS analysis is the terrain visualization. We can not use plot function for proper visualization is 3D. Fortunately R has all packages you need for any type of problem. I will use rgl package which can be downloaded using install.packages("rgl") command.

Once you have the rgl package, generating pseudo 3D terrains as we did for 2D is a trivial thing.

3D Terrain with RGL

You can use the following R script to generate your 3D terrains like above.

library(rgl)

# 3D Terrain Function
height <- function (x,y) {
  sin(x)+0.125*y*sin(2*x)+sin(y)+0.125*x*sin(2*y)+0.25
}

# Terrain boundaries -8<=x<=8 and -8<=y<=8
boundary <- c(-8,8)

# Terrain grid with a step size of 0.1 units
xy<-seq(from=boundary[1],to=boundary[2],by=0.1)

# Evaluate all heights for all grid points
z<-outer(xy,xy,height)

# A few visualization staff
zlim <- range(z)
zlen <- zlim[2] - zlim[1] + 1
colorlut <- terrain.colors(zlen) # height color lookup table
col <- colorlut[ z-zlim[1]+1 ] # assign colors to heights for each point

# Draw the terrain
rgl.open()
bg3d("gray")
rgl.surface(xy, xy, z,
            color=col)

A new function in this script is outer function which generates the product of a vector and a row-vector to have a matrix (product of a row-vector with a vector/column-vector is obviously a scalar value and named to be dot/inner product). The third parameter of the function provides us the mechanism to apply a given function (height in our case) for each element of this matrix. Obviously you can play with height function to have fancier 3D terrains and to have best visualization you may need viewpoint routine in rgl package .

LoS in 3D Terrain

Line of Sight analysis on 3D terrain uses the same principles as it does in 2D. Use the following R script to decide on status of a point (invisible, visible, visible but far away)

library(rgl)
##################
# Functions
##################
# 3D Terrain Function
height <- function (x,y) {
  sin(x)+0.125*y*sin(2*x)+sin(y)+0.125*x*sin(2*y)+0.25
}

# Linear Function
linear <- function (x, observer, target) {
  v <- observer - target

  y <- ((x - observer[1])/v[1])*v[2]+observer[2]
  z <- ((x - observer[1])/v[1])*v[3]+observer[3]

  data.frame(x=x,y=y, z=z)
}

# Linear Function
distance <- function (p0,p1) {
  sqrt(sum((p0-p1)^2))
}

##################
# Input
##################
# Observer location
observer<-c(10,10,1)

# Target on terrain
target <- c(5, 5, height(5,5))

# Max visible distance
maxVisibleDistance = 4

# Generate points with a step size of 0.1
x <- seq(from=min(observer[1],target[1]),
         to=max(observer[1],target[1]),
         by=0.1)

# All points on line
line <- linear(x, observer, target)

# Terrain Height
h <- height(line$x,line$y)

# LoS Analysis
aboveTerrain <- round((line$z-h),2) >= 0.1

# First Rule
visible <- !is.element(FALSE,aboveTerrain)
if (visible){
  # Second Rule
  d <- distance(observer, target)
  if(d <= maxVisibleDistance){
    status <- "LoS"
  }else{
    status <- "non-LoS due to Distance"
  }
}else{
  status <- "non-LoS due to Blocking"
}

Obviously there are a few changes in the script with compared to 2D version.  The first one is linear function(Code Lines 10-18). New version not only evaluates second (y) but also the third dimension (z). Notice that z is our height dimension by convention. We have also utilized data.frame function to concatenate all dimensions to form a table of point dimensions

The second difference is on height function (Code Lines 5-8). It is no longer a mapping from x to y but a mapping from x,y to z.

Rest of the 3D version of script is pretty much the same or trivial to discuss more.

Visualizing LoS on 3D Terrain

Until this point we have analyzed LoS of a single point on 2D-3D terrains. But usually network analists wish to know LoS map of the terrain with respect to a given observer.  In other words we need to visually understand which regions on 3D terrain are visible by the observer, invisible by the observer due to blocking, or further than the limit from the observer.

Here the LoS map of our pseudo 3D terrain with respect to an observer with a given set of coordinates and maximum service range(green vs yellow regions).

