Cartesian to Polar Coordinates
We will be interested in describing
the motion of a satellite in terms of its coordinates. We will later see that
the satellite will move in a plane. So it will be appropriate to describe the
positions of the satellite at each point in time with plane coordinates.
What are plane coordinates?
It will be simpler first to say what "line coordinates" are. Euclidean
geometry is all that is needed to establish the principle of measurement that
is a common feature of every aspect of our daily lives. We measure certain sorts
of substances (or durations of time) in the following way. Consider a substance
such as mass. We imagine that two quantities of mass, say 
may be combined to form a new quantity of mass 
. Pictorially:
This process is, we assume,
reversible. And we assume the substance is such that it is indestructible. Nothing
is ever gained or lost in the processes of association and disassociation pictured
above. Now we use the term substance loosely here. It is meant
to represent some recognizable element of our experience. But we do not use
it too loosely. It cannot represent such elements of experience as "joy"
or "fear" or "love." We know better than to attempt seriously
to measure such things. Indeed, we do not represent them to ourselves, however
whimsically, with pictures such as the above.
We do not immediately associate
numbers with the quantities, however. Rather, we refer them to a geometric substance
that we call "extension." A quantity of extension can be visualized
as a line segment. These can be combined and separated by joining them end to
end, and extension is a substance for anybody's money.
Now this association with
extension has the following intuitive properties:
Each quantity of substance S, say Q(S) is associated with a line segment, I(S). We can compare quantities of S and we insist that
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also
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If we represent the combination
of quantities of S or extension with the symbol 
then
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where 
is obtained by joining the line segments end-to-end.
In pictures again:
This picture brings us to
the threshold of measurement. Of all the quantities of S we choose a unit
quantity: 
. This is associated with a unit interval: 
.
In this way, given any quantity Q(S), its relation (or ratio) with 
is the same as the relation (or ratio) of I(S) with the unit interval 
.
In this way, we can represent
the various amounts of substance S with specific "numbers," these
numbers correspond to the ratios of intervals to the unit interval.
Now the Greeks realized that not all intervals had ratios in the sense that
they would correspond to what we call rational numbers, or fractions.
But with the invention of algebra, and in particular, of the notion of limit,
we now believe that all intervals have ratios (that we call real
numbers) with a fixed interval. And so, once we determine a unit interval
on a line, we can associate a number to any other interval. In this way, any
quantity of substance S can be measured.
This process is instantiated
in many measuring instruments in ordinary life. We use rulers to measure distances,
for example. One finds on the face of weigh scales, a representation of the
interval associated to any mass that is placed on the scale. We think of durations
of time as substance and measure such durations on the (circular) intervals
on the face of analog clocks. Car speedometers measure a different sort of substance
-- the distance traveled in a unit time -- and this is represented also on a
line. This last is an example of a "mixed ratio" a ratio of distance
to time. While the Greeks would not have approved, it led to the many advances
that we shall explore with Calculus. In all of these examples, the actual measuring
devices -- the intervals that they show us -- are demarcated with numbers, so
that we may record the result of our measurement easily.
Now the geometric strategy that sits behind ordinary measurement may be summed up in the following way:
For a given line, choose three things to make it a ruler:
1) A special point O called its origin
2) A positive direction, usually represented with an arrow
3) Another point called U, which determines the unit interval from O in the positive sense.
Once these things are determined,
then every point on the line is represented by a unique number. The origin O
is associated to the number 0, and the unit point U is associated
to the number 1. All other points on this line then have a number associated
with them that we shall call its coordinate. The picture is:
Thus, lines are 1-dimensional. Points on them are described with a single number, called the coordinate, once we equip those lines with the three pieces of structure described above, and make it a ruler.
Now Descartes recognized in
this simple procedure a new way to do Geometry. Why not associate coordinates
with points in the plane ? Well, we quickly see that a single number will not
do to describe a point in the plane. But suppose that we choose two rulers in
the plane that intersect at their origins. We know that two lines that intersect
in a single point determine a unique plane. Thus, we have a picture like this:
Then using the parallel postulate from Geometry, we see how to associate with any point in the plane determined by these rulers two numbers.
