Download Mathematical Billiards

Emily Miller
Capstone Thesis
2014
Mathematical Billiards
Introduction
Mathematics is a very broad subject. It can be related to just about anything in the world
today. It is found in nature, the Fibonacci sequence, in stores, with sales and prices, and even in
sports and games. Billiards is one specific game that has been studied by mathematicians for
years. One type of billiards game, mainly known as pool, is played on a rectangular table with 6
pockets, 16 balls and a cue stick. The object of this game is to knock every ball into a pocket.
This is done by using the cue stick to hit the cue ball at other balls to reflect them into the
pockets. The concept of knocking balls into pockets, reflections of the balls and the paths that the
billiard balls take are all studied thoroughly in mathematics.
“Billiards is not a single mathematical theory… it is rather a mathematicians playground
where various methods and approaches are tested and honed.”
-Serge Tabachnikov
There are many big ideas in mathematics that are tested in billiards. This past year I have
focused on algebra, trigonometry, geometry, and calculus, with each idea taking a different
approach to the study of billiards. When studying billiards mathematically, it is important to
know certain properties of each topic. Properties such as; sine, cosine, tangent, angles of a
triangle, intersecting lines, normal lines, tangent lines, and properties of different shapes can all
be applied to mathematical billiards. The main focus of mathematical billiards is on the
reflection of the ball off the wall, or side of table, and the different paths the billiard ball creates
after these reflections. We look at the effect of different shapes on these reflections as well as
different types of billiards games using precise angles and reflections.
Rectangular Billiards
An important concept in mathematical billiards is the fact that the angle of incidence is
equal to the angle of reflection. This concept is helpful because it can be used to prove theorems
and understand paths of billiard balls. Mathematician’s focus on the bounces of a billiard ball
and the concept that the angle of incidence equals the angle of reflection will give them the
knowledge of the angle it bounces off at. Say we are given the following problem:
Determine where a ball should strike the long side of a table so that a two-cushion bank
shot will put the ball on a path that ends at the upper left-hand corner.
Keeping in mind that the angle of incidence equals the angle of reflection and the properties of
sine, cosine, and tangent, this problem can easily be solved. We are given that the ball is 2 ft.
from the top and the table is 5 ft. wide. We can draw a line bisecting the second bounce of the
ball. This will create two right triangles (the right angle being made with the left side of the
table) (see Figure 1). We know that the orange triangle and the green triangle share the same
angle from the first bounce. Since these triangles have two of the same angles, the right angle
and the angle of incidence/reflection, the third angles must be equal because all the angles of a
triangle must add up to 180, according to the properties of a triangle. These two triangles are
proven similar by AAA (angle, angle, angle), a triangular property. We know the larger triangle
is 5 ft. across and the smaller triangle has a base of 2.5 ft. because the ball starts in the middle of
the table (width wise). With this, we can say that the larger triangle is double the smaller triangle
because one side of the smaller triangle is half the size of the corresponding side of the larger
triangle and the angles are all the same. The base of the larger triangle (left side on the table) is
double the base of the smaller triangle, , giving us
. The third triangle also has a base of
because of the bisecting line between the two larger triangles. We then get the equation
, therefore
added together;
side of the table
. With x, we can solve for the bases of the two larger triangles
. This gives us our answer that the ball must strike the long
from the top of the table.
If we want to continue, we can use trigonometry to find the angle in which the ball should
be shot at in order to hit this spot on the table. With the other two angles of the smaller triangle
and the equation for tangent, we find the third angle;
gives us the angle to shoot the ball at so that it will strike the table
radians. This
from the top.
Knowledge of trigonometry and algebra can solve problems such as this one along with others in
mathematical billiards.
Figure 1: A ball bounces off 2 of the
cushions and hits the top left corner.
