Goals: The student will|
- Learn to use cartesian coordinates (x,y) for defining the position of a point in 2 dimensions.
- Learn to use cartesian coordinates (x,y,z) in 3-dimensional space. [Optional: learn to appreciate there exist two ways of defining the z axis, and which of them is used.]
- Become familiar with some of the tools and terms used by surveyors: theodolite, azimuth, elevation, zenith, [nadir].
Optional items below are for students familiar with trigonometry and with
the theorem of Pythagoras (areas of math also covered by web files
linked to httm://www.phy6.org/stargaze/Smath
Polar coordinates (r, φ) on the plane (2-dimensions)
- Converting (r, φ) to (x,y) and vice versa, at least for r.
- "Spherical" polar coordinates in 3-dimensional space. We could use (r,λ,φ) with λ the latitude (as used in marking places on Earth, going from –90° to +90°), but mathematicians prefer (r,θ,φ), where θ=90°–λ is the "co-latitude" going from 0 to 180°.
Terms: Cartesian coordinates, axes, origin (of coordinates) [polar coordinates]
Theodolite, azimuth, elevation, zenith, [nadir].
Stories and extras: René Descartes.
Guiding questions and additional tidbits
(Suggested answers in parentheses, brackets for comments by the teacher or "optional")
The ancient Greeks were quite clever in mathematics--even though the way they wrote numbers was very clumsy, a bit like the Roman numerals. They also had a very sophisticated geometry--the study of lines, angles, triangles, circles and so forth, and of the laws which governed them.
But interestingly, these two areas were completely separate. The theorem of Pythagoras, for instance, was proved by a method using not numbers but triangles, rectangles and squares. Even today students in schools sometimes learn that method, because it is clever--but it is also rather complicated and hard.
For about 15 centuries this separation continued, and then, in the early 1600s, a French philosopher, René Descartes ("Day-cart") suddenly brought the two together. He introduced a simple but clever scheme of labeling each point on a flat plane by numbers. We call these numbers the "cartesian coordinates" of the point--"cartesian" from "Descartes".
As we will see--some of it today, some in a later class--
this not only allowed points to be labeled by numbers, but also to use numbers for describing straight lines in various directions, as well as circles, ellipses and even curves and shapes which Greek geometry never dreamed of.
Start covering section 5c of "Stargazers" up to polar coordinates (handled separately later), using the questions below in the lesson, in the review afterwards or in both:
-- What are "systems of coordinates"?
Methods of labeling points in space by a set of numbers called their "coordinates."
-- Can you say that the latitude and longitude of a point on Earth are its coordinates?
They are coordinates in a system designed for use on the surface of a sphere. Declination and right ascension are used as celestial coordinates.
-- What are the "cartesian coordinates" of a point on a flat plane?
The point's distances from two straight axes: the "x axis" usually drawn horizontally, and the "y axis" perpendicular to it.
-- How did the name "Cartesian" arise with regards to coordinates?
They were introduced by the French mathematician--also philosopher and soldier--René Descartes [Ray-nay De-cart], 1596-1650.
Tell the class that Descartes also tried to answer the question "how do I know that I exist?" by declaring "I think, therefore I am,", or in Latin, which scholars used in those days, "Cogito, ergo sum." Much later, an American poet joked:
Descartes said: I extol
Myself above the animal
Because I have a soul! --
He put Descartes before the horse.
Just to make sure (we live in the age of the horseless carriage!) ask the class what the old phrase "putting the cart before the horse" meant. (Putting your priorities in the wrong order).
-- What is "the origin of the coordinates?"
That is the point where the x and y axis meet, often denoted by the letter O.
-- What are the x and y coordinates of a point on a flat plane?
They are the two numbers, (x,y), which give its position.
X is the distance measured parallel to the x-axis. It is measured from the y-axis--to the right it is positive, to the left, negative.
Y is the distance measured parallel to the y-axis. It is measured from
the x-axis, above it, it is positive, below it, negative.
-- What are the coordinates of the origin O?
They are ( 0, 0), that is ( zero, zero)
-- How are cartesian coordinates used to define a point in 3-dimensional space?
Three coordinates define the point--x,y and z--defined by 3 perpendicular axes. The coordinates x and y are defined as before, in some plane, and z is measured from the origin in a direction perpendicular to the (x,y) plane.
