Maths SL IA
Short Description
Maths SL IA...
Description
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Mathematics SL Internal Assessment 049046-0003
Introduction The great-circle distance1 is the shortest distance shortest distance between two points two points on the surface of a sphere, a sphere, measured measured along the surface of the sphere. Euclidean geometry states that on a flat map the length of the t he straight line between two points can be used to calculate the distance between the points. While in Non-Euclidean, there are no straight lines on sphere rather they are substituted by geodesics which on sphere is referred to shortest course course between two two points on earth. Along the two two points located on sphere, which are not face to face, there always lies a unique great circle. Though earth is not an accurate sphere, but still the same formulas used for spherical calculations can be applicable in determining the great circle distance. Since I love travelling, last time when I travelled on an airplane, I came across a question that is how pilots determine their route. So I researched about what is that the pilots do to navigate around in the sky. This was when I came to know about the way pilots use spherical geometry and spherical trigonometry to find the shortest distance between two places on Earth. So this was why I chose this topic. With this investigation I am sure I will find how pilots determine their route with solid, more scientific, mathematical theory and come to conclusions that could help me answer my questions. A
1
Introduction includes a clear description of the aim and rationale
https://en.wikipedia.org/wiki/Great-circle_distance
Mathematics SL Internal Assessment 049046-0003
Statement of task In my exploration, I am going to find the shortest distance between two places on Earth with help of spherical geometry and spherical trigonometry. I will be using globe for my numerical calculations. I will be choosing four countries on globe. Then I am going to draw the great circle and find the longitude and latitude of the countries on the great circle. Therefore with the use of spherical trigonometry I will be calculating and finding the shortest distance. Mathematical exploration relates
C
to personal Interest.
What is great circle distance and how does it matter? Latitude2 - Latitude is the angular distance of a place north or south of the earth's equator, or of the equator of a celestial object, usually expressed in degrees and minutes.
B
Key terms are well defined
Longitude3 – Longitude is the angular distance of a place east or west of the Greenwich meridian, or west of the standard meridian of a celestial object, usually expressed in degrees and minutes. Figure 1 shows a part of world map. If we need to travel from Point A to Point B on a world map, we can take the road from A to south till C and then from C to east till B and then we reach the destination. The total distance travelled would then be length of AC + length of CB. Also, if there is any straight road from Point A to Point B, then we can simply travel through that road and the distance would be length of AB.
2 3
https://en.oxforddictionaries.com/definition/latitude https://en.oxforddictionaries.com/definition/us/longitude
Mathematics SL Internal Assessment 049046-0003
B
Useful explanation appropriately supported by the diagram
FIGURE 1 If we are travelling on an aircraft and if we want to move from one city to another city we will draw a straight line between those cities. This can be seen in figure 2 where a straight line is drawn between point X and point Y on the flat world map. And if we locate the same straight line between two points on a globe, the distance shown on globe will not be equal to distance shown on a map. The reason for this is that the straight line on a sphere when drawn on a sphere turns into a great circle. A great circle basically is that circle which intersects and its plane travels through the centre of the earth. Hence, any distance calculated on the Earth’s surface is fundamentally a spherical distance or great circle distance. That is why it’s important to properly and accurately calculate the shortest distance usin g because any difference between the distance on map and great circle distance can mislead and misguide the person extremely while travelling. So to calculate an accurate distance between two places on earth, the great circle distance method is used.
Mathematics SL Internal Assessment 049046-0003
A
Figure 2
Appropriate explanation
How do we calculate great circle distance? There are many methods to calculate distance but many of those are done on computers using programming. So in my exploration I have chosen those two methods where mathematical implementation can be done using mathematical formulas. These methods are not very accurate as a GPS because Earth is not an exact sphere. A perfect sphere4 has the same radius from the centre of the sphere to every single point on the surface. As the earth is geographically assorted and is spinning on its axis, it is actually an oblate sphere which is flat at the poles and whose radius at the equator is slightly longer than the radius at the poles. 1. Northing/easting: This is the method where we firstly have to locate the centre
of the earth and the two places between which we will calculate distance. Then
4
https://en.wikipedia.org/wiki/Sphere
Mathematics SL Internal Assessment 049046-0003
we will draw a line from the centre of the earth to both the places. After that, we will be calculating the difference between the latitude and longitude of the chosen two places. This can be seen in the figure 3 where the difference is represented by north south distance(
) and east west distance(
). Once we
have calculated the difference between the longitudes and latitudes of the two places, we can use these to calculate the angle A and angle B which further which help us to calculate the north-south distance and east-west distance.
