Coordinate Geometry
The coordinate plane The points on a line can be referenced if we choose an origin and a unit of
distance on the axis and give each point an identity on the corresponding number line. We can also give
each point in a plane an identity using an ordered pair of real numbers called Cartesian coordinates.
We choose an origin o for our plane and draw a pair of perpendicular number lines intersecting at o
which we call axes. Usually one of the axes is horizontal with positive numbers on the right and we
call it the x-axis. The other axis is usually vertical with positive numbers above the x-axis. We usually
call the vertical axis, the y-axis. This divides the plane into four quadrants, labelled I − IV as shown
below. Note that the points on the axes are not in any of the four quadrants.
20
18
16
14
12
10
y
8
II
I
6
P(a, b)
b
4
2
– 10
–5
5
o
a
10
x
–2
III
IV
–4
–6
10
–8
Any point P in the plane can be given a unique address or label with an ordered pair of numbers
P (a, b). Here a is the x-coordinate which we find by constructing a line from the point to the x-axis
which is perpendicular to the x-axis. The point where this line cuts the x axis is the x-coordinate of
P (in this case a). We find the y-coordinate of the point (in this case b) by constructing a line from
the point to the y-axis which is perpendicular to the y-axis. The point where this line cuts the y axis
is the y-coordinate of P (in this case b). For a point on either axis, the coordinate corresponding to
that axis is determined by their position on the axis. Here are some examples:
–910
– 12
8
– 14
7
6
5
4
3
(0, 3)
2
(-2, 1)
(4,1)
1
(0, 0)
– 6
– 4
– 2
(2, 0)
2
4
6
8
10
– 1
(-2, -1)
– 2
(1, -2)
– 3
– 4
Conversely, given an ordered pair of numbers there is a point on the plane corresponding to that ordered
pair.
– 5
– 6
Example Plot the following points on the Cartesian plane:
– 7
(0, 5)
(−3, 1)
(2, −3)
– 8
(−2, −2)
– 9
– 10
1
(0, −3),
(−3, 0).
Kinematic data Analysis
A major branch of sports science involves kinematic data analysis. Kinematics is a branch of mechanics
that describes the spatial and temporal components of motion. The description of motion involves the
position, velocity and acceleration of a body with no consideration of the forces causing the motion.
With the development of user friendly software, athletes, coaches and health professionals in addition
to sports scientists often use video analysis to track and analyze the movements of various points on the
body during sports activities. The advantage of using video analysis is that it gives precise measurements
of distances, speeds and acceleration. Although most movements in sport are three dimensional and
would require a three dimensional co-ordinate system to accurately track the movement, much insight
can be gained from a two dimensional representation of movement in sport. One must choose a frame
of reference, usually a Cartesian co-ordinate system, and which points to track. Data may be collected
from raw video using digitizing software or motion sensors may be placed on various points of a moving
body and their paths tracked using specialized video cameras. The origin of the co-ordinate system
may also be placed at a joint or point which moves with the body and an axis may be aligned with a
segment of the body so that the movement of the body relative to that point or that body segment may
be analyzed. The use of cameras, software and interpretation and management of the large amount of
data generated requires much time spent learning to use the equipment and software and is outside the
scope of this course. We will look at some much simplified hypothetical examples below to get a feel
for how we might track and analyze movement using a Cartesian co-ordinate system.
Example The following data was collected for the position of the head of the club in the golf swing
of the individual below. Plot the data given to get a rough picture of the path of the head of the club
throughout golf swing. The starting position is shown in the picture (you may wish to add directional
arrows to the points you draw).
Time = seconds from the start x
y
0.1
100 11
0.2
60 50
0.3
55 100
0.4
50 150
0.5
75 200
0.6
150 220
0.7
160 200
0.8
140 190
0.9
70 150
1
50 50
1.1
150 10
1.2
200 50
1.3
250 100
1.4
210 150
2
200
150
100
50
0
0
50
100
150
200
250
(a) How does the path of the head of the above golf club compare to that of the golf club of Mr.