Suppose the point is called
P. Then there is a unique line through P that is
parallel to ruler y. That line is not parallel to ruler x, otherwise
ruler x would be parallel to ruler y. Thus it meets ruler x in a single point:
and this point is determined by a unique number -- since ruler x is a ruler!
Call that number 
.
In a similar way, there is
a unique line through P that is parallel to ruler x. This line
meets ruler y in a unique point: 
. Since this point is on ruler y, it is associated with a unique number, since
ruler y is a ruler. Call that number 
. The picture is as follows:
Thus to a point P we have a pair of numbers: 
and 
,
or x and y. These numbers are determined by P as
we saw, but we have a more interesting fact.
Question 1: Show that if for a pair of points in the plane: P and Q, we have
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then P = Q.
We are thus justified in calling
and 
the coordinates of P, and simply calling 
: x and 
:
y and representing the point simply: 
the x and y being the coordinates of P. The "rulers"
are then called "axes" -- the x-axis and the y-axis.
This is the Cartesian representation
of points in the plane. Notice that it is not necessary to require that the
rulers be perpendicular. Nor is it necessary to require that the unit lengths
of each ruler be equal. Usually, however, as you will see in our experiment,
they are represented that way. Once we choose a pair of rulers with a common
origin, we have a way of assigning coordinates to each point P.
Now what is special about
this assignment leads us to the notion of vectors, and may be summarized very
easily. Suppose we think of a point in the plane as the directed "arrow"
starting at the origin and terminating at the point, thus.
This "arrow" really represents a parallel motion 
of the entire plane that carries the origin to P and that has this property:
If A and B are points in the
plane, then the segment connecting 
is parallel to the segment connecting 
and these segments have the same length.
Question 2: Show that if such a parallel motion has the property that
for a single A, then it is the identity mapping, that is 
for all P.
Vector addition and the parallelogram law
When
we think of a point in the plane P as its associated mapping: 
we will call it a "vector." These vectors will be very important for
all that follows. Here, we take up one basic fact about them. Consider two points
in the plane P and Q. They have associated to them
parallel translations 
where
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Question 3: Show, using only Geometry and properties of parallelograms that the compositions are equal:
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This last fact allows us
to define the sum of two points in the plane (vectors) P and Q
as the vector R, where
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This definition does not yet
depend on the choice of rulers and their coordinates. Pictorially this definition
is easy to visualize in terms of the parallelogram generated by the vectors:
Now suppose we introduce a pair of "axes" so that we can associate coordinates to these points.
It is an interesting exercise
in Geometry (and only Euclidean Geometry) to show the following:
Question 4: Suppose that the coordinates are given by 
. Then prove that the coordinates of 
are given by
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This is the basic law of vector
addition. What is especially interesting (and worth reflection) is the fact
that if the axes (rulers) are changed, then the coordinates will all change:
and we will have
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In other words, the law of vector addition itself is independent of the choice of axes (rulers). The addition law retains its form whatever coordinate system we use. In more modern language, we would say that the addition law is covariant. This is a very important principle in Physics. A physical law must not depend on the "point of view" (replace "rulers") of the observer.
Now the Cartesian representation
of points in the plane, and their representation as vectors, will be very important
in all that follows. But there is another representation with which we will
want to be familiar because it is natural for the study of satellite orbits.
Let us explore the "polar"
representation of points in the plane. The Earth is, as we will see, at one
focus of the "ellipse" (defined later) that describes the orbit. Further,
as will be developed further in this chapter, and in Harmony of the Spheres,
the conservation of angular momentum has a simple interpretation
as Kepler's second law when we use a polar representation of the motion.
What is the polar representation?
Each point in the plane is described by two coordinates: 
in the following way. Choose a point P, and R is
its distance from the origin O. We insist that 
so that the origin itself is not assigned coordinates. The number A
is the radian measure of the angle from the positive ray (pointing to the East
from O) to the point P, measured positive in the counterclockwise
sense. In this microworld, we will report the degree measure of A
rather than the radian measure for simplicity.
It is true that points have multiple representations in this scheme (different pairs of coordinates correspond to the same point) but this is not generally a problem for the simple uses we make here of this coordinate system.
Exploration: Cartesian and Polar Coordinates
The exercise on this page
is simple. Click a point in the Cartesian screen Graph2D on the right and the
coordinates, both in Cartesian and in Polar coordinates will be reported on
the left.
