[MWForum]Circle and arc help
David Longman
mwforum@lists.mathcats.com
Sun, 20 Feb 2005 17:53:25 -0000
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Another circle solution. This one treats the Turtle more like a robot =
than a graphics pen.
=20
No trigonometry at all!=20
=20
(Using MSWLogo)
=20
Here is version 1:
to crawl =20
if (distance :c) < :r [lt 1] ;if too close to centre of object turn =
away
if (distance :c) > :r [rt 1] ;if too far from centre turn towards
fd 1
crawl
end
=20
Where
:c is the centre of the object (i.e. the circle)
:r is the size of the object (i.e. the radius)
In this procedure the Turtle checks its distance from the centre of the =
object. Assuming that the Turtle is travelling in a clockwise direction, =
the Turtle turns LEFT if it is too close so as to move away from the =
object. If it is too far from the centre the Turtle turns RIGHT so as to =
move towards it.=20
NOTE that for this procedure to produce a circle the turtle must be =
positioned appropriately. For example this line will draw a circle:
=20
CS PU FD :R RT 90 PD CRAWL
NOTE the use of RT 90 to orient the Turtle appropriately. What happens =
if you use LT 90 instead? What happens if you use RT 30? Try swapping =
the use of LT and RT inside the procedure.
=20
Varying the starting point also produces some beautiful and interesting =
effects.
=20
What happens if you start CRAWL without moving the Turtle from the =
centre?=20
=20
What happens if you start CRAWL further out? E.g.
=20
CS PU FD :R+50 RT 90 PD CRAWL
=20
The procedure can be made more flexible by the use of more variables:
=20
to crawl
if (distance :c) < :r [lt :a]
if (distance :c) > :r [rt :a]
fd :step
crawl
end
=20
The following are sample values :
=20
Make "c [100 100]
Make "r 100
Make "a 1
Make "step 1
=20
A start line like the following would position the Turtle appropriately =
before beginning CRAWL:
=ADcs pu setpos :c fd :r rt 90 pd crawl=20
Experiment with these variables.
CRAWL can written in a slightly different way in order to maintain a =
constant distance.
This example is named ORBIT:
to orbit
seth towards :c ;face towards centre
localmake "move ((distance :c) - :r) ;calculate error
pu fd :move ;move towards or =
away from centre
rt 90 pd fd :step ;orient and draw
orbit
end
In this procedure the Turtle uses a simple strategy to ensure a constant =
distance from the centre. The initial position of the Turtle is =
irrelevant - this procedure always finds the right starting point.
=20
=20
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~=20
David Longman, Senior Lecturer in ICT
School of Education University of Wales, Newport
Caerleon Campus, PO Box 179
Newport
South Wales NP18 3YG
Tel: +44 (0)1633 432629
Fax: +44 (0)1633 432074
Web site: http://education.newport.ac.uk=20
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
________________________________
From: mwforum-admin@lists.mathcats.com on behalf of Jacques Mathil
Sent: Fri 18/02/2005 2:27 PM
To: mwforum@lists.mathcats.com
Subject: Re: [MWForum]Circle and arc help
Hello,
here is another solution. This procedure draws a centered polygon :
=20
to centered_polygon :number_sides :radius
make "alpha 360 / :number_sides
make "side 2 * :radius * sin (:alpha / 2)
pu
fd :radius
rt 90 + :alpha / 2
pd
repeat :number_sides [fd :side rt :alpha]
pu
left 90 + :alpha / 2
bk :radius
end
=20
it uses some trigonometrics not for young children, but you can use it =
as a tool.
To obtain a circle, try many sides : b.e centered_polygon 90 100
you could modify this procedure to obtain arcs.
=20
<-- Some tricks-->
to fig1 :r
if :r > 150 [stop]
centered_polygon 90 :r
fig1 :r + 20
end
=20
to fig2 :r
if :r > 150 [stop]
centered_polygon 3 :r
rt 10
fig2 :r + 5
end
=20
to fig3 :r
if :r > 150 [stop]
centered_polygon 90 :r
rt 20 fd 8
fig3 :r + 3
end
=20
Try them like this : fig1 0 fig2 0 fig3 0
=20
Jacques Mathil
Chemin de la Moniasse 7
1254 Jussy
GENEVA SWITZERLAND
T=E9l. : +41 22 759 81 81
----- Original Message -----=20
From: Susan Einhorn <mailto:susan@lcsi.ca> =20
To: mwforum@lists.mathcats.com=20
Sent: Thursday, February 17, 2005 5:14 PM
Subject: RE: [MWForum]Circle and arc help
This has been a great discussion on how to create different procedures =
to draw a circle. This is a great challenge for students. Here's another =
circle-drawing procedure from a new book about Math and MicroWorlds =
that's being developed by LCSI (authored by Wendy Petti):
=20
to circle
pu
repeat 360 [ fd 50 pd fd 0 pu bk 50 rt 1]
end
=20
With an input:
to circle :r
pu
repeat 360 [fd :r pd fd 0 pu bk :r rt 1]
end
=20
If you're not familiar with the pd fd 0 trick, it creates a dot without =
displacing the turtle. If the radius for this circle is too large, there =
are gaps between the dots in the circumference. To make the dots closer =
together decrease the size of the turn (e.g., rt 1 / 2)and increase the =
input to repeat accordingly (repeat 360 * 2).
