CSci 5715, Fall 15: Homework 4
Table of Participation
Question ID
1
2
3
4
5
Answer drafted by
Answer reviewed by
Q1: The coordinate system we will use in the following questions is the first quadrant in a Cartesian coordinate
system as shown in Figure 1 where Y axis points to the North and X axis points to the East. The angle used in the
questions is counter-clockwise from the East vector. All the numbers are unitless.
Y
North vector
East vector
angle
X
(0,0)
Figure 1 Coordinate system
Q1: Part A: Student A is trying to obtain the coordinates (X, Y) of his current location. There is a landmark B at
(100, 60) as shown in Figure 2. Student A has a laser range finder and compass to measure the distance to B and
the direction (i.e. the counter-clockwise angle from the East). Assume the distance is 200 and the angle from East
vector is 60 degrees. Determine the coordinates of student A.
Y
100
North vector
B
60
50
0
50
100
East vector
X
Figure 2 Landmark B
Q1: Part B: The epicenter of an earthquake sends out two types of shockwaves that travel at different speeds. By
measuring the time delay between the first and second shockwave, geologists can determine how far away the
earthquake occurred. Suppose there are three earthquake monitoring centers A, B and C, and they all identified an
earthquake just happened. The information reported by the three centers is shown in Table 1. 1) Compute the 2dimension surface coordinates of the epicenter of this earthquake assuming a flat 2-dimension Euclidean space. 2)
How many monitoring centers are needed to estimate the depth as well as the surface coordinates of the epicenter
of the earthquake using a spherical model of a planet?
Center
A
B
C
Table 1 Information from towers
Coordinate (X, Y)
(0, 0)
(0, 120)
(80, 0)
Distance
100
100
60
Q2: Part A: To monitor fires in a forest stand, two towers A and B are built as shown in Figure 3. Fire location
man be determined by measuring the direction (i.e., counter clockwise angle from East vector) from each tower
and the intersection will be the fire site. Suppose a fire in on sight and the information reported by tower A and B
is shown in Table 2. Calculate the location of the fire.
Y
100
50
0
North vector
45°
East vector
Fire tower in Itasca
State Park, MN
Fire site
B 45°
50
campsitereports.com
100
X
Figure 3 Forest fire monitoring
Table 2 Information from towers
Tower
Coordinate (X, Y)
A
(0, 100)
B
(50, 0)
Direction (angle from east)
-45
45
Q2: Part B: Sailors on a ship need to determine how far away they are from the shore before pulling into the
harbor. There are two lighthouses A and B on the shore as shown in Figure 4. The distance between A and B is
100. The direction (counter clockwise angle from East vector) is 60 degrees from A and 150 degrees from B.
Determine the distance between the ship and the shore.
Y
100
North vector
East vector
Ship
50
A
0
60°
B
50
150°
100
X
Figure 4 Ship pulling into the dock
Q3: Part A: Figure 5 shows a few of cities in a planar (Euclidean 2-dimensional space) map of the United States.
It also shows the locations (projection on a 2-dimensional planar map) of four satellites {A, B, C, D} at a selected
time. For more details please visit: http://www.gps.gov/multimedia/tutorials/trilateration/. The map scale is given
(lower-right) and can be used to determine the X, Y coordinates of each location on the map. For your reference,
the map coordinates of satellites are given in Table 3. The length of the scale bar is calibrated to be 1 cm in a
letter size print-out (L: 11”, W: 8.5”) of this whole page.
Suppose the speed of light c = 299,792,458 m / s, and the time taken for the signal to reach the receiver is t = |tsat treceiver|, where tsat is the time on satellites when signal is sent and treceiver is the time on receiver when signal is
received, assuming all the clocks are perfectly synchronized. The distance between the each satellite and a
receiver can then be calculated by Distance = c * t. Now a receiver is determining his location (i.e., closest city)
from the information obtained from the four satellites in Table 4. Show which city has the smallest Euclidean
distance to the receiver and include necessary steps for calculation. Hint: You may verify your result by drawing
distance-circles on the map with a compass or ruler.
Saint Paul
Minneapolis
Origin
(0,0)
Scale
1000 km
Figure 5 Distribution of cities and satellites
Table 3 Satellite map coordinates
Satellite A
B
C
D
X (cm) 4.4 4.49 2.02 11.2
Y (cm) 6.35 1.45 7.3 0.49
* The values assume the map being printed out with standard letter size (L: 11”, W: 8.5”).
