WHAT IS IT? -----------
 The model displays two basic conic sections: circles and
ellipses. The figures are generated behaviorally as opposed to algebraically- the
turtles attempt to behave like points on the specified shape. The partner to this
model is called 'Conic Sections-hyperbola¶b.'

A circle is the set of all points at a certain distance (radius) from a central
point.  An ellipse is the set of points such that the sum of the distances to two
points is constant.  These two points are called foci.  The CONSTANT slider
corresponds to the radius for circles and to the sum of distances to the foci for
each turtle.

As an illustration of this, imagine a string loosely looped around two nails,
each representing a focus.  If you pull the string tight with a pencil point and
move the pencil point around the foci, you will draw an ellipse.

The ancient Greeks discovered that each conic section can be found by taking a
cross section of one or two cones with their points pointing toward each other. 
A circle results from taking a slice that is perpendicular to the axis, while an
ellipse results from taking a slice of one cone that is not perpendicular to the
axis. Similiarly, a parabola results from a cross section that passes through one
cone in a vertical fashion, such that the plane of the cut is parallel to one
face. A hyperbola results from a vertical section that passes through both cones.

The turtles use feedback to make decisions about how they behave. They set out in
random directions, and then they receive information as to whether or not they
are getting closer to where they want to be. If they are getting closer, they
continue moving forward in the direction they are going. If they are moving
farther away, they set out in a new random direction. This process is akin to the
children's game of "Hot & Cold", in which players are told whether they are
getting "hotter" or "colder" in relation to a hidden goal.

HOW TO USE IT ------------- 
*Circles: -Select the number of turtles with the
TURTLES slider. -Press SETUP. -Make sure the SECOND-FOCUS switch is set to 1.
-Press the CHANGE-FOCUS button. CHANGE-FOCUS waits for a mouse click in the
display window, and sets the circle's center to be that point. The center appears
as a white dot. -Press the MOVE-TURTLES button. Adjust the radius of the circle
with the CONSTANT slider. The turtles will automatically correct themselves as
you change both CONSTANT and the location of the center. (You can change the
circle's center by clicking on a new point as long as CHANGE-FOCUS is down.)
*Ellipses: -Select the number of turtles with the TURTLES slider. -Press SETUP.
-Make sure the SECOND-FOCUS switch is set to 0. -Press the CHANGE-FOCUS button.
Then select two foci by clicking in two different areas. -Press the MOVE-TURTLES
button. As for a circle, the size of the ellipse can be modified with CONSTANT,
and new foci can be picked while MOVE-TURTLES is running.

THINGS TO NOTICE ---------------- 
Notice that if CONSTANT is too large, the
turtles will wrap onto the other side of the screen. This may give you some
unsettled shapes- try lowering CONSTANT or enlarging the screen-size with the
'Settings' window in the Edit menu above.

When forming a circle, turtles try to attain a distance of CONSTANT (a value
determined by the user with a slider in the interface window) from a center that
the user determines by pointing and clicking (as explained above).

When forming an ellipse, turtles try to attain a combined distance of CONSTANT
from the two foci, again determined by the user's points and clicks.

THINGS TO TRY ------------- 
You may be able to get a better feeling for the
turtles' behavior if only a few turtles are on-screen at one time. Try setting
num to a small value (like 16 or 1) and watching the turtles.

Both of these conic sections can be observed by shining a flashlight at a cone
and looking at its shadow.  Can you figure out at what angles the cone must be

EXTENDING THE MODEL ------------------- 
Look at the StarLogoT model
'emergent-circle'. Watch how the turtles react with each other- something that is
missing from 'Conic Sections'. Implement this emergent behavior for one or both
of the conics in this project.

STARLOGOT FEATURES ----------------- 
Like more traditional programming languages
(e.g. C++), StarLogoT can have functions that return a value to the calling
function. The command used is called 'output'- it has one argument, the value to
be returned. Look at the function 'minus'. It takes one argument, num, and
returns its negative, 0 - num. (The colon preceding num indicates that num is a
local variable sent to the called function.)

; This model was created as part of the project: CONNECTED MATHEMATICS: ; MAKING
(OBPML). ; The project gratefully acknowledges the support of the National
Science ; Foundation (Applications of Advanced Technologies Program) -- grant
numbers ; RED #9552950 and REC #9632612. ; ; Copyright 1996 by Uri Wilensky. All
rights reserved. ; ; Permission to use, copy, or modify this software and its
documentation for ; educational and research purposes only and without fee is
hereby granted, ; provided that this copyright notice and the original authors'
names appear ; on all copies and supporting documentation. For any other uses of
this ; software, in original or modified form, including but not limited to ;
distribution in whole or in part, specific prior permission must be ; obtained
from Uri Wilensky. These programs shall not be used, rewritten, or ; adapted as
the basis of a commercial software or hardware product without ; first obtaining
appropriate licenses from Uri Wilensky. We make no ; representations about the
suitability of this software for any purpose. It ; is provided "as is" without
express or implied warranty.