=20
That makes three (or four, although the old Apple Logo method is a =
variation of Shawn's method) different ways to draw a circle. If you =
think about it, I'm sure you could come up with even more.
=
=20
Susan
-------------------------------
Susan Einhorn
LCSI
www.MicroWorlds.com
susan@lcsi.ca
=20
=
=20
-----Original Message-----
From: mwforum-admin@lists.mathcats.com =
[mailto:mwforum-admin@lists.mathcats.com]On Behalf Of Logo Computer =
Systems Inc.
Sent: Thursday, February 17, 2005 10:34 AM
To: mwforum@lists.mathcats.com
Subject: Re: [MWForum]Circle and arc help
=20
Hi!
Could you program the circle with a variable (or 2) to make it easy to =
make different sized circles? And how about the arc?
Would the spell checker from MSWord work with Pro or EX on Mac OS =
10)?
=20
Hi Kate,
=20
I used the following procedure in a -tools-.txt file and was able to =
create different sized circles.
=20
to cir :r
repeat 360 [fd 2 * pi * :r / 360 rt 1]
end
Unfortunately, the spell checker features are only included with =
Windows versions of Pro and EX. I used to use an OS 9 application =
called, SpellTools which is available at no charge for download from the =
following address: =
http://mac.softpedia.com/progDownload/SpellTools-Download-1097.html =
It's a universal spell checker that works with all of your text oriented =
applications. I have never tried it under OS X.
=20
Regards.
=20
Shawn
=20
--=20
=20
Logo Computer Systems Inc.
1 Westmount Square, Suite 300
Montreal,QC Canada
H3Z 2P9
info@lcsi.ca
www.lcsi.ca
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<DIV id=3DidOWAReplyText81066 dir=3Dltr>=0A=
<DIV dir=3Dltr><FONT face=3DArial color=3D#000000 size=3D2>Another =
circle solution. This =0A=
one treats the Turtle more like a robot than a graphics pen.</FONT></DIV>=0A=
<DIV dir=3Dltr><FONT face=3DArial size=3D2></FONT> </DIV>=0A=
<DIV dir=3Dltr><FONT face=3DArial size=3D2>No trigonometry at all! =
</FONT></DIV>=0A=
<DIV dir=3Dltr><FONT face=3DArial size=3D2></FONT> </DIV>=0A=
<DIV dir=3Dltr><FONT face=3DArial size=3D2>(Using MSWLogo)</FONT></DIV>=0A=
<DIV dir=3Dltr><SPAN =0A=
style=3D"FONT-SIZE: 11pt; FONT-FAMILY: Arial"></SPAN> </DIV>=0A=
<DIV dir=3Dltr><SPAN style=3D"FONT-SIZE: 11pt; FONT-FAMILY: Arial">Here =
is version =0A=
1:<?xml:namespace prefix =3D o ns =3D =
"urn:schemas-microsoft-com:office:office" =0A=
/><o:p></o:p></SPAN></DIV>=0A=
<P class=3DMsoPlainText style=3D"MARGIN: 12pt 0cm 0pt 36pt"><SPAN =0A=
style=3D"FONT-SIZE: 11pt; FONT-FAMILY: Arial">to crawl<SPAN =0A=
style=3D"mso-tab-count: =
2"> &nbs=
p; </SPA=
N><SPAN =0A=
style=3D"mso-spacerun: yes"> </SPAN><SPAN =0A=
style=3D"mso-spacerun: yes"> </SPAN></SPAN></P>=0A=
<P class=3DMsoPlainText style=3D"MARGIN: 0cm 0cm 0pt 36pt"><SPAN =0A=
style=3D"FONT-SIZE: 11pt; FONT-FAMILY: Arial">if (distance :c) < :r =
[lt 1] =0A=
<SPAN style=3D"mso-spacerun: yes"> </SPAN>;<I =0A=
style=3D"mso-bidi-font-style: normal">if too close to centre of object =