Table 4 Received data from satellites
Satellite
A
Time t (second)
0.01457675
Distance d (km)
4370
Scaled map distance (cm) 4.37
B
0.00346906
1040
1.04
C
0.02068097
6200
6.2
D
0.02031405
6090
6.09
* For scaled map distance, the values assume the map being printed out with standard letter size (L: 11”, W: 8.5”).
Q3: Part B: Now consider a new location of the receiver and the data is shown in Table 5. Determine the mean
location of the receiver ignoring positional inaccuracy.
Notice that the time given has 8-digit precision. Positional inaccuracy occurs since the 9th digit is truncated and it
affects the accuracy. Table 6 shows the positional inaccuracy brought by the missing digits under a baseline of
Nano second precision. To quantitatively evaluate the effect of time inaccuracy, determine the location and error
of the receiver by dropping: 1) only the last digit and 2) the last three digits of the original time data. You will
need to re-compute the distances. Compare your “inaccurate” results with the original result and show the effect
on positional accuracy.
(Note: The updated numbers below do not affect the procedure in your solution, just minor differences in
numbers. So if you already computed the results before this update, you can simply use that result and no need for
changes and it will be graded using that result. Otherwise, use the updated version)
Table 5 Received data from satellites
Satellite
A
Time t (second)
0.00700485
Distance d (km)
2100
Scaled map distance (cm) 2.1
B
0.01267543
3800
3.8
C
0.01367613
4100
4.1
Table 6 Positional inaccuracy by time inaccuracy - example
Satellite
A
Error EA B
Error EB
Original time t (ns)
7004850 +/- 5
13676130 +/- 5
Missing 1 digit (ns) 7004900 +/- 50
13676100 +/- 50
Missing 2 digits (ns) 7005000 +/- 500
13676000 +/- 500
Missing 3 digits (ns) 7000000 +/- 5000 13680000 +/- 5000
Missing 4 digits (ns) 7000000 +/- 50000 13700000 +/- 50000
D
0.02301592
6900
6.9
Q4 Douglas-Peucker algorithm is widely used to simplify lines and polygon boundaries in the interest of simpler
visualization. The idea of the algorithm is to recursively remove the small “bumps” in a line or polygon (Figure
6). Suppose we create a straight line L using the start and end points of a line. A point is considered part of a
small bump if its distance to the straight line Lis smaller than our threshold. The pseudocode of the algorithm is
shown below Figure 6. Follow the steps and simplify the blue line shown in Figure 7 with a threshold t = √2 ≈
1.414, which is the diagonal distance of each grid in Figure 7. Fill out Table 7 to show the trace of simplification.
Simplification
Input
Output (black)
Figure 6 Douglas-Peucker simplification
The algorithm proceeds as follows:
Input:
LineString LS[] – each entry is a point;
distance threshold t;
Pseudo code:
FUNCTION DP_Simplification(LS[], t):
len = length(LS[]);
IF(len<3) RETURN;
straight_line[] = new line(LS[1], LS[len]);
/**Returns (node ID, distance) with maximum distance to the line.*/
[node_ID, dist] = max_distance(LS[2 : len - 1], straight_line[]);
IF(dist <= t)
LS[] = straight_line[];
ELSE
LS1[] = LS[1 : node_ID];
LS2[] = LS[node_ID : len];
LS[] = concatecate( DP_Simplification(LS1[]) ,
DP_Simplification(LS2[]) );
RETURN;
Figure 7 Nodes on a LineString
Table 7 Execution trace of line simplification
#th call of
Start Node End Node
Node with
DP_Simplification
max distance
1
1
13
9
2
1
9
5
3
4
5
6
7
8
9
10
*The result might not fill the whole empty table.
First two lines of Table 7 are given for your reference.
Distance <= 1.414
FALSE
FALSE
Q5 Visit the following web maps:
1) Google Maps: https://www.google.com/maps
2) Bing Maps: https://www.bing.com/mapspreview
3) Open Street Map: https://www.openstreetmap.org/#map=5/51.500/-0.100
4) Minnesota Topology Map: http://arcgis.dnr.state.mn.us/maps/mntopo/
5) Map of Antarctica: http://lima.usgs.gov/view_lima.php
Determine which projection is used for each of the above web maps.