Unlike the popular game of billiards known as pool, most studies in mathematical
billiards are interested in the path of the ball rather than the pocket it goes in. This is because of
the variety of mathematical techniques and methods that can be studied while looking at the
billiard path. Say we have a rectangle with the cue ball in the center of the table (width wise) and
2 ft. from the bottom (see Figure 2). We want to know the path this cue ball will take when we
shoot it at a specific angle, say 45 degrees. Rather than measuring out each angle and figuring
out where the ball will hit on the table, we can use the unfolding method. This method starts with
the original rectangle and then adds another rectangle of the same size on the right side of that
and another one above the second. In Figure 2, it shows three rectangles added to the original
rectangle. We can then draw a straight line from the ball in the original rectangle through the
other rectangles at the given angle of 45 degrees (see Figure 2). This pattern of “unfolding” right
and up is continued until the straight line reaches a corner (in this case, Corner A). When the ball
reaches the given point we can reflect each line over the x-axis or y-axis, respectively, to project
it onto the original rectangle. When put together, we will see the path of the ball when shot at 45
degrees (see Figure 3).
Corner A
Corner A
Figure 2: The path of a billiard ball
using unfolded tables. *not drawn to
scale
Figure 3: The path of a billiard ball
when the tables are folded back up.
*not drawn to scale
Elliptical Billiards
The mathematical study of billiards continues far beyond a rectangular table. There are
many methods and theorems that have been defined and proven for billiards on non-rectangular
tables. One of the shapes I studied was an ellipse. When studying elliptical billiards the key
interest is again the path of a billiard ball. In an ellipse there are certain patterns the path of a
billiard ball can have. The path is described by a sequence of points; P0, P1, P2, P3,... Pn, and the
line segments between these points. In order to study the path of the billiard ball, the starting
point, P0, must always be on the edge of the ellipse. The following points, P1, P2, P3,... Pn, are the
positions where the ball is reflected off the edge of the ellipse. A path is an n-cycle depending
on the number of times it hits the edge and returns back to the starting position. But just how
many times does a ball have to bounce before it returns back to the starting position? Can a
billiard ball return back to the starting position in an ellipse? How many n-cycles are there in an
elliptical billiard table? These questions along with others are answered through theorems and
their proofs.
It is important to remember that the angle of incidence equals the angle of reflection. In a
rectangular table this angle is made with the side of the table. What about in an ellipse? The
angle is measured with the tangent line to the ellipse at that point. Looking at Figure 4, we can
see the angle made by line segment P0P1 is equal to the angle made by line segment P1P2. Also,
drawn in this figure is the normal line at P1. The normal line is perpendicular to the tangent line.
This line bisects the angle made by these two line segments according to the properties of a
normal line.
P0
P2
P1
Figure 4: The path of billiard ball, P0, P1, P2.
The angle formed with the tangent line.
The normal line at P1.
Theorem 1
Let F0 and F1 denote the foci of an ellipse and F0F1 the line segment between the foci.
(1) If a path intersects F0F1 once but not F0 or F1, every line segment in the path is tangent
to one of the two branches of a common hyperbola. The foci of the hyperbola are the foci
of the original ellipse
(2) If one line segment in a path does not intersect F0F1, every line segment in the path is
tangent to a common ellipse. The foci of the ellipse are the foci of the original ellipse.
Part one of this theorem is stating that if a billiard ball is shot in between the foci, every
line segment in that path will also pass in between the foci. Not only will the reflection intersect
the line segment between the foci, F0F1, but each segment on the path will additionally be
tangent to a point on one of the two sides of a hyperbola. The hyperbola shares the same foci as
the original ellipse (see Figures 5-7).
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Figure 5: The first shot through the line
0 1
segment between the foci, F F
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Figure 6: The path through the line
0 1
segment between the foci, F F , after the
second reflection.
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Figure 7: The hyperbola formed by a path
that intersects the line segment between
the foci, F0F1. This is the projection at 50
reflections.
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On the other hand, the second part of this theorem states that if a ball is shot outside of
the line segment, F0F1, each following line segment in the path will also be outside of F0F1. This
will create a path that circles around the ellipse. The second part of this theorem also states that
each of the line segments on the path will be tangent to a smaller ellipse. This smaller ellipse
shares the same foci as the original ellipse (see Figures 7-9).