3D LoS Analysis

You can obtain this visualization using following R script.

library(rgl)
##################
# Functions
##################
# 3D Terrain Function
height <- function (point) {
  sin(point$x)+0.125*point$y*sin(2*point$x)+sin(point$y)+0.125*point$x*sin(2*point$y)+3
}

# Linear Function
linear <- function (px, observer, target) {
  v <- observer - target

  y <- ((px - observer[1])/v[1])*v[2]+observer[2]
  z <- ((px - observer[1])/v[1])*v[3]+observer[3]

  data.frame(x=px,y=y, z=z)
}

# Linear Function
distance <- function (terrain, observer) {
  sqrt((terrain$x-observer[1])^2+(terrain$y-observer[2])^2+(terrain$height-observer[3])^2)
}

LoS <- function(terrain, observer, maxVisibleDistance){
  status = c()
  for (i in seq(1:nrow(terrain))) {
    if (observer[1] == terrain$x[i] && observer[2] == terrain$y[i]){
      if(observer[3] >= terrain$height[i]){
        if (terrain$dist2observer[i] > maxVisibleDistance){
          status <- c(status,"yellow")
        }else{
          status <- c(status,"green")
        }
      }else{
        status <- c(status,"red")
      }
    }else{
      # All points on line
      line <- linear(seq(from=min(observer[1],terrain$x[i]),
                         to=max(observer[1],terrain$x[i]),
                         by=0.1),
                     observer,
                     c(terrain$x[i],terrain$y[i],terrain$height[i]))
      # Terrain Height
      h <- height(line)

      # LoS Analysis
      aboveTerrain <- round((line$z-h),2) >= 0.00

      visible <- !is.element(FALSE,aboveTerrain)
      if (visible){
        # Second Rule
        if(terrain$dist2observer[i] <= maxVisibleDistance){
          status <- c(status,"green")
        }else{
          status <- c(status,"yellow")
        }
      }else{
        status <- c(status,"red")
      }
    }
  }

  status
}

##################
# Input
##################
# Observer location
observer<-c(0.835597146302462, -1.71025141328573, 6)

# Max visible distance
maxVisibleDistance = 8

# Generate points with a step size of 0.1
x <- seq(from=-8,to=8,by=0.4)

xygrid <- expand.grid(x=x,
                      y=x)

terrain <- data.frame(xygrid,
                      height=height(xygrid)
                      )

terrain <- data.frame(terrain,
                      dist2observer=distance(terrain, observer)
                      )

terrain <- data.frame(terrain,
                      status = LoS(terrain, observer, maxVisibleDistance))

rgl.open()
rgl.surface(x, x,
            matrix(data=terrain$height,nrow=length(x),ncol=length(x)),
            col=matrix(data=terrain$status,nrow=length(x),ncol=length(x))
            )
bg3d("gray")
# Mark the observer
spheres3d(c(observer[1]),
          c(observer[3]),
          c(observer[2]),
          radius=0.5,
          color="white"
          )

rgl.viewpoint(-60,30)

For a better visualization R allows you to implement spinning 3D terrains using play3d function and record it in gif format using movie3d function as I did below.

3D LoS Analysis 360

Line of Sight (LoS) Analysis: Basics (Part 1)

Introduction

Line of Sight analysis is a commonly used technique in telecommunication industry for A/I (Air Interface) equipment planning and allocation. With the simplest terms LoS is the question whether a point on N-dimensional space is visible by an other observer point. The question can be used to answer where to locate a transceiver on terrain so that it can serve customers on some region A.

Before relatively more complicated problems, let’s start with an easy example focusing on two dimensional terrains. Throughout the post, we will use R for coding which is my favorite option for any mathematical problem (statistics, plotting, linear algebra, optimization, etc.). But you can easily adapt coding material to Mathlab, Python,or your favorite language.

We will start by defining a mathematical function to be used to generate our pseudo terrains. For this purpose trigonometric functions (sin, cos) and polynomial functions are the best ones because of their wavy shapes. Here is an example of trigonometric terrain

Figure 1 Trigonometric Terrain

In order to generate this two dimensional one use the following code piece

x <- seq(from=4,to=10,by=0.01)
y <- sin(x)+cos(2*x)+sin(3*x)+cos(4*x)+3

windows()
plot(x,y,'l',
      main="y=sin(x)+cos(2x)+sin(3x)+cos(4x)+3",
      ylab="height",col="blue")
You may choose to use a polynomial terrain also

Figure 2 Polynomial Terrain

To obtain this terrain, use the following R script piece

x <- seq(from=0,to=6,by=0.01)
y <- x*(x-1)*(x-2)*(x-3)*(x-4)*(x-5)*(x-6)+100

windows()
plot(x,y,'l',
     main="y=x(x-1)(x-2)(x-3)(x-4)(x-5)(x-6)+100",
     ylab="height",col="blue")

Combining polynomial terrain functions with trigonometric ones will give you fancier ones.

