THE GEOMETRY OF TENNIS
Swanberg is a former collegiate player at Livingston
University and each month he will give us a new tennis
Okay, in this second
installment of "Bird's Tennis Instruction,"
we will be discussing one of the most important facets
of tennis tactics. It is a system by where you
will know where to hit the ball and also where the ball
will be hit.
Okay, okay, I know. No system can really give
you all that. But this is important stuff! This
is where it's at, people. So, without further ado,
I give you:
The Geometry of Tennis
Back in the early 80s, when I was but a lad, playing
tennis nearly every day, entering upwards of 30 tournaments
a year, my mother decided to move me to a different
club. The one I had been taking lessons at was
pretty far away, and there was another facility much
At the time, the teaching pro at this club was a man
by the name of Bob McKenna. I took one lesson from
Bob, and then he went to New York to open a fried chicken
store. But before he left, he gave me the greatest
hour of instruction I've ever had. This is part
one of a three-part instructional series on the geometry
of this game we call badminton
er, I mean, tennis.
Wherever you are, Bob, thank you.
Every tennis court has the same dimensions. It's
a rectangle, 78 feet long, 27 feet wide (36 feet if
you include the doubles alleys). Your singles court
consists of three boxes, the deuce service box, the
ad service box, and the backcourt, or "no-person's
land." It used to be called no-mans
land but Congress changed its name in 1987.
These three boxes, in their entirety, comprise the
sum of all possible shots your opponent can hit, while
still remaining in the point. Therefore, these
are the shots that you are responsible for getting to
The service boxes are 21 feet long, leaving 18 feet
for the backcourt. The "T" is the spot
where the dividing lines between the three boxes meet. The
precise center of your singles court, therefore, is
18 inches from the T along the center service line toward
But Ill tell you what. This isn't school
geometry. There won't be a test at the end of this. So,
let's make things easy. Let's just say that the
center of the court is the "T". I suppose
a case could be made that the 2 feet along the net aren't
realistically playable anyway, so the "T"
is actually a very good representation of the center
of your singles court, or "The Center of All Possible
Shots Your Opponent Can Hit."
The "Average" Shot
Since the center of all the possible shots your opponent
can hit is the "T", it stands to reason that
we could call this the "average" shot. If
your opponent simply hit random shots for the entire
match, and then we looked at where all those shots landed,
and then took the geometric center of all of them
bingo! The "T".
Going forward, since your opponents shots are traveling
the distance from his racquet to your court, it seems
logical that the ball will continue to travel in that
same direction (by the way, for you smarties out there,
we are dumbing this down a lot
we're not talking
spins or wind or earth's rotation or gravimetric shear
or burps in the space-time continuum
Newtonian stuff here, you Relativists!). This gives
us an arrow, or a ray, as my geometry teacher called
it. The origin of the ray is your opponent's racquet
and the direction of the ray is the ball's direction
Where is all this leading? you may be asking
yourself. Well, it's all very simple. Imagine
that your opponent is standing slightly behind the middle
of his baseline. He is hitting balls your direction. The
average ball will bounce on the "T" and continue
onward past the center of your own baseline. So,
where would you stand, knowing that this is the case?
Exactly! The center of your own baseline. Give
yourself 10 points if you got that right.
Okay, so not every shot lands on the "T". In
fact, your opponent is probably going to avoid that
shot as much as his abilities will allow. But where,
then, will he hit the ball?
Well, let's go back to the bit about random shots and
all the places they landed. Let's stop looking
inwardly (toward the norm) and start looking outwardly
(toward the extreme). What are the most outlandish
shots your opponent can hit?
Well, he can hit those spots on your singles sidelines
about 2 feet from the net (remember, we said that the
2 feet along the net is a near-impossible shot to hit). This
takes the ball on a path 18.2 degrees off the line that
leads straight toward you. But it can go to your
forehand side or your backhand side.
Bisection of Angles
No, this section is not about alternative lifestyles
in the heavenly realm. Angles, not angels.
So, given this 18.2 degrees to either side, where do
Exactly, right in the same spot: the middle of your
baseline. What you've done here is bisected the
angle. Essentially, you are telling yourself, "he
can go here, or he can go there, or he can go anywhere
in between, so I am going to stand in the middle of
all those possibilities."
Back to Reality
Okay, now we have to interject a little reality into
all this. Your average player -- heck, the top
pros in the world -- cannot hit the point on your singles
sideline 2 feet from the net. And even those that
can hit there are going to be smart enough to know that
they don't have to go for that much to win a tennis
So, for the sake of keeping things somewhat real, we
are going to say that any shot that approaches your
sidelines is most likely going to be farther from the
net. See the illustration. The red areas are
realistically possible shots. The yellow areas
do not need to be bothered with.
"Why are we ignoring those shots, Bird?"
I hear you asking. "Won't my opponent just
go for those shots knowing I am not covering them?"
To put it bluntly, if you are playing against someone
who can hit those shots with any consistency, then your
best strategy is to put your racquets in your bag and
leave the court, because Mr. Federer is going to crush
What Does This Mean?
Okay, I think you are getting the gist of all of this. The
point is that the best place to be when looking to get
to your opponents shot is to bisect the possible
angles that he can hit. When your opponent is in
the center of his backcourt, you need to be in the center
of your backcourt.
But what if he is out wide? What then? How
do you know how to bisect the angles?
Well, it goes back to our center of the court
ideal. Remember the T? If you
draw an imaginary line from your opponents point-of-contact
through the T, that is a great approximation
of the bisection. So, where should you be? Answer:
on that imaginary line!
Lets look at an example. Say, for arguments
sake, that your opponent has been pulled wide to his
(right-handed) forehand. Where would you stand
to maximize your chances of getting to his next shot?
Well, he is pulled wide to his right. Draw a line
from his point-of-contact through the T
on your side of the court and you find that you should
stand a few feet to the right of the center mark (the
small mark protruding toward the net from the center
of the baseline).
Take a look at the illustration here. The angle
shows your opponents likely possible shots. Note
where the bisection is. And as you can see, your
opponents best down-the-line shot (to your backhand)
doesnt take you off the court at all. However,
his best crosscourt shot takes you far off the court
on your forehand side (assuming you are right-handed
like your opponent).
I am sure many of you have heard your coach or instructor
or perhaps a more-experienced player say, when
at net, you have to cover the down-the-line. This
is a direct result of that. If you are between
your T and your opponents shot, then
it stands to reason that you will be on the same side
of the center service line as your opponent.
The intuitive reason for this is simple and pretty
obvious. Your opponents best shots from his
forehand corner cross the net, and your position, at
points to the left of the center. If he tries to
pass you down-the-line, his shot may well still be out
over the doubles alley as it crosses the net. But
if he tries to pass you cross-court, the ball will cross
the net in the interior of the court, much closer to
the center of your court than the down-the-line possibility. So,
you need to cover that down-the-line when at the net,
and this is the geometrical explanation as to why.
Okay, this has shown you where you need to be in response
to your opponents possible shots and also his
position on the court. But there is more
oh so much more.
Note that where your opponent is on the court is a
direct result of where you chose to hit your shot. The
next two installments of The Geometry of Tennis
are going to explore this notion. First, we will
look at what shots to choose, when, and why. We
will discuss defensive tactics in the next lesson. And
then we will take it to the next level and discuss offensive
As for what you should do from here
level shouldnt matter much. Beginners and
advanced players alike should be aware of their own
center of the court and use it to position themselves
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