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mjsandy@[EMAIL PROTECTED]
made some interesting points.
The Ancient Greeks linked a cone to a circle and
ellipse (also parabola and hyperbola) . The circle, by
cutting the cone along a plane perpendicular to the main
axis. If the angle of the plane is < 90 (> 0) an
ellipse is formed. If the apex of the cone is moved to
infinity, the cone becomes a cylinder of circular
cross-section and an oblique cut again produces and
ellipse. Try it with a cardboard cylinder. If such a
cardboard boundary is examined one diameter of the
circle remains unchanged in the ellipse, but the
corresponding perpendicular diameter of the circle
has elongated in the ellipse. In other words the
y-value of the circle remains unchanged in ellipse but
the x-value of the circle (xc) have been increased in
the ellipse (xe). In fact every xe has increased in
the same proportion to the corresponding xc (it's a
matter of similar right-angled triangles.
xe = k*xc where k is a constant > 1. So the ellipse
with minor axis b is directly linked with the circle
of radius b and its major axis is b*k. k, in fact
is 1/sin ang, where ang is the angle the plane makes
with the central axis.
So if circle of radius b is drawn, the corresponding
ellipse is formed by plotting yc and xc*k.
Logo_programmer's suggestion for constructing an ellipse
from a rectangle is possible, though I'm not sure what
method of increasing the number of sides of the
polygon was intended. Starting with a rectangle and
joining the centre sections of each side and
repeating gives a good approximation (see .jpg s).
The central section is ~0.428 of side for the values
of the major and minor axes I've tried. Why this
value I've yet to work out.
My original method for an ellipse, (the sum of the
distances from the foci being constant), follows
naturally from the method of drawing a circle using 2
ends of a piece of string tied to the same point
with the pencil keeping the string taut.
Child friendly approach?
The further evolution to drawing ovals also follows.
When James Clerk Maxwell(of Maxwell's Laws of Physics)
was 14, he drew ellipses using the string method but
then presumably asked himself the 'what if' question -
what is produced if the end of the string, tied to
one focus, is instead, continued round the focus
and tied to the pencil, giving 2*d1+d2=s. He
generalised to m*d1+n*d2=s ((m, n) integers).
He wrote a paper on the result which was read
at a meeting of the Royal Society of Edinburgh!
The "what if" question is very important for teachers.
Mike
----- Original Message -----
From: logo_programmer
To: LogoForum@[EMAIL PROTECTED]
Monday, January 07, 2008 6:43 PM
Subject: [LogoForum] Re: making ovals
--- In LogoForum@[EMAIL PROTECTED]
"Wendy Petti" <wpetti@[EMAIL PROTECTED]
> wrote:
>
> Hi, I have a fourth-grade student who has been asking
> me how to draw ovals with Logo (preferably MicroWorlds).
> Is this something that can be accomplished with
> math that a bright nine-year-old could understand?
I believe this is something that can be accomplished
with IMAGINATION ... by a bright nine-year-old.
> If not, is it something that can be accomplished
> with high-school level math that could be explained in
> kid-friendly (and teacher- friendly) terms?
Though I and others have responded to this quite a bit, it
has occurred to me that this question could have and perhaps
should have been addressed at this level of communication in
kid-friendly and teacher-friendly terms. Thus I have
chosen to respond back up at the top of this thread.
Just as a picture is supposed to be worth 1K words, a
depiction -- or visualization -- IS often better than
a description/explanation. Logo and photographs both
allow us to transcend the limitations of a mere stream
of words. Picturing `things' in our mind's eye also
allows us to re-frame a situation or `problem.
Oval ... or ellipse ... or circle viewed at a angle?
I suspect that both kids and teachers can view a hula hoop
from different angles and `understand', first person, that
a circular hoop APPEARS elliptical, just as a polygon
with enough sides drawn on computer screen via logo
APPEARS circular.