turn =0A=
away<o:p></o:p></I></SPAN></P>=0A=
<P class=3DMsoPlainText style=3D"MARGIN: 0cm 0cm 0pt 36pt"><SPAN =0A=
style=3D"FONT-SIZE: 11pt; FONT-FAMILY: Arial">if (distance :c) > :r =
[rt 1] =0A=
<SPAN style=3D"mso-spacerun: yes"> </SPAN>;<I =0A=
style=3D"mso-bidi-font-style: normal">if too far from centre turn =0A=
towards</I><o:p></o:p></SPAN></P>=0A=
<P class=3DMsoPlainText style=3D"MARGIN: 0cm 0cm 0pt 36pt"><SPAN =0A=
style=3D"FONT-SIZE: 11pt; FONT-FAMILY: Arial">fd 1<o:p></o:p></SPAN></P>=0A=
<P class=3DMsoPlainText style=3D"MARGIN: 0cm 0cm 0pt 36pt"><SPAN =0A=
style=3D"FONT-SIZE: 11pt; FONT-FAMILY: Arial">crawl<o:p></o:p></SPAN></P>=0A=
<P class=3DMsoPlainText style=3D"MARGIN: 0cm 0cm 0pt 36pt"><SPAN =0A=
style=3D"FONT-SIZE: 11pt; FONT-FAMILY: Arial">end<o:p></o:p></SPAN></P>=0A=
<P class=3DMsoNormal =0A=
style=3D"MARGIN: 0cm 0cm 0pt; mso-layout-grid-align: none"><SPAN =0A=
style=3D"FONT-SIZE: 11pt; FONT-FAMILY: =
Arial"><o:p> </o:p></SPAN></P>=0A=
<P class=3DMsoNormal =0A=
style=3D"MARGIN: 0cm 0cm 0pt; mso-layout-grid-align: none"><SPAN =0A=
style=3D"FONT-SIZE: 11pt; FONT-FAMILY: Arial">Where<o:p></o:p></SPAN></P>=0A=
<P class=3DMsoNormal =0A=
style=3D"MARGIN: 0cm 0cm 0pt 36pt; mso-layout-grid-align: none"><SPAN =0A=
style=3D"FONT-SIZE: 11pt; FONT-FAMILY: Arial">:c is the centre of the =
object =0A=
(i.e. the circle)<o:p></o:p></SPAN></P>=0A=
<P class=3DMsoNormal =0A=
style=3D"MARGIN: 0cm 0cm 0pt 36pt; mso-layout-grid-align: none"><SPAN =0A=
style=3D"FONT-SIZE: 11pt; FONT-FAMILY: Arial">:r is the size of the =
object (i.e. =0A=
the radius)<o:p></o:p></SPAN></P>=0A=
<P class=3DMsoNormal =0A=
style=3D"MARGIN: 12pt 0cm 0pt; mso-layout-grid-align: none"><SPAN =0A=
style=3D"FONT-SIZE: 11pt; FONT-FAMILY: Arial">In this procedure the =
Turtle checks =0A=
its distance from the centre of the object. Assuming that the Turtle is =0A=
travelling in a clockwise direction, the Turtle turns LEFT if it is too =
close so =0A=
as to move away from the object. If it is too far from the centre the =
Turtle =0A=
turns RIGHT so as to move towards it. <o:p></o:p></SPAN></P>=0A=
<P class=3DMsoNormal style=3D"MARGIN: 12pt 0cm 0pt; =
mso-layout-grid-align: none"><B =0A=
style=3D"mso-bidi-font-weight: normal"><SPAN =0A=
style=3D"FONT-SIZE: 11pt; FONT-FAMILY: Arial">NOTE </SPAN></B><SPAN =0A=
style=3D"FONT-SIZE: 11pt; FONT-FAMILY: Arial">that for this procedure to =
produce a =0A=
circle the turtle must be positioned appropriately. For example this =
line will =0A=
draw a circle:<o:p></o:p></SPAN></P>=0A=
<P class=3DMsoNormal =0A=
style=3D"MARGIN: 0cm 0cm 0pt; mso-layout-grid-align: none"><SPAN =0A=
style=3D"FONT-SIZE: 11pt; FONT-FAMILY: =
Arial"><o:p> </o:p></SPAN></P>=0A=
<P class=3DMsoNormal =0A=
style=3D"MARGIN: 0cm 0cm 0pt 36pt; mso-layout-grid-align: none"><SPAN =0A=
style=3D"FONT-SIZE: 11pt; FONT-FAMILY: Arial">CS PU FD :R RT 90 PD =
CRAWL<BR =0A=
style=3D"mso-special-character: line-break"><BR =0A=