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Figure 7: The first shot outside of the
line, F0F1
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Figure 1: The second reflection outside of
0 1
the segment, F F .
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Figure 2: The ellipse formed by a billiard
0 1
ball path outside of the foci, F F , after
50 reflections.
Theorem 2
The only 2-cycles in an ellipse are along its axes.
Say we have
and
at the center of the ellipse. We have the path between
as the horizontal axis and
starts at –
to
as the vertical axis. It is clear that if a ball
and it is shot directly across at a 90 degree angle, it will reflect at
return to the starting point at the same angle (the angle of incidence equals the angle of
reflection). Similarly if a ball starts at
and is shot directly down it will reflect off
and
and return to the starting point as well (Figure 11). These are the only two 2-cycles in an ellipse
because there are no other points on an ellipse where the ball will bounce off at a 90 degree angle
due to the curvature of an ellipse.
The points on the ellipse that are on the axes are the only ones that have parallel tangent
lines. We can prove that these points have parallel tangent lines using calculus and implicit
differentiation. The equation for an ellipse is
, where (x,y) is a point on the ellipse
and a is the x-radius along the x axis while b is the y-radius along the y-axis. Using implicit
differentiation on this equation, we can find the tangent lines at points (-x, 0), (x, 0), (0, y), and
(0, -y). Say we have a=5 as the x-radius and b=2 as the y-radius. We first have to use
the derivative of the equation for an ellipse;
. We want to get
equation
equation we get
alone so we subtract
to take
. This gives us the equation
and divide by
giving us the
to use to find the tangent line. When we plug (x,0), a=5 and b=2 into the
. A tangent line of DNE (Does Not Exist) tells us that it
is a vertical tangent line. If we do the same thing for (-x,0), we get
.
This also gives us a tangent line of DNE, therefore these two points, (x,0) and (-x,0), have
parallel tangent lines. We use the same process for (0,y) and (0,-y). When we plug the point (0,y)
in we get
. Similarly, (0, -y) has a tangent line of
.A
tangent line equal to zero tells us that the line is horizontal. Since both of the derivatives are
equal to zero, (0,y) and (0,-y) have parallel tangent lines.
(0,y)
(-x,0)
(x,0)
(0,-y)
Figure 11: The two 2cycle paths in an ellipse
are on the two axes.
Theorem 3
Suppose P0 and P1 are points on a path P0, P1, P2, P3,…Pn in an ellipse and a focus point
lies on the line segment P0P1 (see Figure 12). Then the points P0, P2, P4,… converge to
one vertex and the points P1, P3, P5,… converge to the other vertex. The vertices on an
ellipse are the left most point and the right most point such as (-x,0) and (x,0) shown in
the Figure 11.
This theorem is saying that if the first shot passes through one of the foci, then each
following point of even index is going to get closer on the ellipse to one vertex and each
following point of odd index is going to get closer on the ellipse to the other vertex. Since each
line segment in the path must pass through one of the foci, the ball will continue to bounce back
and forth over the y-axis so P0, P2, P4,… will be on one side of the y-axis(left or right) and P1, P3,
P5,… will be on the opposite side of the y-axis (Figure 12).
Say we looked at vertex,
be the magnitude of the angle
, which is closest to focus point,
(See Figure 12). Let
, created by the focus point, vertex and point. The
sequence
is decreasing due to
being on the side of the triangle,
, which is
extended beyond the triangle. This sequence is bonded by 0; therefore, it converges to a limit A.
The map
is continuous because it is the result of two reflections and A is considered
a fixed point of the two reflections. The first one being from
and the second from
. Since zero is the only angle that is fixed, A must be equal to zero and
converges towards
while
converges towards
. We can demonstrate this using the
ellipse program on Wolfram alpha. If we manipulate the starting point and the angle of the first
shot we can create an image of a billiard path through the foci. Figure 12 shows a billiard path
with 10 reflections. The first line segment, P0P1, passes through the foci, F0. From the image, you
can tell that P1, P3 and P5 are all converging towards the left vertex,
converging towards the right vertex,
.
while P0, P2 and P4 are
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F0
P3
0.0
P2
F1
P0
PP4
0
PP5
0
P1
P0
P0
P1
P0
P0
P
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1
P
P0
0
P0
1.0
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Figure 12: A shot through the foci will
cause the remaining points of a path to
converge towards a vertex. This shows a
billiard path after 10 reflections through
the foci.
