As In the World Rugby Laws, it is

As someone who has recently been introduced to the
game of Rugby, kicking has always been somewhat of a mystery to me, as I play
the forward position, and hence have little chance to attempt kicking in an
actual game. Many kicks take place during a game of Rugby, as it is the only
way to get the ball to move forward without needing to crash through an
opposing player or pass backwards down the line. The kick that gets the most
attention, however, takes place in a controlled scenario, during stoppage of the
game – the conversion kick, also known as the place kick.

The place kick is an integral gameplay element many
sports around the world, such as rugby, American football and Hockey. In Rugby
Union, when a player scores a try, their team given the right to “convert” the
try, by kicking the ball over the crossbar and through the uprights (the large
“H”-shaped stand). However, the kick can only be taken in line the point where
the try was scored, parallel to the touch line (line of play).

 

 

 

 

 

 

Figure
1: A diagrammatic of representation of the angle seen by the kicker

From
the diagram, the distance between uprights are represented by the distance
between points A and B, and the dotted line depicts the line where the
conversion can be taken.

In the World Rugby Laws, it is stated in Law 9.B.1 (b)
that “the kick is taken through the line where the try was scored in the field
of play”. However, the kicker is able to decide the position he/she will take
on that line where they will take the kick.

The central aim of this IA emerges from this scenario:
what is the best position to take along this line in order to kick the ball
through the uprights? There is a simple solution: ensure that the angle KAB
(from the diagram above) is the largest possible.           

This is a variation of Regiomontanus’ Angle
maximisation problem, in which the question is asked: where is best position to
stand in order to best view a painting which is pinned above the viewer’s head?
In other words, where to stand in order to maximise the angle at which the
painting is viewed.

(TAKE PICTURE OF RUGBY FIELD OR SCREENSHOT FROM VIDEO)

Applying this is rugby, we can see that the same
concept applies – where does the kicker stand in order to maximise their
kicking angle? By doing so, they will ensure that they have the highest chance
of kicking it through the uprights and scoring points, as they then have the
largest target to aim at.

However, there are many factors in the rugby that
affect how and where the shot is taken, and hence a few assumptions that must
be made in order to properly assess this situation and mathematically model it.

Firstly, kicking strength, range and accuracy vary
between each individual kicker, as well as their technique and experience. All
of this will be assumed to be constant, and the kicker will be able to kick the
ball along the same flight path, with the same strength and accuracy, every
single time.

Secondly, the ball is assumed to be a particle, and
hence variations in optimal position due to its size, shape or mass will be
nullified.

Thirdly, the flight path of the Rugby ball during
flight is usually curved, but this will affect the optimal position and
distance at which the kick is taken at, and hence the flight path will be
assumed to be a straight line.

At the same time, the movement of the ball during
flight will be assumed to be straight, and no spin will be applied to the ball.

Lastly, weather conditions such as wind speed and
direction will be assumed to be negligible.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Figure 2: Official World Rugby plan
of a standard rugby field.

To obtain certain values
for this IA, some context is required on the game of Rugby.

The Rugby field is approximately around 144m x 70m
field.

The try line is the width of the field, and is around
70m.

 

 

 

 

 

 

 

 

 

Figure 3: Official World Rugby Plan
of standard rugby uprights

The dimensions of the Rugby uprights are depicted in
the figure above. The distance to the crossbar, from the floor is 3m, while the
distance between both posts is 5.6m.

The height of the two uprights is a minimum of 3.4m.

In order to “convert” the try, the kick must go in
between the two uprights, and above the crossbar

 

Figure
5: Geometric diagram of a conversion kick

Where,

F=
Far post

G=
Near post

T=
Point where try line meets conversion line

C=
Point where conversion is taken

d= Distance between posts    

?=
Viewing angle of kicker

In the diagram, we can see that the angle between both
posts increases with distance from T, the point where the try line meets the
conversion line.

The further away the kicker takes the kick, the larger
the angle that is presented to him. However, he must be able to kick the ball a
significantly longer distance.

Conversely, if the kicker were to take the kick closer
to point T, he would only need to kick the ball a shorter distance, but would
have to keep it within a much smaller angle.

From a few first-hand sources, many kickers say that
their prime angle of kicking would be around 30 to 35 degrees. This is purely
from intuition and practice.

One way of approaching the problem of finding the best
position to take the kick would be the geometric method. The above diagram
depicts the conversion kick in a geometric diagram. In order to relate the
length CT and the angle ?, we can draw a circle through points F, G and C,
which brings us to the diagram below.