style=3D"mso-special-character: line-break"><o:p></o:p></SPAN></P>=0A=
<P class=3DMsoNormal style=3D"MARGIN: 0cm 0cm 0pt; =
mso-layout-grid-align: none"><B =0A=
style=3D"mso-bidi-font-weight: normal"><SPAN =0A=
style=3D"FONT-SIZE: 11pt; FONT-FAMILY: Arial">NOTE </SPAN></B><SPAN =0A=
style=3D"FONT-SIZE: 11pt; FONT-FAMILY: Arial">the use of RT 90 to orient =
the =0A=
Turtle appropriately. What happens if you use LT 90 instead? What =
happens if you =0A=
use RT 30? Try swapping the use of LT and RT inside the =0A=
procedure.<o:p></o:p></SPAN></P>=0A=
<P class=3DMsoNormal =0A=
style=3D"MARGIN: 0cm 0cm 0pt; mso-layout-grid-align: none"><SPAN =0A=
style=3D"FONT-SIZE: 11pt; FONT-FAMILY: =
Arial"><o:p> </o:p></SPAN></P>=0A=
<P class=3DMsoNormal =0A=
style=3D"MARGIN: 0cm 0cm 0pt; mso-layout-grid-align: none"><SPAN =0A=
style=3D"FONT-SIZE: 11pt; FONT-FAMILY: Arial">Varying the starting point =
also =0A=
produces some beautiful and interesting effects.<o:p></o:p></SPAN></P>=0A=
<P class=3DMsoNormal =0A=
style=3D"MARGIN: 0cm 0cm 0pt; mso-layout-grid-align: none"><SPAN =0A=
style=3D"FONT-SIZE: 11pt; FONT-FAMILY: =
Arial"><o:p> </o:p></SPAN></P>=0A=
<P class=3DMsoNormal =0A=
style=3D"MARGIN: 0cm 0cm 0pt 36pt; mso-layout-grid-align: none"><SPAN =0A=
style=3D"FONT-SIZE: 11pt; FONT-FAMILY: Arial">What happens if you start =0A=
CRAWL without moving the Turtle from the centre? =
<o:p></o:p></SPAN></P>=0A=
<P class=3DMsoNormal =0A=
style=3D"MARGIN: 0cm 0cm 0pt 36pt; mso-layout-grid-align: none"><SPAN =0A=
style=3D"FONT-SIZE: 11pt; FONT-FAMILY: =
Arial"><o:p> </o:p></SPAN></P>=0A=
<P class=3DMsoNormal =0A=
style=3D"MARGIN: 0cm 0cm 0pt 36pt; mso-layout-grid-align: none"><SPAN =0A=
style=3D"FONT-SIZE: 11pt; FONT-FAMILY: Arial">What happens if you start =
CRAWL =0A=
further out? E.g.<o:p></o:p></SPAN></P>=0A=
<P class=3DMsoNormal =0A=
style=3D"MARGIN: 0cm 0cm 0pt 36pt; mso-layout-grid-align: none"><SPAN =0A=
style=3D"FONT-SIZE: 11pt; FONT-FAMILY: =
Arial"><o:p> </o:p></SPAN></P>=0A=
<P class=3DMsoNormal =0A=
style=3D"MARGIN: 0cm 0cm 0pt 36pt; mso-layout-grid-align: none"><SPAN =0A=
style=3D"FONT-SIZE: 11pt; FONT-FAMILY: Arial">CS PU FD :R+50 RT 90 PD =0A=
CRAWL<o:p></o:p></SPAN></P>=0A=
<P class=3DMsoNormal =0A=
style=3D"MARGIN: 0cm 0cm 0pt; mso-layout-grid-align: none"><SPAN =0A=
style=3D"FONT-SIZE: 11pt; FONT-FAMILY: =
Arial"><o:p> </o:p></SPAN></P>=0A=
<P class=3DMsoNormal =0A=
style=3D"MARGIN: 0cm 0cm 0pt; mso-layout-grid-align: none"><SPAN =0A=
style=3D"FONT-SIZE: 11pt; FONT-FAMILY: Arial">The procedure can be made =
more =0A=
flexible by the use of more variables:<o:p></o:p></SPAN></P>=0A=
<P class=3DMsoNormal =0A=
style=3D"MARGIN: 0cm 0cm 0pt; mso-layout-grid-align: none"><SPAN =0A=
style=3D"FONT-SIZE: 11pt; FONT-FAMILY: =
Arial"><o:p> </o:p></SPAN></P>=0A=
<P class=3DMsoPlainText style=3D"MARGIN: 0cm 0cm 0pt 36pt"><SPAN =0A=
style=3D"FONT-SIZE: 11pt; FONT-FAMILY: Arial">to =
crawl<o:p></o:p></SPAN></P>=0A=
