


Chapter
1: Dimension two
Hipparchus
explains how two numbers can describe the position of a point on a
sphere.
He then explains stereographic projection: how can one draw a picture
of the Earth on a piece of paper?



Hipparchus
is the first hero of our story, but we must not take what he says too
literally! He claims to have been the inventor of both geography and
astronomy. This seems a bit exaggerated; who could really lay claim to
this? Haven't travelers always described their voyages, and shepherds
always gazed at the stars? It is extremely rare that a single
individual invents an entire science. Nevertheless, let us pay homage
to Hipparchus as one of the greatest sages of antiquity.
We know very little
about the life of Hipparchus. He was born about 190 B.C. and died about
120 B.C. This article
contains a
short description and this site
a more extensive
biography. However, there is no doubt that our sage was one of the very
first to produce a catalog of the stars and to measure their positions
in the celestial sphere, in fact with astonishing precision.
Astronomers have honored Hipparchus by naming one of the craters of the
moon after him. Hergé sent Tintin to
this crater in Explorers on the Moon, noting that "le
cirque d'Hipparque n'a pas besoin de clowns, donc vous ne pouvez pas
faire l'affaire..." (For the context of this remark, read Explorers on the
Moon!)




The second role in
this chapter is played by Ptolemy, who lived three
centuries later, between 85 and 135 A.D. Widely recognized as a great
astronomer and geographer, he was inspired by the works of Hipparchus,
but historians do not agree on the extent of this influence. Did
Ptolemy use Hipparchus' measurements instead of making his own? A
difficult question, which we will leave to the specialists.
For a biography of
Ptolemy, click here,
and for a more
detailed analysis consult this site.
And never fear,
Ptolemy also has his own crater on the moon!


2. Longitude and latitude
What will Hipparchus and Ptolemy
teach us in this first chapter? They will explain the concept known
today as a coordinate system.


The Earth is round. We
have known this for a very long time, and even before anyone had
circumnavigated it clever Greek geometers found a way to measure the
circumference quite accurately (see, for example, this page).
The Earth makes one
revolution each day around an axis which joins two points, called the north
and south poles. It also makes one revolution each year
around the Sun, but neither Hipparchus nor Ptolemy knew this; instead
they believed that the Sun revolved around the Earth. It wasn't until Copernicus,
in the
sixteenth century, that we began to understand that it is really the
Earth that revolves around the Sun.
Precisely determining the
shape of the Earth took much longer,
and in fact it was only a few dozen years ago that we could measure the
dimensions to within a centimeter. The Earth is only slightly different
from a sphere: it is a little flattened at the poles but the polar
radius (6,356 km) and the
equatorial radius (6,378 km) are very close. See this page
to find out more.



Hipparchus invites us to assume
that the Earth is exactly a sphere, and then explains the rudiments of
spherical geometry. By definition, a sphere
is the set of points in space which are all at equal distance from a
single point called its center. A line which goes
through the center of a sphere cuts the sphere in two points and gives
an axis of symmetry for the sphere. If we choose such a line, we can
think of it as the axis of rotation of the Earth, and the two points of
intersection are then called the north and south
poles.



A plane passing through
the center of a sphere intersects the sphere in a circle called a great
circle, which cuts the sphere into two hemispheres.
If the plane passing through the center is perpendicular to the chosen
axis, the
circle of intersection is called the equator, and
the hemispheres are called the southern and northern
hemispheres. A plane which
contains the axis cuts the sphere in a great circle which goes through
the two poles. These circles consist of two semicircles joining the
poles; these are called meridians.
Every point on the surface of the Earth, except the poles, is on a
unique meridian. Since we are assuming that the Earth is a sphere, all
meridians have the same length: this is the distance which one must
travel over the surface of the Earth if one wants to go from the North
pole to the South pole, i.e. 20,000 km (approximately).
Of all meridians on the
Earth, exactly one of them serves as an origin, namely the one which
goes through the observatory of Greenwich in
England; however one could just as well have taken any other for an
origin (and the French would have liked very much for it to have been
the one which goes through Paris!). Any other meridian is described by
an angle (indicated in red on the figure below) which we call its longitude. Geographic
tradition demands that this angle measure between 0 and 180 degrees to
the east or to the west of Greenwich's meridian.
Click on the image at left for a film.


Planes perpendicular to
the axis cut the sphere in circles called parallels.
It is possible
they are called that because they do not intersect each other, like
parallel lines. The parallels become smaller and smaller as they get
closer to the poles. The equator is a special parallel, halfway
between the two poles; it is the longest of the parallels. Each other
parallel is either to the north or south of the equator, and is
described by an angle called the latitude,
indicated in
green on the figure.
Every point of the Earth,
except the poles, is situated at the intersection of a parallel and a
meridian, and we can therefore assign a latitude and a longitude to it. Conversely,
if we are given a latitude and a longitude we can locate the point on
the Earth...
The important thing to
remember is that in order to describe a point on the surface
of the earth we need two numbers. For
that reason we say that the surface of the earth is a twodimensional
object. This reasoning applies not only
to the surface of the Earth but also to the plane of a table or the
surface of a soccer ball.