Theorem 4
If n is odd, there are no n-cycles passing through F0F1.
Remember that in order to be n-cyclic the path must return to the starting position. So, if
P0P1 passes through F0F1 then P0,P2,P4,… are on one side of the ellipse and P1,P3,P5… are on the
other side. Assuming that the ellipse is cut in half through the foci (along the x-axis), the arc of
the top half is one side of the ellipse and the arc of the bottom half is the other side of the ellipse.
Since all odd n’s are on the opposite side of the starting point, P0, there is no possible way that an
“odd” path can return to the starting point. Therefore, if n is odd there are no n-cycles when P0P1
passes through F0F1. If the path is an n-cycle through F0F1, n must be even.
Circular Billiards
Similar to an ellipse, another geometric shape that has been studied in mathematical
billiards is a circle. The properties of a circle, such as a circle has rotational symmetry, can be
helpful when looking at circular billiard trajectories. Since the circle has rotational symmetry and
the rule that the angle of incidence equals the angle of reflection still applies, the angle remains
the same after each reflection in the circle. Unlike an ellipse, when rotated the circle will look the
same because the curvature at each point is the same all the way around. In an ellipse, however,
the arc angle, or curvature of the ellipse, is different at each point so each spot creates a different
angle for the ball to bounce at. This property of a circle creates patterns in the paths of billiard
balls.
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Figure 13: 5-cycle billiard trajectory in a
circle when p=2
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Figure 14: 5-cycle billiard trajectory in a
circle when p=1
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When looking at a circle, we let
billiard ball is n-periodic if
each integer
where
be the angle of reflection in a path. The path of a
, where is a reduced fraction. If
is a positive integer, for
is relatively prime to , there is a distinct n-periodic orbit. There are
two 5-cycles in a circle, the pentagon when
(Figure 14) and the pentagram when
(Figure 13). This means there are two ways to shoot a billiard ball and have it return back to P0
in four reflections. Also, there are infinitely many 2-cycles in a circle. If a ball is shot directly
through the center of a circle it will return back to the starting point after one bounce. Since a
circle has rotational symmetry, this is true for every point on a circle. If
return back to the starting point creating an n-periodic orbit because
is an integer, it will
is rational. Some numbers,
like 5, can create multiple n-periodic orbits (shown above in Figures 13 and 14). We know that if
is an integer it will return back to the starting point creating an n-periodic orbit but what if
is
irrational, specifically, π-irrational? π-irrational means that the number cannot be written in any
form of, or as a multiple of, π.
denotes the circle rotation through angle θ.
Theorem 5
If θ is π–irrational (cannot be written as a rational multiple of π) then Tθ-orbit of every
point is dense on the circle. In other words, every interval contains points of this orbit.
This theorem is saying that if we start at point, P0, and traverse the circle making steps of
length θ, we will eventually return back to a point arbitrarily close to the starting point. A step of
length θ means that the next point on the circle will fall at an arc length θ away from the previous
point. Since θ is π–irrational, we will never return to the exact starting point on a circle but the
path will hit a point very close to it. When the theorem says the orbit of every point is dense, it is
referring to the periodic orbit the ball takes. An orbit is periodic if it repeats the shape of the
path. The points are dense because for every point, x, there is another point that is arbitrarily
close to that x. When the path continues around the circle in the orbit, for a large number of
iterations, the circle becomes thick with points. We then get a path that is a sequence of rotations
through π-irrational angles. The path passes through every interval in the circle. The ball will hit
an infinite number of points in each interval when θ is π-irrational (Figure 15). This is because it
never hits the same point twice but it will hit a point that is extremely close to another one, just
off by a hair.
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Figure 15: 200 π-irrational iterations of
Tθ.