<P class=3DMsoPlainText style=3D"MARGIN: 0cm 0cm 0pt 36pt"><SPAN =0A=
style=3D"FONT-SIZE: 11pt; FONT-FAMILY: Arial">if (distance :c) < :r =
[lt =0A=
:a]<o:p></o:p></SPAN></P>=0A=
<P class=3DMsoPlainText style=3D"MARGIN: 0cm 0cm 0pt 36pt"><SPAN =0A=
style=3D"FONT-SIZE: 11pt; FONT-FAMILY: Arial">if (distance :c) > :r =
[rt =0A=
:a]<o:p></o:p></SPAN></P>=0A=
<P class=3DMsoPlainText style=3D"MARGIN: 0cm 0cm 0pt 36pt"><SPAN =0A=
style=3D"FONT-SIZE: 11pt; FONT-FAMILY: Arial">fd =
:step<o:p></o:p></SPAN></P>=0A=
<P class=3DMsoPlainText style=3D"MARGIN: 0cm 0cm 0pt 36pt"><SPAN =0A=
style=3D"FONT-SIZE: 11pt; FONT-FAMILY: Arial">crawl<o:p></o:p></SPAN></P>=0A=
<P class=3DMsoPlainText style=3D"MARGIN: 0cm 0cm 0pt 36pt"><SPAN =0A=
style=3D"FONT-SIZE: 11pt; FONT-FAMILY: Arial">end<o:p></o:p></SPAN></P>=0A=
<P class=3DMsoPlainText style=3D"MARGIN: 0cm 0cm 0pt"><SPAN =0A=
style=3D"FONT-SIZE: 11pt; FONT-FAMILY: =
Arial"><o:p> </o:p></SPAN></P>=0A=
<P class=3DMsoPlainText style=3D"MARGIN: 0cm 0cm 0pt"><SPAN =0A=
style=3D"FONT-SIZE: 11pt; FONT-FAMILY: Arial">The following are sample =
values =0A=
:<o:p></o:p></SPAN></P>=0A=
<P class=3DMsoPlainText style=3D"MARGIN: 0cm 0cm 0pt"><SPAN =0A=
style=3D"FONT-SIZE: 11pt; FONT-FAMILY: =
Arial"><o:p> </o:p></SPAN></P>=0A=
<P class=3DMsoPlainText style=3D"MARGIN: 0cm 0cm 0pt 36pt"><SPAN =0A=
style=3D"FONT-SIZE: 11pt; FONT-FAMILY: Arial">Make “c [100 =0A=
100]<o:p></o:p></SPAN></P>=0A=
<P class=3DMsoPlainText style=3D"MARGIN: 0cm 0cm 0pt 36pt"><SPAN =0A=
style=3D"FONT-SIZE: 11pt; FONT-FAMILY: Arial">Make “r =
100<o:p></o:p></SPAN></P>=0A=
<P class=3DMsoPlainText style=3D"MARGIN: 0cm 0cm 0pt 36pt"><SPAN =0A=
style=3D"FONT-SIZE: 11pt; FONT-FAMILY: Arial">Make “a =
1<o:p></o:p></SPAN></P>=0A=
<P class=3DMsoPlainText style=3D"MARGIN: 0cm 0cm 0pt 36pt"><SPAN =0A=
style=3D"FONT-SIZE: 11pt; FONT-FAMILY: Arial">Make “step =
1<o:p></o:p></SPAN></P>=0A=
<P class=3DMsoPlainText style=3D"MARGIN: 0cm 0cm 0pt"><SPAN =0A=
style=3D"FONT-SIZE: 11pt; FONT-FAMILY: =
Arial"><o:p> </o:p></SPAN></P>=0A=
<P class=3DMsoPlainText style=3D"MARGIN: 0cm 0cm 0pt"><SPAN =0A=
style=3D"FONT-SIZE: 11pt; FONT-FAMILY: Arial">A start line like the =
following =0A=
would position the Turtle appropriately before beginning =0A=
CRAWL:<o:p></o:p></SPAN></P>=0A=
<P class=3DMsoPlainText style=3D"MARGIN: 12pt 0cm 0pt 36pt"><SPAN =0A=
style=3D"FONT-SIZE: 11pt; FONT-FAMILY: Arial">­cs pu setpos :c fd :r =
rt 90 pd =0A=
crawl <o:p></o:p></SPAN></P>=0A=
<P class=3DMsoPlainText style=3D"BACKGROUND: silver; MARGIN: 12pt 0cm =
0pt"><SPAN =0A=
style=3D"FONT-SIZE: 11pt; FONT-FAMILY: Arial"><o:p></o:p></SPAN></P>=0A=
<P class=3DMsoNormal =0A=
style=3D"MARGIN: 12pt 0cm 0pt; mso-layout-grid-align: none"><SPAN =0A=
style=3D"FONT-SIZE: 11pt; FONT-FAMILY: Arial">Experiment with these =0A=
variables.</SPAN></P>=0A=
<P class=3DMsoNormal =0A=
style=3D"MARGIN: 12pt 0cm 0pt; mso-layout-grid-align: none"><SPAN =0A=
style=3D"FONT-SIZE: 11pt; FONT-FAMILY: Arial">CRAWL can written in a =