C
Evidence of Personal en a ement
Figure 3
The north-south distance will be represented by
Where
∆
NS depicts
=
∆
the North and south difference
between the latitudes of two places. And re stands for mean radius of the earth
The east-west distance would be represented by
Where
∆
Ew depicts
=
∆
the East and West difference
between longitudes of two places and re stands for mean radius of the earth
Mathematics SL Internal Assessment 049046-0003
After we have figured the angles, we can simply use the Pythagorean Theorem because we will have calculated the north-south distance (length AB) and the eastwest distance (length BC). With the help of Pythagorean Theorem formula we can now calculate the distance (length AC) which can be represented by formula by
B
Good use of notation
=
(
) +(
)
Mathematics SL Internal Assessment 049046-0003
Practical implementation of the method For my practical implementation of the Northing/Easting method to calculate the shortest distance firstly I will be taking New York and Johannesburg as two cities between which I will calculate the distance and then I will be calculating the distance between Dakar and Cape Town. Mean radius of the earth = 6371 km C
Good use of an example
Distance between New York and Johannesburg In degrees
NEW YORK, USA
JOHANNESBURG, Africa
Latitude = 40.714˚
Latitude = -26.202˚
Longitude = -74.006˚
Longitude = 28.044˚
In Radians
NEW YORK, USA
JOHANNESBURG, Africa
Latitude = 0.710593 ͨ
Latitude = -0.457311 ͨ
Longitude = -1.291648 ͨ
Longitude = 0.489460
Actual distance between New York and Johannesburg = 12831.2997 km
Mathematics SL Internal Assessment 049046-0003
Northing/Easting
Difference between the latitude of New York and Johannesburg = 40.714-(-26.202)
= 66.916˚ Difference between the longitude of New York and Johannesburg = ( -74.006)-28.044 = -102.05
˚
North-South distance =
∗ ∗
2π 6371 66.916 360
= 7440.720 km
East-West distance =
∗ ∗−
2π 6371 ( 102.05) 360
Mathematics SL Internal Assessment 049046-0003
= Mod(-11347.442 km) =11347.442km Shortest
distance
between
New
York
and
Johannesburg
=
(7440.720)2 + (11347.442)2
= 13569.405 km By this we can conclude that the distance calculated by this method is not very accurate as there is 7.68% difference between the actual distance and the calculated distance. Even though the percentage is small but in terms of distance this percentage is huge as the difference between both cities is around 1000 km which is a very large value.
Distance between Dakar and Cape Town In degrees
Dakar, Senegal
Cape Town, Africa
Latitude = 14.6937000˚
Latitude = -33.9258400˚
Longitude =-17.4440600˚
Longitude = 18.4232200˚
In Radians
Dakar, Senegal
Cape Town, Africa
Latitude = 0.256453 ͨ
Latitude = -0.592117 ͨ
Longitude =-0.304457 ͨ
Longitude = 0.321545 ͨ
Mathematics SL Internal Assessment 049046-0003
Actual distance between Dakar and Cape Town= 6,407 km
Northing/Easting
Difference between the latitude of Dakar and Cape Town = 14.6937-(-33.9258)
=48.62˚ Difference between the longitude of Dakar and Cape Town = (-17.4441)-18.4232 = -35.867˚
North-South distance =
∗ ∗
2π 6371 48.6195 360
= 5406.2417 km
East-West distance =
∗ ∗−
2π 6371 ( 35.8673) 360
11
Mathematics SL Internal Assessment 049046-0003
=Mod( -3988.2618 km) =3988.2618
Shortest distance between Dakar and Cape Town =
(5406.2417)2 + (3988.2618)2
= 6718.160574 km By this we can conclude that the distance calculated by this method is not very a ccurate as there is 4.86% difference between the actual distance and the calculated distance.