Similar to the ellipse, each line segment in a billiard ball’s path in a circle is tangent to a
concentric circle, assuming that the first shot is not directly through the circle’s center. This is
because the angles of the bounces will be the same around the whole circle. Therefore, the path
will continue to go around the center at a specific angle. The radius of this concentric circle
depends on the angle α that the line segments makes with the unit circle. The equation for this
radius is
. Looking at Figure 15, you can see the concentric circle that is made by this
specific angle α.
Triangular Billiards
We can also look at mathematical billiards in polygons, such as the triangle. There are
multiple types of triangles, such as isosceles, acute, and right. Each triangle holds different
properties. These properties result in specific patterns in the path of a billiard ball. It is helpful to
have prior knowledge of triangles and their properties in order to understand proofs of the
billiard trajectories in a triangle. Figure 16 shows a billiard trajectory in an acute triangle.
Lemma 1
The triangle connecting the base points of the three altitudes is a 3-periodic billiard
trajectory.
It is a 3-periodic billiard trajectory because of the following: The quadrilateral BPOR has
two right angles, ˂BRO and ˂BPO. This quadrilateral can be inscribed into a circle shown in
Figure 17. The angles APR and ABQ are supported by the same arc on this circle, meaning that
the arms of these angles intercept the circle at the same two points. Therefore, these angles are
equal. The two points and the arc are highlighted in Figure 17. The angles APQ and ACR are
equal and ˂ABQ is equal to ˂ACR. Both angle ABQ and ACR are complements of the angle
BAC to π/2.
Figure 16: Billiard trajectory in an
acute triangle *not drawn to scale
Figure 17: Quadrilateral BPOR inscribed in
a circle. *not drawn to scale
Theorem 6
In a right triangle, almost every billiard trajectory that starts at a side of the right angle in
the perpendicular direction returns to this side in the same direction.
This theorem is saying that the starting position of a billiard trajectory, P0, is on one of
the sides that forms the right angle. When the ball is shot in a perpendicular direction from this
side it will reflect off the other sides and return to the starting position at a 90 degree angle
(perpendicular). This is because when it is shot, it bounces off the hypotenuse at the same angle
as it hits it. This causes the ball to then approach the third side (the other side that forms the right
angle) at a 90 degree angle (see Figure 18). Again, this bounces off and follows the same path it
took back to the starting position. This can be visualized through a similar unfolding method as
mentioned above in rectangular billiards (see Figure 19). This theorem may not work if the ball
starts directly in the right angle corner of the triangle and is shot straight so that is bounces
against another corner. This may cause the ball to go a different direction because it will hit two
sides of the triangle at the same time.
Figure 19: The unfolding of a right
triangle to show the billiard ball path.
Figure 18: A perpendicular trajectory in a
right triangle starting at one of the sides
forming the right angle
Carom Billiards
Carom billiards is a specific game of billiards played between two people. This game is
played on a 5x10ft pocketless table with three balls. Rather than knock balls into the pockets
with the cue ball, the object of this game is to use angles to knock into the other two balls on the
table with the cue ball in one shot. The use of the cushions or sides of the table to reflect the cue
ball is an important concept of this game. This is because on most shots, the cue ball will have to
bounce off of at least one cushion in order to hit both balls. Figure 20 shows a randomly set up
carom billiards game. If the player hits the ball correctly to hit the other two balls, the player
scores a point and gets to go again until they miss. An average for each player can be calculated
at the end of the game by dividing the number of points by the number of innings. An inning,
similar to baseball, is the division of the game in which both players alternate on the table to take
their turn. A player’s turn is over when they fail to hit both object balls with the cue ball. One
way to look at the success of a player in carom billiards is to assume a Bernoulli process. The
Bernoulli process can be defined by
λ is the success rate, for all
, where
is the probability to score n points, and
However, this process does not account for position play in
carom billiards; therefore, we use the Markov process. Position play is when a player has an easy
shot and shoots the cue ball so that it is in a favorable position for the next shot. This can create a
series of easy shots and long runs for the player.