slightly =0A=
different way in order to maintain a constant distance.</SPAN></P>=0A=
<P class=3DMsoNormal =0A=
style=3D"MARGIN: 12pt 0cm 0pt; mso-layout-grid-align: none"><SPAN =0A=
style=3D"FONT-SIZE: 11pt; FONT-FAMILY: Arial">This example is named =0A=
ORBIT:<o:p></o:p></SPAN></P>=0A=
<P class=3DMsoNormal =0A=
style=3D"MARGIN: 12pt 0cm 0pt 36pt; mso-layout-grid-align: none"><SPAN =0A=
style=3D"FONT-SIZE: 11pt; FONT-FAMILY: Arial">to =
orbit<o:p></o:p></SPAN></P>=0A=
<P class=3DMsoNormal =0A=
style=3D"MARGIN: 0cm 0cm 0pt 36pt; mso-layout-grid-align: none"><SPAN =0A=
style=3D"FONT-SIZE: 11pt; FONT-FAMILY: Arial">seth towards :c<SPAN =0A=
style=3D"mso-tab-count: =
3"> &nbs=
p;  =
; =0A=
</SPAN><I style=3D"mso-bidi-font-style: normal">;face towards =0A=
centre<o:p></o:p></I></SPAN></P>=0A=
<P class=3DMsoNormal =0A=
style=3D"MARGIN: 0cm 0cm 0pt 36pt; mso-layout-grid-align: none"><SPAN =0A=
style=3D"FONT-SIZE: 11pt; FONT-FAMILY: Arial">localmake "move ((distance =
:c) - =0A=
:r)<SPAN style=3D"mso-tab-count: 1"> </SPAN><I =0A=
style=3D"mso-bidi-font-style: normal">;calculate =
error<o:p></o:p></I></SPAN></P>=0A=
<P class=3DMsoNormal =0A=
style=3D"MARGIN: 0cm 0cm 0pt 36pt; mso-layout-grid-align: none"><SPAN =0A=
style=3D"FONT-SIZE: 11pt; FONT-FAMILY: Arial">pu fd :move <SPAN =0A=
style=3D"mso-tab-count: 1"> </SPAN><SPAN =0A=
style=3D"mso-tab-count: =
3"> &nbs=
p;  =
; =0A=
</SPAN>;<I style=3D"mso-bidi-font-style: normal">move towards or away =
from =0A=
centre<o:p></o:p></I></SPAN></P>=0A=
<P class=3DMsoNormal =0A=
style=3D"MARGIN: 0cm 0cm 0pt 36pt; mso-layout-grid-align: none"><SPAN =0A=
style=3D"FONT-SIZE: 11pt; FONT-FAMILY: Arial">rt 90 pd fd :step<SPAN =0A=
style=3D"mso-tab-count: =
3"> &nbs=
p;  =
; =0A=
</SPAN><I style=3D"mso-bidi-font-style: normal">;orient and =0A=
draw</I><o:p></o:p></SPAN></P>=0A=
<P class=3DMsoNormal =0A=
style=3D"MARGIN: 0cm 0cm 0pt 36pt; mso-layout-grid-align: none"><SPAN =0A=
style=3D"FONT-SIZE: 11pt; FONT-FAMILY: Arial">orbit<o:p></o:p></SPAN></P>=0A=
<P class=3DMsoNormal =0A=
style=3D"MARGIN: 0cm 0cm 0pt 36pt; mso-layout-grid-align: none"><SPAN =0A=
style=3D"FONT-SIZE: 11pt; FONT-FAMILY: Arial">end<o:p></o:p></SPAN></P>=0A=
<P class=3DMsoNormal =0A=
style=3D"MARGIN: 12pt 0cm 0pt; mso-layout-grid-align: none"><SPAN =0A=
style=3D"FONT-SIZE: 11pt; FONT-FAMILY: Arial">In this procedure the =
Turtle uses a =0A=
simple strategy to ensure a constant distance from the centre. The =
initial =0A=
position of the Turtle is irrelevant - this procedure always finds the =
right =0A=
starting point.<o:p></o:p></SPAN></P>=0A=
<DIV dir=3Dltr><FONT face=3DArial size=3D2></FONT> </DIV>=0A=
<DIV dir=3Dltr><FONT face=3DArial color=3D#000000 =
size=3D2></FONT> </DIV></DIV>=0A=
<DIV id=3DidSignature85861 dir=3Dltr>=0A=
<DIV RE>~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ </DIV>=0A=
<DIV RE>David Longman, Senior Lecturer in ICT</DIV>=0A=