Figure
6: Diagram to show the application of the angles in same segment theorem to the
problem

Where,

F= Far post

G= Near post

T= Point where try line meets conversion line

C= Point where conversion is taken

D= Centre of circle

d= distance between posts
    

?= Viewing angle of kicker/Angle in same segment

If the kicker were to take the kick
anywhere else along the circle, the angle would remain the same, as angles
subtended in the same arc are the same. Outside of this circle, any other
angles will be smaller than the angle created at C. Therefore, point C is the
optimum position to take the conversion kick. However, by this diagram alone,
we cannot find the length GT and the angle ? as we do not have enough
information about the diagram.

Figure
7: Diagram that has the optimum position indicated with the addition of the
circle

By adding a circle to the diagram, we ensure that the
optimum angle is obtained, and that the optimum position is maintained. In
addition, by adding a line parallel to CT, DE, we can see a method emerging to
find the length y.

 

Further developing the diagram to be able to find
distance y, we connect the points and
DG and DF, forming right-angled triangles DFE and DGE.

Where,

A= Far post

B= Near post

C= Point where conversion is taken

E= Midpoint of the two posts

T= Point where try line meets conversion line

d= distance between the two posts

x= distance between
midpoint of the posts and the conversion line

y= distance from try line
to the conversion point

Figure
8: A further developed geometric diagram that depicts the relationship between x, y and d.

From the figure above, we can use Pythagoras’ theorem
to relate x,y and d:

By rearranging the equation to make y the subject, we end up with:

Hence, with this equation we can now input different
values for the distance between the midpoint and the try line, x, to find what is the optimum distance
from the try line to the conversion point, y,
to take the kick at.

From the figure 2, the standard length of the goal
line is around 68- 70m. For this exploration we shall assume the length is 70m.
The distance between the goal posts is 5.6m.

Therefore, . By dividing the goal line into 2
halves, there is a total of 35m of goal line where it is possible for the try
to be scored and a conversion taken.

Sample calculation:

Distance between midpoint of the posts and the conversion
line (x)/m

Distance from try line to the conversion point (y)/m

5.00

4.14

10.00

9.60

15.00

14.74

20.00

19.80

25.00

24.84

30.00

29.87

35.00

34.89

This is illustrated in the table below:

Table 1: Processed data for the equation

Graphing out the equation, we can
model the best distance to take the kick at from every possible position on the
goal line. This is shown in the figure below.

 Figure
9: Graphical representation of the optimal position to take the conversion kick
for each position on the goal line

 

 

Now that the optimal distance is found, all that is
left is to find the angle at each position.

This can be done with a few
adjustments to figure 8.

 

 

 

 

 

 

 

 

 

Figure
10: Modified figure 8, which includes angle ?

Angle AOB is equal to 2? as angles at the centre of a
circle are twice the angle at the circumference of the circle. Since OE the
bisector of triangle AOB, angle AOE is equals to ?.

Therefore, to find ?, we can express it in terms of
sin:

To find ?, we take the inverse of this function:

Inserting the values, we obtained in Table 1, we can
obtain the value of ?.

For a distance of 10.00m between the midpoint of the posts
and conversion line, the optimum distance, x,
is 9.60m.

Thus, the value of ?, at x=10.00m, is 16.96°; the
kicker has maximum viewing angle of 16.96° at the optimum position 9.60m in
front of the goal line when the try is scored 10.00m from the midpoint of the
posts.

The rest of the angles are shown in the table below.

Distance between midpoint of the posts and the conversion line (x)/m

Distance from try line to the conversion point (y)/m

Maximum viewing angle (?)/°

5.00

4.14

42.53

10.00

9.60

16.96

15.00

14.74

10.95

20.00

19.80

8.13

25.00

24.84

6.47

30.00

29.87

5.38

Table
2: Processed data for maximum viewing angle

 

 

 

Conclusion:

At the end of this mathematics exploration, I have
been able to find the mathematically optimum position to take a conversion kick
in Rugby, and the maximum viewing angle for each position. However, these
results are not conclusive, as many factors were taken out of consideration in
order to make the experiment feasible. For example, one important consideration
that occurs in actual Rugby would be the spin of the ball. Many professional
kickers are able to put a spin on the ball whilst taking a kick, affecting its
flight path and allowing them to take sharper angles as compared to a
straight-line flight path, which is assumed here.

In addition to this, this exploration categorises
“optimum position” as the position that gives the kicker the maximum viewing angle.
This may not be true in some situations. An example would be the fact whilst
taking the kick, the opposing team is allowed to advance towards the kicker’s
team during the run-up to the kick. Hence, for the value of y, in an actual Rugby match, it may be a
better decision to take the kick further behind, even though it provides a
smaller viewing angle, as at least there will be no interference by the
opposing team.

Through this mathematical exploration, I have learned
about the application of mathematics in real life. Using geometry, we can find
solutions to real life problems and questions. Most of the time, we feel that
mathematics is not applicable, but this is a clear example that it is.