2. Haversine formula: In this method too, the first step is to locate the two cities and the centre of the earth and then we will draw a line from the centre of the earth to both the cities. Under this we will just find the central subtended angle, the angle which is subtended at the centre of the earth by the two lines connecting the angle is found, and it can be used to find distance. For any two points on a sphere, the Haversine 5 of the central angle between them is given by
− ∅ −∅ − hav
d r
= hav φ2
φ1 + cos φ1 cos φ2 hav(
where hav is the haversine function: hav θ = sin2 B
θ 2
=
1 cos (θ) 2
Variables are well defined
d is the distance between the two points,
r is the radius of the sphere,
φ1 , φ2 : latitude of point 1 and latitude of point 2, i n radians 5
https://en.wikipedia.org/wiki/Haversine_formula
12
2
1)
Mathematics SL Internal Assessment 049046-0003
∅∅ 1 ,
2:
longitude of point 1 and longitude of point 2, in radians
On the left side of the equals sign
d r
is the central angle.
Solve for d by applying the inverse Haversine or by using the arcsine function: d = rhav
− 1
Where h is hav C
Good demonstration of personal involvement
h = 2r arcsin( h )
− ∅−∅ − ∅ − ∅ d r
or more explicitly:
=
+
(
)
Or
d = 2r arcsin( hav φ2
φ1 + cos φ1 cos φ2 hav(
13
2
1)
Mathematics SL Internal Assessment 049046-0003
Practical implementation of the method For my practical implementation of the Haversine formula to calculate the shortest distance firstly I will be taking New York and Johannesburg as two cities between which I will calculate the distance and then I will be calculating the distance between Dakar and Cape Town. Mean radius of the earth = 6371 km
Distance between New York and Johannesburg In degrees
NEW YORK, USA
JOHANNESBURG, Africa
Latitude = 40.714˚
Latitude = -26.202˚
Longitude = -74.006˚
Longitude = 28.044˚
In Radians
NEW YORK, USA
JOHANNESBURG, Africa
Latitude = 0.710593 ͨ
Latitude = -0.457311 ͨ
Longitude = -1.291648 ͨ
Longitude = 0.489460
ͨ
Actual distance between New York and Johannesburg = 12831.2997 km
14
Mathematics SL Internal Assessment 049046-0003
Haversine Formula
E
Good mathematical manipulation of formula
− − − − − − − − − =
( 0.457311) 0.710593 ͨ
( 101679) ͨ
=
=
.
=
=
+
+
0.710593
0.457311
0.710593
0.457311
+
0.710593
.
+ . )( .
.
=
( .
)
+ .
.
15
0.457311
(
(
( .
0.489460 ( 1.291648)
1.78111
)
)
)
Mathematics SL Internal Assessment 049046-0003
∗ ∗∗ =
.
=
.
=
.
=
.
km
This value is still inaccurate but we can say that this value is more accurate than northing/easting method as the difference between the actual and the calculated distance is 5.75%.
Distance between Dakar and Cape Town In degrees
Dakar, Senegal
Cape Town, Africa
Latitude = 14.6937000˚
Latitude = -33.9258400˚
Longitude =-17.4440600˚
Longitude = 18.4232200˚
In Radians
Dakar, Senegal
Cape Town, Africa
Latitude = 0.256453 ͨ
Latitude = -0.592117 ͨ
Longitude =-0.304457 ͨ
Longitude = 0.321545 ͨ
Actual distance between New York and Johannesburg = 12831.2997 km C
Student does a good work by showing examples of different countries
16
Mathematics SL Internal Assessment 049046-0003
Haversine Formula
Good demonstration of
C
learning and describing unfamiliar mathematics
=
− − − − − − − − − ( 0.592117)
0.256453 ͨͨ
( 0.84857)ͨ
=
=
.
+
0.256453ͨ
0.592117
(
0.626002
+
0.256453
0.592117
(
+
0.256453
0.592117
( .
17
0.321545 ( 0.304457)
)
)
)
Mathematics SL Internal Assessment 049046-0003
∗ ∗∗ .
=
=
+ . )( .
