Part of a high school course on astronomy, Newtonian mechanics and spaceflight
by David P. Stern
This lesson plan supplements: "Coordinates," section #5b: on disk Scelcoor.htm, on the web |
"From Stargazers to Starships" home page and index: on disk Sintro.htm, on the web
Goals: The student will|
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
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"?
-- Can you say that the latitude and longitude of a point on Earth are its coordinates?
-- What are the "cartesian coordinates" of a point on a flat plane?
-- How did the name "Cartesian" arise with regards to coordinates?
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:
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?"
-- What are the x and y coordinates of a point on a flat plane?
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?
-- How are cartesian coordinates used to define a point in 3-dimensional space?
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?
(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
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
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?
--Describe one such system, the polar coordinates.
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)?
-- Given (x,y), how can you derive r? What function of φ can you express?
--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?
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.
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Author and Curator: Dr. David P. Stern
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Last updated: 11 September 2004