A teacher might be aware of different notions or conceptions
of a circle. Some teachers are aware that mathematicians
make much out of NOTHING ... literally. Points -- `being'
infinitesimally small, smaller than an atom of any sort -- `are'
used as building blocks for composing line segments, circles, and
other figments of imagination made out of the NOTHING which these
imaginary -- beyond material science, beyond physics-physical -- math
objects are imagined. Though a `bright nine-year-old' might not
have been exposed to the something-from-nothing metaphysics of
mathematicians yet, and this is not necessary for either kid or
teacher to `get it'. How many kids can't recognize their own shadows
.... or don't know how silhouettes are formed? I'd venture that our
bright nine-year-old could use a hula hoop, circular `ring', or
circular piece of paper to form silhouettes by tilting any of these
circular objects experimentally and observing the oval/elliptical --
(s)he knows-not-what ... nomenclature wise -- shapes that occur.
It seems a rather small leap from this rather Montessori-like exercise
to using logo -- a 3D-capable one preferably -- to draw something
circular on-an-angle (EG non-perpendicular, to us math biased adults)
so as to produce a desired or acceptable shape vis-a-vis whatever his
or her aesthetics or requirements may compel.
---------------------
Here is another way of approaching the situation.
Is there a circle which is not an ellipse ... the way there is a
square which is not a rectangle?
If a circle inscribed within a square IS an ellipse within a rectangle
how might this help us re-think and re-frame?
If would-be, `looks-like' circles in logo are typically drawn as
polygons -- via REPEAT n [FD <side_length> RT (360 / n)] -- then
can we inscribe our circle-cum-ellipse in not just a square, but any
regular polygon?
Hold this thought!
---------------------
Circle bounded by an outer square and inner diamond ... viewed non
perpendicularly ... becomes
ellipse bounded by outer rectangle and inner parallelogram
If an inner `diamond' (EG square, really) is drawn between
the 4 points where a outer square intersect with our circular ellipse
of interest an inner bounding polygon is formed.
Q: if the side count of both our inner and outer bounding polygons are
increased from 4 to 8 to 16 and so on, is our ellipse ever not bounded
by its outer and inner polygons?
If not, then might we graphically display/present an ellipse as a
polygon of n sides as we depict a circle as a polygon of n sides?
Is there ANY regular polygon viewed non-perpendicularly which does not
approximate an ellipse ... the way any regular polygon viewed
perpendicularly approximates, however crudely, a circle?
-------------------------
An intuitive solution to the generation of depiction and approximation
of an ellipse:
1) Draw the first approximation as a diamond whose vertexes
are the midpoints of a bounding rectangular parallelogram.
2) Increase the side count of the internal regular polygon
(EG our diamond originally) bounded by the rectangle
3) repeat step 2 until a close-enough approximation is obtained.
----------------------
While on the topic, if our bright 9-year-old is interested in the
theme of ovals, ellipses, and circles viewed from an angle ... then
why not present the concepts of ...
* elliptical orbits
http://en.wikipedia.org/wiki/Elliptic_orbit
* How *any* regular solid can be used as a template to approximate
a sphere ... by replacing the regular polygon from which it is
constructed with a circular ellipse
Tetrahedron ==> 4 circular ellipses
cube ==> 6 "
octahedron ==> 8 "
Dodecahedron ==> 12 "
icosohedron ==> 20 "
(aside: I sent a picture containing a dodeca-ellipse shape
and several tetra-ellipse shapes, all constructed from hula
hoops, to my friend Pavel a couple of years ago. He quickly
transcribed the dodecaHulaHoop into an Elica program which
is now both viewable online at
<http://www.elica.net/site/museum/Hoola%20Hoops.jpg>
and as a runnable Logo program as one of the 800+ samples
included with Elica, available for free at
http://www.elica.net
)
* comparisons between 2D approximates of ellipses
and 3D approximations of ellipsoids
* comparisons and constrasts between ovoids and
ellispoids (including the special case of spheroids)
http://en.wikipedia.org/wiki/Ovoid_(projective_geometry)
http://en.wikipedia.org/wiki/Ellipsoid
All that has come to mind on ovals, ellipses,
and such of late, Wendy, et al.
Cheers!
Gene
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