[Dean's World] Andrew Cory:

[notify at powerblogs.com]

Posted by Andrew Cory:

http://www.deanesmay.com/posts/1150708077.shtml


   Barnabas Truman (1:08:13 AM): Most of the equations you deal with in
   graphing are functions. A few are not. Barnabas Truman (1:08:29 AM):
   F'r instance, a circle is not a function, because there are some
   x-values with TWO associated y-values.

   Andrew (1:08:53 AM): right! but it can be described by equation?

   Barnabas Truman (1:08:58 AM): Yes.

   Andrew (1:09:17 AM): heh are you having a sudden fear that some day
   I'll be setting your budget?

   Barnabas Truman (1:09:23 AM): Hah.

   Andrew (1:09:33 AM): 'cause that's why my degree is _really_ useful
   for...

   Barnabas Truman (1:09:38 AM): What is it?

   Andrew (1:09:42 AM): Political Science...

   Barnabas Truman (1:09:52 AM): Aha. Barnabas Truman (1:10:14 AM): Want
   to learn about the equation for a circle?

   Andrew (1:10:23 AM): yeah. I've got a +2 understanding budget and a +3
   in not falling asleep while talking about them... Andrew (1:10:25 AM):
   sure! Andrew (1:10:42 AM): (and a +18 in wanting to learn more about
   how the world works)

   Barnabas Truman (1:10:53 AM): That's the spirit! Barnabas Truman
   (1:11:03 AM): Got some paper? It might be easier to understand if you
   sketch it out as I describe it.

   ([1]Nifty, tell me more!)

   Andrew (1:12:17 AM): *nod* ok, paper and pen handy. shoot.

   Barnabas Truman (1:12:36 AM): Start by sketching in the x and y axes.

   Andrew (1:12:49 AM): done

   Barnabas Truman (1:13:06 AM): Marks at -2, -1, 0, 1, and 2 on each
   should suffice for now.

   Barnabas Truman (1:13:22 AM): Then sketch a circle centered at the
   origin (0,0), with radius one.

   Andrew (1:13:32 AM): ok

   Barnabas Truman (1:13:47 AM): So it should go through (0,1), (1,0),
   (0,-1), and (-1,0).

   Andrew (1:14:04 AM): *nod*

   Barnabas Truman (1:14:51 AM): We want, ultimately, an equation
   describing the relationship between x and y for ANY point (x,y) on the
   circumference of the circle.

   Andrew (1:15:08 AM): *nod* but not within the circle...

   Barnabas Truman (1:15:24 AM): Right, just on the circumference--that
   curve is the graph in question.

   Andrew (1:15:30 AM): *nod*

   Barnabas Truman (1:16:48 AM): This relation has to work for ANY point
   on the circle, so pick an arbitrary point anywhere on the
   circumference, and mark it (x,y). Barnabas Truman (1:17:08 AM): Now,
   officially, we don't know anything at all about that point except that
   it's on the circle.

   Andrew (1:17:15 AM): right!

   Barnabas Truman (1:17:32 AM): My computer's acting funny; I'm going to
   restart--meanwhile, you try to figure out a geometric relation between
   that x and that y. Barnabas Truman signed off at 1:17:35 AM. Barnabas
   Truman signed on at 1:26:53 AM. Barnabas Truman (1:27:08 AM): Got
   anything yet?

   Andrew (1:27:26 AM): heh I'm not even sure I understand the
   question....

   Andrew (1:27:42 AM): I mean, I get that any point on the line is 1R
   from (0,0) but... Andrew (1:27:49 AM): any point on the circle, that
   is...

   Barnabas Truman (1:28:11 AM): Okay, we're trying to set up an equation
   relating x and y that will be true for every point on the circle.

   Andrew (1:28:16 AM): right

   Barnabas Truman (1:28:18 AM): We can use ideas from geometry to figure
   this out. Barnabas Truman (1:28:28 AM): Specifically: Barnabas Truman
   (1:28:45 AM): Draw in a diagonal line segment from the center (0,0) to
   your point on the circumference (x,y).

   Andrew (1:29:02 AM): ok

   Barnabas Truman (1:29:13 AM): Is your point (x,y) above or below the
   x-axis?

   Andrew (1:29:22 AM): below

   Barnabas Truman (1:29:40 AM): Okay, draw another segment from (x,y)
   straight up to the x-axis, and stop there.

   Andrew (1:29:43 AM): it's roughly half-way between (-1,0) and (0,1)
   Andrew (1:30:17 AM): ah-ha! a triangle...

   Barnabas Truman (1:30:22 AM): Bingo! Barnabas Truman (1:30:25 AM):
   What sort of triangle?

   Andrew (1:30:43 AM): a right angle. Andrew (1:30:55 AM): of course,
   since I know not the sides or the hypotenuse... Andrew (1:31:01 AM):
   but Let's see where this goes...

   Barnabas Truman (1:31:27 AM): You DO know the hypotenuse. Barnabas
   Truman (1:31:33 AM): It's the radius of the circle.

   Andrew (1:31:39 AM): oh shit Andrew (1:31:40 AM): duh! Andrew (1:31:42
   AM): heh

   Barnabas Truman (1:31:59 AM): So what do you know about the sides of a
   right triangle, from our old buddy Pythagoras?

   Andrew (1:32:09 AM): Ok, Pythagoras was A^2+B^2=C^2? Andrew (1:32:21
   AM): (it's been about 15 years since him)... Andrew (1:34:54 AM): so I
   need either 2 more equations, or a value for one of the non hypotinal
   sides..