<DIV RE>School of Education University of Wales, Newport</DIV>=0A=
<DIV RE>Caerleon Campus, PO Box 179</DIV>=0A=
<DIV RE>Newport</DIV>=0A=
<DIV RE>South Wales NP18 3YG</DIV>=0A=
<DIV RE>Tel: +44 (0)1633 432629</DIV>=0A=
<DIV RE>Fax: +44 (0)1633 432074</DIV>=0A=
<DIV RE>Web site: <A =0A=
href=3D"http://education.newport.ac.uk">http://education.newport.ac.uk</A=
> </DIV>=0A=
<DIV RE>~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~</DIV>=0A=
<DIV RE>=0A=
<DIV RE><PRE></PRE></DIV></DIV></DIV>=0A=
<DIV dir=3Dltr><BR>=0A=
<HR tabIndex=3D-1>=0A=
<FONT face=3DTahoma size=3D2><B>From:</B> =
mwforum-admin@lists.mathcats.com on behalf =0A=
of Jacques Mathil<BR><B>Sent:</B> Fri 18/02/2005 2:27 PM<BR><B>To:</B> =0A=
mwforum@lists.mathcats.com<BR><B>Subject:</B> Re: [MWForum]Circle and =
arc =0A=
help<BR></FONT><BR></DIV>=0A=
<DIV>=0A=
<DIV><FONT face=3DArial size=3D2>Hello,</FONT></DIV>=0A=
<DIV><FONT face=3DArial size=3D2>here is another solution. This =
procedure draws a =0A=
centered polygon :</FONT></DIV>=0A=
<DIV><FONT face=3DArial size=3D2></FONT> </DIV><PRE>to =
centered_polygon :number_sides :radius</PRE><PRE>make "alpha 360 / =
:number_sides</PRE><PRE>make "side 2 * :radius * sin (:alpha / =
2)</PRE><PRE>pu</PRE><PRE>fd :radius</PRE><PRE>rt 90 + :alpha / =
2</PRE><PRE>pd</PRE><PRE>repeat :number_sides [fd :side rt =
:alpha]</PRE><PRE>pu</PRE><PRE>left 90 + :alpha / 2</PRE><PRE>bk =
:radius</PRE><PRE>end</PRE><PRE> </PRE><FONT face=3D"Courier New" =
size=3D2>=0A=
<DIV><FONT face=3DArial>it uses some trigonometrics</FONT> <FONT =
face=3DArial>not =0A=
for young children, but you can use it as a tool.</FONT></DIV>=0A=
<DIV><FONT face=3DArial>To obtain a circle, try many sides : =
b.e =0A=
centered_polygon 90 100</FONT></DIV>=0A=
<DIV><FONT face=3DArial>you could modify this procedure to obtain =0A=
arcs.</FONT></DIV>=0A=
<DIV><FONT face=3DArial></FONT> </DIV></FONT><PRE><-- Some =
tricks--></PRE><PRE>to fig1 :r</PRE><PRE>if :r > 150 =
[stop]</PRE><PRE>centered_polygon 90 :r</PRE><PRE>fig1 :r + =
20</PRE><PRE>end</PRE><PRE> </PRE><PRE>to fig2 :r</PRE><PRE>if :r =
> 150 [stop]</PRE><PRE>centered_polygon 3 :r</PRE><PRE>rt =
10</PRE><PRE>fig2 :r + 5</PRE><PRE>end</PRE><PRE> </PRE><PRE>to =
fig3 :r</PRE><PRE>if :r > 150 [stop]</PRE><PRE>centered_polygon 90 =
:r</PRE><PRE>rt 20 fd 8</PRE><PRE>fig3 :r + 3</PRE><PRE>end</PRE>=0A=
<DIV><FONT face=3DArial size=3D2></FONT> </DIV>=0A=
<DIV><FONT face=3DArial size=3D2>Try them like this : fig1 =
0 =0A=
fig2 0 fig3 0</FONT></DIV>=0A=
<DIV><FONT face=3DArial size=3D2></FONT> </DIV>=0A=
<DIV><FONT face=3DArial size=3D2>Jacques Mathil<BR>Chemin de la Moniasse =
7<BR>1254 =0A=
Jussy</FONT></DIV>=0A=
<DIV><FONT face=3DArial size=3D2>GENEVA =
SWITZERLAND<BR>T=E9l. : +41 22 759 =0A=
81 81</FONT></DIV>=0A=
<BLOCKQUOTE dir=3Dltr =0A=
style=3D"PADDING-RIGHT: 0px; PADDING-LEFT: 5px; MARGIN-LEFT: 5px; =
BORDER-LEFT: #000000 2px solid; MARGIN-RIGHT: 0px">=0A=
<DIV style=3D"FONT: 10pt arial">----- Original Message ----- </DIV>=0A=