Cue stick
Cue ball
Figure 20: A carom billiards table with a cue ball and two other balls
In order to account for the correlation between position play and probability to score n
points, one can use the Markov process. This process considers there to be
possible positions
of the balls on the table. The likelihood of scoring and the position the balls will be depends on
the position of the cue ball and the other two balls at the beginning of the shot.
In carom billiards, one can also calculate the probability of scoring at least n points. Say
we have an N-dimensional vector,
. This vector has components i and n, where i is the
probability of having the balls in a position of type i and n is the type of shot. These types of
shots cannot be random because then we could use the Bernoulli process. Shots of type 1 are
generally followed by another shot of type 1. A shot of type 1 is considered a difficult shot,
therefore this is a difficult shot followed by another difficult shot and are less favorable to the
player. Shots of type 2 are considered easy shots and are generally followed by other shots of
type 2. For all
,
follows
, where K is an NxN matrix with component
being the probability to score from a type j shot followed by a type i shot. More clearly, this is
the probability of a player to score after shooting a shot of type j then shooting a shot type i. Let
the eigenvalues of K be
divided by K to get
where
and
are its eigenvectors.
can be
alone and find the probability. We then get the equation written as follows
with N=2:
This is found because the matrix K is diagonalizable meaning that the equation can be
written as
. A matrix is diagonalizable if an invertible matrix K exists, such that
is a diagonal matrix. A diagonal matrix means that there are numbers along the diagonal
from the top left to the bottom right of a matrix but not in any other spot. There are zeros in the
other spots of the matrix. An example of this is
points is equal to the sum of components of
and
.
, the equation for
. The probability to score at least n
. Given that Y is a scalar known as a function of
is shown below for all
as a linear combination of
and
Then the players average can be found by calculating the sum over n of
, given by the
equation of m.
We can find the equation for Y in terms of m,
into the equation for probability given by
. Then the probability of scoring at least n points is
dependent on the two unknown parameters,
After plugging Y back into
(
(given below) and plug that back
.
we get the following equation:
)
Thus, we have presented a model that accounts for position play, using the Markov
process. We can use this process to find the correlation between the score and the difficulty of
the previous shot as well as find the probability to score at least n points using two geometric
sequences. Similar to the sports of track and swimming, players in carom billiards are ranked and
matched up to compete against one another using these statistics. This will create interesting
games played by people of similar ranks because the players will be similar in their strength of
game play. Some carom billiards games last multiple days because the players are so similar in
their game play. These players can continue to shoot the ball in using type 1 or type 2 shots for
hours.
Conclusion
There are many big ideas found in mathematical billiards. Some of which include
algebra, trigonometry, geometry and calculus. These topics are first learned about in early
schooling. The properties and rules that go along with each topic are used when studying
billiards mathematically. The main focus of mathematical billiards is on the path of a billiard
ball. We first looked at this in a rectangular table. We can apply trigonometry concepts to the
shape in order to find the angles that create specific paths. We then focussed on elliptical
billiards. There are very interesting patterns that are formed by the path of a billiard ball in an
ellipse. Two of which were shown through Theorem 1. It is facinating to see how a hyperbola
and an ellipse are formed just by changing the angles and starting points of a billiard trajectory.
Using Wolfram Mathematica, we can see a great visual of the reflections in an ellipse and in a
circle. Since the circle has rotational symmetry, every angle in a billiard path is the same. This
creates interesting patterns in circular billiards. Other than the popular game of pool that most
people have seen and played, there are other games that have similar concepts such as carom
billiards. Carom billiards is played with a cue ball and two other balls on a table without pockets.
The object of the game is to hit the cue ball so it knocks into both of the other balls. The
probabilty of taking multiple shots in a row can be calculated using mathematics. The correlation
between the type of shot and the number of shots taken in a row can also be found with
mathematics using diagnolization from linear algebra. There are multiple concepts in billiards
that can be studied in depth using a variety of mathematics including algebra, geometry and
trigonometry. The concepts that can be studied are countless, some of which include: angles of
shots, the path of billiard balls in different shapes such as circles and triangles, and different
types of game play of billiards like carom billiards.