Click on the image for a
film. 

Of course we live only
approximately on the surface of the Earth! We may find ourselves at
times flying in an airplane, for example. The two numbers latitude and
longitude are not enough to give our position precisely...we also need
to give our altitude, so in fact we need three numbers to describe our
point in space.
For this reason we say that space has three
dimensions, and we will come back to this point later...


3. Projections
In the second part of this
chapter, Hipparchus explains one of the greatest ideas in mathematics,
that of a projection. The Earth is round, but we
would like to represent it on a plane, such as a piece of paper, in
order to make a map which we can put in an atlas.



There are many methods for
drawing a map the Earth. The general principle is to choose a region of
the Earth and to associate to each point p of this
region a point F(p) on the plane. We have then
represented the region in question on a piece of the plane. Choosing
the representation F is the essence of the art of
cartography; different choices emphasize different characteristics of
the region. The ideal would be an isometric map, in which one could
measure the distance between two points p and q
by measuring the distance between their representations F(p)
and F(p). Unfortunately there is no such ideal map,
and we must compromise in some way or another. For example, some maps
try to preserve areas. Cartography is a
fascinating subject with a long history often parallel to that of
mathematics, and which has recently made considerable progress thanks
to very precise modern measurements and to the help of computers. Here
are two
sites
which can serve as
points of entry to the study of this science.


The map which Hipparchus
presents to us bears an erudite name: stereographic projection. We
should say that stereographic projection is not much used in maps
today, except to represent polar zones. But we will see little by
little in the course of this film that this projection has considerable
mathematical interest and that it is in fact extremely useful.


The definition is very
simple. We consider a plane P
tangent to the Earth at the south pole. For every point p
of the sphere which is different from the north pole, we draw the line pn
which joins p to the north pole.
This line hits the tangent plane P
in another point, F(p).
Stereographic projection is thus a representation of the sphere (with
the north pole removed) in the plane P.
Who invented this
projection? This is another contentious historical debate...Some think
it was Hipparchus, others Ptolemy, and others claim that Hipparchus did
indeed invent stereographic projection but did not understand its
properties.



Stereographic projection has
three essential properties, closely related to one another.
The first
property, nicely illustrated in the film, is that the projection
transforms a circle on the sphere into a circle or a line in the plane.
If you have the patience to wait until the final chapter, you will
understand why.



In order to demonstrate
this, Hipparchus rolls the Earth on the tangent plane at the south
pole. As a result, it is no longer the south pole that touches the
plane, and the projection is no longer from the north pole, but rather
from the "highest point" onto the tangent plane at the "lowest point."
It may not be realistic to think of rolling the Earth in this way, but
it gives very nice projections!
Click
on the image at left
for a film.
The second
property of stereographic projection, which is not illustrated in the
film, is that it respects angles. This means that if we take two curves
on the sphere which intersect at a certain angle at some point, then
the projections of the curves intersect in the same angle. In the image
at left you can see that the projections of the meridians and parallels
intersect at right angles, just as they do on the sphere. This feature
is useful for a navigator who is charting his course...he would like
very much for the angles he is measuring with his instruments to be
exactly the same as those on his map.


The third
property of stereographic projection is that, even though it does not
realize the ideal of preserving distances, it "does the best it can."
Take a point p on the sphere and
consider a very small region R
surrounding p. Stereographic
projection sends the region R to
a region F(R) around the point F(p).
The smaller R is, the better F
preserves the form of R. What
this means mathematically is that there is a constant k,
which we might call the scale of the map in R,
such that if q_{1}
and q_{2}
are two points of R, then the
ratio between the distances from q_{1}
and q_{2}
(in the sphere) and F(q_{1})
and F(q_{2})
(on the plane) is almost equal to k.
What does "almost" mean here? It means that this ratio becomes closer
and close to k as R
becomes smaller and smaller. In less precise language this means that
the map respects the form of very small regions, and for that reason it
is called conformal. This is the principal feature
of stereographic projection: it is almost perfect for a user who
employs it only in a small neighborhood of himself!



After this first excursion, let
us reemphasize the lesson of Hipparchus: the sphere is twodimensional
because we can describe its points by two numbers: latitude and
longitude. Furthermore,
it is very useful to represent the sphere in a plane using
stereographic projection...
These lessons will come in very
handy for exploring the third dimension...and then the fourth!