Mannequin shown in the advertisement for a tracking device shown below? I’ve superimposed the
pictures on the next page for a more detailed comparison, however you should see what you can derive
without co-ordinates for Mr. Mannequin before you look at them.
(b) In addition to tracking the path of the head of the golf club, what other points on the body might
be important to track for the coach or athlete?
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(c) Can you see any more differences between the two golfers using the superimposed images in the
co-ordinate system below?
200
200
200
150
150
150
100
100
100
50
50
50
0
0
50
100
150
200
250
0
0
50
100
150
200
250
0
0
50
100
150
200
250
Example Note how the origin of a co-ordinate system is given in the following pictures, but a unit of
measurement is not. Choose a unit of measurement and give approximate co-ordinates for the end of
the upswing and the end of the follow through for each picture.
What are the main differences between the three golf swings.
Note that movements occur over time and we can get more information from our data by noting the
time taken to perform a task or the time taken to cover a given distance. We will elaborate on this in
the next few lectures, when we introduce functions, speed, velocity and acceleration.
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Distance Formula
We can derive a formula for the distance between two point in the plane using Pythagoras’ theorem.
On can see form the following picture that the distance between the points A and B, denoted d(A, B)
is given by
p
d(A, B) = (x2 − x1 )2 + (y2 − y1 )2 ,
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16
where A = (x1 , y1 ) and B = (x2 , y2 ).
p
Note that this tells us that the distance of a point (x1 , y1 ) from the origin (0, 0) is given by r = x21 + y12 .
14
12
B(x2,y2)
y2
10
d(A,B)
8
|y2 - y1|
6
y1
4
A(x1,y1)
|x2 - x1|
2
–5
5
10
x1
–2
x2
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Example Find the distance between
(a) the points P (−2, 1) and Q(1, 1) and
–4
–6
–8
– 10
(b) the points P (0, 1) and Q(−3, 4).
(c) Find the distance of the following points from the origin: P (0, 1), Q(−3, 4).
Units of measurement
When choosing units of measurement we can use either the British system or the metric system based
on the Systeme International d’Unites (SI). In the British system, we use pounds (lbs), feet (ft.) and
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seconds (s.) for weight, distance and time. In the metric system, we use kilograms (kgs), meters (m.)
and seconds for mass, distance and time.
Graph of an equation
Equations Sometimes the graph of the data we collect, closely matches the graph of a known curve
resulting from an equation. The graph of an equation relating x and y is the set of all points in the
plane which fit the equation.
Example the graph of the equation y = x2 , is the set of all points (x1 , y1 ) in the xy-plane for which
y1 = x21 . Note that for this equation, the value of y is completely determined by the value of x. We
can get some idea of what the graph looks like by plotting a few points on the graph. Fill in the table
below and plot the resulting points to get a picture of the corresponding graph.
x
y = x2
(x, y)
2
−2 (−2) = 4 (−2, 4)
−1
0
1
2
Analyzing Motion
We break our analysis of motion into three categories:
Translational or linear motion occurs when all points on a body or object move the same distance
over the same time. The motion might be in a a straight line (straight-line motion) such as when an
ice skater maintains a pose and glides across the ice, or it may occur along a curved path ( curvilinear
motion) such as the path that a ball follows when thrown. Although a body such as that of an athlete
running around a track may not be strictly within the realm of our definition of curvilinear motion, we
can apply our analysis of curvilinear motion to their path to derive some global results.
After studying linear motion, we will also study angular motion) which occurs when all parts of a
body move through the same angle but do not undergo the same linear displacement. Angular motion
occurs around an axis of rotation which is a line perpendicular to the plane in which the rotation
occurs. For example, if you spin a bicycle wheel around its axis, each point on a spoke of the wheel
rotates through the same angle in the same amount of time, however a point closer to the edge travels
further than a point nearer the axle.
When a combination of rotation and translation occurs, it is described as general motion. Most human
movements involve a combination of translational and angular motion.
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