.
+ .
=
.
.
=
.
=
B
)
.
=
=
( .
.
.
No explanation of how this value is determined .
With this formula, the difference between the actual distance and the calculated distance is 4.87%. So, we can know that any method to calculate distance will always result in inaccurate distance. These methods were used in the past when the technology was not that advanced. However, today the technology has developed to the point where we can find accurate distance through GPS.
18
Mathematics SL Internal Assessment 049046-0003
Limitations of Great distance formula:
Not exact distance can be calculated. There is always some difference between calculated distance and actual distance.
D
Differences in radians and degrees values can lead to calculation of
Discussed limitation of
inaccurate distance.
the result
High chance of math error while calculating distance.
Conclusion This exploration was really fascinating for me as I could link my exploration to the experiences of life by applying my conceptual knowledge of mathematics to develop theories. This exploration helped me to clear my doubts and made it easy for me to understand the concept of navigation. With the help of geodesic, I was able to understand the spherical geometry and trigonometry in a better way and could actually use it with ease in my exploration. The biggest lesson I learnt from this research is that any work will look hard till the time you put your effort on researching on it. The same thing happened during my project. At the starting of my project I was not at all confident on whether I will be able to complete my project or not but as I investigated on it, I came across a totally new field, learned many new things and was finally able to do my project. So, overall I enjoyed doing this exploration as through this not only I learned non-Euclidean geometry and broadened my view on maths but also it was a fun activity for me as I gathered information about different countries through this exploration.
A
Student relates the mathematical ideas and consider the significance of the result.
19
Mathematics SL Internal Assessment 049046-0003
Bibliography
Great Circle Distance. (2013). YouTube. Retrieved from http://www.youtube.com/watch?v=634GucAdzzA
Great Circle ( GC) Track. Calculation of GC dist, initial and final course Part 1. (2015). YouTube. Retrieved from https://www.youtube.com/watch?v=3LEpjVsKEU4
World Map: A clickable map of world countries :-). Geology.com. Retrieved from http://geology.com/world/world-map.shtml
Inc,
G. Calculating
Distance
Coordinates. Support.groundspeak.com.
between
Two
Sets
Retrieved
of from
https://support.groundspeak.com/index.php?pg=kb.page&id=211
York, N. Geographic coordinates of New York, New York, USA. Latitude, longitude, and elevation above sea level of New York. Dateandtime.info. Retrieved from http://dateandtime.info/citycoordinates.php?id=5128581
Geographic coordinates of Johannesburg, South Africa. Latitude, longitude, and elevation above sea level of Johannesburg. Dateandtime.info. Retrieved from http://dateandtime.info/citycoordinates.php?id=993800
Town, C. Geographic coordinates of Cape Town, South Africa. Latitude, longitude, and elevation above sea level of Cape Town. Dateandtime.info. Retrieved from http://dateandtime.info/citycoordinates.php?id=3369157
Geographic coordinates of Dakar, Senegal. Latitude, longitude, and elevation above
sea
level
of
Dakar. Dateandtime.info.
http://dateandtime.info/citycoordinates.php?id=2253354
20
Retrieved
from
Mathematics SL Internal Assessment 049046-0003
Assessment Criterion
A
B
C
D
E
Total
Achievement level awarded by teacher
3
2
3
2
4
14
Maximum Possible Achievement level
4
3
4
3
6
20
Comments Criteria A :Communication: A3:The work is concise and easy to follow. It fulfills the aim and its complete. CriteriaB:Mathematical representation B2:There is good definition of terms. Good use of photography to generate the visual view of the problem. Criteria C:Personal Engagement C3:The student has created some examples. Some personal interest is expressed with the evidence of thinking independently. Criteria D:Reflection D2:It links with the area of maths. Student needs to put some more efforts for meaningful reflection. Criteria E:Use of Mathematics: E4:Relevent mathematics
is used. Thorough knowledge and understanding is
demonstrated.
21
Mathematics SL Internal Assessment 049046-0003
Background Information Before beginning with an exploration, student has made a deep study about the spherical geometry and mathematical term associated with it. Student has also gone through the Euclidean and non Euclidean geometry.
22
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