   Barnabas Truman (1:35:10 AM): You know the hypotenuse, C, is 1 (radius
   of the circle).

   Andrew (1:35:15 AM): right Barnabas Truman (1:35:22 AM): The other two
   sides you also know, though you might not realize it.

   Andrew (1:35:43 AM): ok, how do I figure it out? Andrew (1:35:55 AM):
   I mean, what am I missing that will make it all fall into place?

   Barnabas Truman (1:39:02 AM): The vertical leg is the distance from
   the x-axis up to the point (x,y). Andrew (1:39:10 AM): right...

   Barnabas Truman (1:39:18 AM): The horizontal leg is the distance from
   the y-axis to the point (x,y).

   Andrew (1:39:22 AM): *nod* Andrew (1:39:29 AM): I got that far...
   Barnabas Truman (1:39:29 AM): It's x and y!

   Andrew (1:39:32 AM): yup Andrew (1:39:48 AM): but how do they relate
   to 1? or is that irrelevant?

   Barnabas Truman (1:39:50 AM): So the legs are x and y, and the
   hypotenuse is 1.

   Andrew (1:40:01 AM): or 1= X+ Y

   Barnabas Truman (1:40:01 AM): A^2 + B^2 = C^2. Barnabas Truman
   (1:40:11 AM): Leg squared plus leg squared equals hypotenuse squared.

   Andrew (1:40:14 AM): right Andrew (1:40:15 AM): duh

   Barnabas Truman (1:40:18 AM): x squared plus y squared equals one
   squared. Barnabas Truman (1:40:26 AM): And that's the equation.

   Andrew (1:40:26 AM): 1=X^2+y^2 Andrew (1:40:30 AM): oh

   Barnabas Truman (1:40:31 AM): Yep.

   Barnabas Truman (1:40:48 AM): So we've got the equation for a circle
   centered at the origin with radius one.

   Andrew (1:40:52 AM): heh I was so busy trying to solve for X and Y
   that I missed the point of the exercise...

   Barnabas Truman (1:40:55 AM): Heh.

   Barnabas Truman (1:41:10 AM): But we need to be able to get the
   equation of any circle--any center, any radius.

   Andrew (1:41:17 AM): *nod*

   Barnabas Truman (1:41:22 AM): First of all, what part of that equation
   refers to the radius?

   Andrew (1:41:29 AM): Z Andrew (1:41:33 AM): well, C... Andrew (1:41:41
   AM): or 1, if you want the value

   Barnabas Truman (1:41:43 AM): And in this specific equation? Barnabas
   Truman (1:41:44 AM): Yeah, 1. Barnabas Truman (1:41:52 AM): So what if
   we want a circle of radius 2?

   Andrew (1:42:00 AM): then c=2.

   Barnabas Truman (1:42:07 AM): So what's the equation?

   Andrew (1:42:19 AM): 4=Y^2+X^2

   Barnabas Truman (1:42:27 AM): Yep.

   Andrew (1:42:37 AM): I can do _some_ math in my head..,. Andrew
   (1:42:46 AM): 2^2 being about my limits...

   Barnabas Truman (1:42:50 AM): In general, X^2 + Y^2 = R^2, where R is
   the radius.

   Andrew (1:42:54 AM): *nod*

   Barnabas Truman (1:43:18 AM): Now we just need to know how to move the
   center left/right and up/down.

   Barnabas Truman (1:43:39 AM): And because it's late and I need to go
   to sleep I'm just going to skip to the solution here.

   Andrew (1:43:44 AM): heh Andrew (1:43:45 AM): ok

   Barnabas Truman (1:45:54 AM): Anyway: you can move the entire graph to
   the right K units by changing every occurrence of X to (X-K).

   Andrew (1:46:18 AM): And to the left by (X+K)?

   Barnabas Truman (1:46:23 AM): Yeah.

   Andrew (1:46:29 AM): similarly for Y... (1:46:31 AM): Similarly, you
   can move the entire graph up H units by changing every occurrence of Y
   to (Y-H).

   Barnabas Truman (1:46:48 AM): Yeah, if you want to move it left or
   down, just use negative values for K and H.

   Andrew (1:47:04 AM): *nod* or, add numbers either negative or
   positive..

   Barnabas Truman (1:47:05 AM): So the final equation is (X-K)^2 +
   (Y-H)^2 = R^2

   Barnabas Truman (1:47:23 AM): That gives you a circle centered at the
   point (K, H) with radius R.

   Andrew (1:47:31 AM): Got it! nifty! Andrew (1:47:38 AM): I have no
   idea when it will become useful, but... By the way, that method for
   moving a graph up/down/left/right works for ANY graph. Barnabas Truman
   (1:49:53 AM): Whether it's a function or not. Andrew (1:50:02 AM):
   *nod* that's good to know also..

   Barnabas Truman (1:50:14 AM): Yep. Andrew (1:50:19 AM): If I ever have
   to go back to school, it's algebra, trig, calc for me...

   Barnabas Truman (1:50:39 AM): And with most graphs, there's some sort
   of "scaling factor" (like the radius, in this case) that can be
   doubled or halved or whatever to change the size of the graph.

   ([2]Now I know!)

References

   1. file://localhost/var/www/powerblogs/deanesmay/posts/1150708077.html
   2. file://localhost/var/www/powerblogs/deanesmay/posts/1150708077.html