What is LoS ?

You can think that we have already answered this question but this was an informal try which is not very useful for solving the problem. In order to solve this problem methodically we need to understand what makes a target visible (within LoS) by the observer.

As you see on Figure 3, green point is within line of sight of observer (blue point). However there is pseudo hill between red point and observer. The difference is that the line connecting observer and green point is always greater than the terrain function whereas this is not valid for the line connecting observer and red point (for x ε [~2.5, ~3.5] red line is under the terrain curve).

Figure 3 LoS vs non-LoS

This was the first point (blocking) we should define. The second one is an easier one related with maximum Euclidean distance between observer and target. The distance between observer and target may cause a phase shift in signal if the distance is sufficiently long or depending on weather conditions and terrain properties you may observer diffraction problems (actually there might be more than those). In return this will cause signal quality issues or call drops.  On Figure 3, although blocking is not an issue between observer and  yellow point, target is out of  visible range (say 8 units) of observer.

You can generate Figure 3 using the following R script

# Terrain Function
height <- function (x) {
  x*x/3+sin(x)+cos(2*x)+sin(3*x)+cos(4*x)+sin(5*x)+cos(6*x)+3
}

# Observer location
observer<-c(1.5,8.9)

# Generate terrain points with a tolerance of 0.1
x<-seq(from=-0.1,to=6.1,by=0.1)
terrainHeight<-height(x)

windows()
# Draw terrain
plot(x,terrainHeight,type='b',
     xlim=range(x),ylim=range(terrainHeight),
     main="Line of Sight (LoS)",
     ylab="Height",xlab="")

# Not LoS
points(x=c(observer[1],x[41]),
       y=c(observer[2],terrainHeight[41]),
       col="red",type='b')

# LoS
points(x=c(observer[1],x[5]),
       y=c(observer[2],terrainHeight[5]),
       col="green",type='b')

# LoS but far
points(x=c(observer[1],x[length(x)]),
       y=c(observer[2],terrainHeight[length(x)]),
       col="yellow",type='b')

# Draw Observer
points(x=c(observer[1]),
       y=c(observer[2]),
       col="blue",pch=10)

Method to Decide LoS

Finally let’s define a method to find all visible, invisible, and “far” points on any terrain. Since it is not “easy” to decide analytically whether the line connecting observer and target “is above” the terrain for any terrain function, we will use a simple numeric method.

We will define a step size small enough (around Spatial Tolerance) to generate all x values between observer and target. seq function is a good choice for doing this (Code Lines 33-36).  Evaluate these x values for line function connecting observer and target and terrain function. Evaluation is simple for terrain function using height function (Code Lines 4-7). Evaluation of line function is held by function linear  using parametric definition of line function (Code Lines 9-14) . Next step is to search for any x value having a line evaluation less than terrain evaluation (Code Line 44-28). The rest is simple as to evaluate euclidean distance and assigning values to status variable.

##################
# Functions
##################
# Terrain Function
height <- function (x) {
  x*x/3+sin(x)+cos(2*x)+sin(3*x)+cos(4*x)+sin(5*x)+cos(6*x)+3
}

# Linear Function
linear <- function (x, observer, target) {
  v <- observer - target

  ((x - observer[1])/v[1])*v[2]+observer[2]
}

# Linear Function
distance <- function (p0,p1) {
  sqrt(sum((p0-p1)^2))
}

##################
# Input
##################
# Observer location
observer<-c(1.5,9)

# Target on terrain
target <- c(5, height(5))

# Max visible distance
maxVisibleDistance = 4

# Generate points with a step size of 0.1
x <- seq(from=min(observer[1],target[1]),
         to=max(observer[1],target[1]),
         by=0.1)

# Terrain Height
h <- height(x)

# y Values
y <- linear(x, observer, target)

# LoS Analysis
aboveTerrain <- round((y-h),2) >= 0.00

# First Rule
visible <- !is.element(FALSE,aboveTerrain)
if (visible){
  # Second Rule
  d <- distance(observer, target)
  if(d <= maxVisibleDistance){
    status <- "LoS"
  }else{
    status <- "non-LoS due to Distance"
  }
}else{
  status <- "non-LoS due to Blocking"
}