Given a point P, we can always construct a box-shape with perpendicular sides, so that one corner is at P and the opposite corner at the origin O.
The (x,y,z) coordinates are then the length of the sides of the box parallel to the (x,y,z) axes, with + or - sign, depending which sides of the axis they are on.
In case the question arises: when z is measured from the (x,y) plane in a perpendicular direction, there exist two such directions, one to each side. Does it make any difference which we choose?
Answer: Yes, it does make a difference. It is customary to choose the (x,y,z) axes in the directions of (thumb, index, middle finger) of the right hand. The other choice would have corresponded to (thumb, index, middle finger) of the left hand.]
-- An additional question: Space only has 3 dimensions. But would you guess that we can mathematically explore what 4-dimensional space would be like (or even 5-dimensional space) by imagining that each point was defined by 4 numbers (x,y,z,u) or even 5 numbers (x,y,z,u,v)?
(Optional, for a class which has already covered the theorem of Pythagoras).
--What does the theorem of Pythagoras say?
In a triangle with sides having length (a,b,c), if the sides a and b make a "right angle (90°), then
Here the class starts going over the material on polar coordinates, using the questions below:
a2 + b2 = c2
(Teacher explains, drawing on the blackboard:) If we are given the coordinates of two points in the plane, the theorem of Pythagoras allows us to calculate the distance between them.
Suppose the points and their values of (x,y) are:
Choose a 3rd point C which takes its x from one point and y from the other. Say
We will see that the triangle (ABC) is a right angled one.
On one side, from A=(1,4) to C=(1,1), x has a constant value, namely x=1. That side is therefore parallel to the x axis. The difference of y equals 3 (ignore the sign here, we only need the length, which is always positive), so if we call this side "a", its length is 3.
On the other side, from B=(5,1) to C = 1,1) y has the same value of 1, so it is parallel to the y axis, and it therefore makes a 90° angle with the side we labeled "a". Its values of x differ by 4, so if we name this side "b", it has length 4.
The third side of the triangle ABC is the distance from A to B. Naming that line "c" we have, by the theorem of Pythagoras
c2 = a2 + b2 = 32 + 42 = 9 + 16 = 25
The square root of 25 is 5, so the distance AB equals 5.
(The point C takes x from one point and y from the other. We could have chosen C=(5,4). Do we get the same result?)
(End of the optional section)
-- Can systems of coordinates, other than the cartesian one, be used to label points on a plane?
Yes, other systems can be used.
--Describe one such system, the polar coordinates.
Polar coordinates also have an "origin" O as reference point, but instead of (x,y), they use the distance r from O to the given point P, and the angle φ ("phi"--Greek f) between the line OP ("radius"--hence the letter "r") and some reference line.
For students familiar with trigonometry
-- If a cartesian system (x,y) has the same origin as a polar system, and the reference line of the polar system is the x axis--given (r,φ) of a point, what are its (x,y)?
Answer   x = r cos φ   y = r sin φ f
-- Given (x,y), how can you derive r? What function of φ can you express?
- For r, take square root of r2 = x2 + y2
- For φ   tanφ = y/x
--Can you label a point P in 3-dimensional space by polar coordinates?
How many numbers are needed for describing each point?
Three numbers are needed.
What are they?
One is the distance from the origin, OP = r.
For the other two points, put a sphere of radius r around the origin, so that P is somewhere on the sphere. Then set up on the sphere a system of latitude λ and longitude φ, the 3 coordinates are then (r,λ,φ) and give the point exactly.
Of course, many choices exist. e.g., where to place the axis? And where on the "equator" should be the reference point from which longitude φ is measured?
-- How do surveyors use this system?
They have a telescope on a horizontal turntable which can also rotate up and down. When they sight on some distant point P, the up-down angle, corresponding to λ is the elevation of the point P. The direction on the turntable, an angle φ measured counterclockwise from north, is its azimuth.
Given these two as measured from two points (A,B) separated by a known distance, surveyors can calculate everything about the triangle (ABP), which is why the method is called triangulation. In particular, they can calculate the distances (AP) and (BP) from either point to P. This technique is useful in making maps.
Add later: actually, when mathematicians define such "spherical coordinates," they prefer to measure "latitude" from the pole and not from the equator, like declination in the sky. It is sometimes called "co-latitude" and marked with a letter θ, the Greek letter "theta" pronounced like T.