<DIV style=3D"BACKGROUND: #e4e4e4; FONT: 10pt arial"><B>From:</B> <A =0A=
title=3Dsusan@lcsi.ca href=3D"mailto:susan@lcsi.ca">Susan Einhorn</A> =
</DIV>=0A=
<DIV style=3D"FONT: 10pt arial"><B>To:</B> <A =
title=3Dmwforum@lists.mathcats.com =0A=
=
href=3D"mailto:mwforum@lists.mathcats.com">mwforum@lists.mathcats.com</A>=
</DIV>=0A=
<DIV style=3D"FONT: 10pt arial"><B>Sent:</B> Thursday, February 17, =
2005 5:14 =0A=
PM</DIV>=0A=
<DIV style=3D"FONT: 10pt arial"><B>Subject:</B> RE: [MWForum]Circle =
and arc =0A=
help</DIV>=0A=
<DIV><BR></DIV>=0A=
<DIV class=3DSection1>=0A=
<P class=3DMsoNormal><SPAN class=3DEmailStyle17><FONT face=3DArial =
color=3Dnavy =0A=
size=3D2><SPAN style=3D"FONT-SIZE: 10pt; FONT-FAMILY: Arial">This has =
been a great =0A=
discussion on how to create different procedures to draw a circle. =
This is a =0A=
great challenge for students. Here’s another circle-drawing =
procedure from a =0A=
new book about Math and MicroWorlds that’s being developed by =
LCSI =0A=
<SPAN> </SPAN>(authored by Wendy Petti):</SPAN></FONT></SPAN></P>=0A=
<P class=3DMsoNormal><SPAN class=3DEmailStyle17><FONT face=3DArial =
color=3Dnavy =0A=
size=3D2><SPAN =0A=
style=3D"FONT-SIZE: 10pt; FONT-FAMILY: =
Arial"></SPAN></FONT></SPAN> </P>=0A=
<P class=3DMsoNormal><SPAN class=3DEmailStyle17><FONT face=3DArial =
color=3Dnavy =0A=
size=3D2><SPAN style=3D"FONT-SIZE: 10pt; FONT-FAMILY: Arial">to =0A=
circle</SPAN></FONT></SPAN></P>=0A=
<P class=3DMsoNormal><SPAN class=3DEmailStyle17><FONT face=3DArial =
color=3Dnavy =0A=
size=3D2><SPAN =0A=
style=3D"FONT-SIZE: 10pt; FONT-FAMILY: =
Arial">pu</SPAN></FONT></SPAN></P>=0A=
<P class=3DMsoNormal><SPAN class=3DEmailStyle17><FONT face=3DArial =
color=3Dnavy =0A=
size=3D2><SPAN style=3D"FONT-SIZE: 10pt; FONT-FAMILY: Arial">repeat =
360 [ =0A=
<SPAN> </SPAN>fd 50 pd fd 0 pu bk 50 rt =
1]</SPAN></FONT></SPAN></P>=0A=
<P class=3DMsoNormal><SPAN class=3DEmailStyle17><FONT face=3DArial =
color=3Dnavy =0A=
size=3D2><SPAN =0A=
style=3D"FONT-SIZE: 10pt; FONT-FAMILY: =
Arial">end</SPAN></FONT></SPAN></P>=0A=
<P class=3DMsoNormal><SPAN class=3DEmailStyle17><FONT face=3DArial =
color=3Dnavy =0A=
size=3D2><SPAN =0A=
style=3D"FONT-SIZE: 10pt; FONT-FAMILY: =
Arial"></SPAN></FONT></SPAN> </P>=0A=
<P class=3DMsoNormal><SPAN class=3DEmailStyle17><FONT face=3DArial =
color=3Dnavy =0A=
size=3D2><SPAN style=3D"FONT-SIZE: 10pt; FONT-FAMILY: Arial">With an =0A=
input:</SPAN></FONT></SPAN></P>=0A=
<P class=3DMsoNormal><SPAN class=3DEmailStyle17><FONT face=3DArial =
color=3Dnavy =0A=
size=3D2><SPAN style=3D"FONT-SIZE: 10pt; FONT-FAMILY: Arial">to circle =0A=
:r</SPAN></FONT></SPAN></P>=0A=
<P class=3DMsoNormal><SPAN class=3DEmailStyle17><FONT face=3DArial =
color=3Dnavy =0A=
size=3D2><SPAN =0A=
style=3D"FONT-SIZE: 10pt; FONT-FAMILY: =
Arial">pu</SPAN></FONT></SPAN></P>=0A=
<P class=3DMsoNormal><SPAN class=3DEmailStyle17><FONT face=3DArial =
color=3Dnavy =0A=
size=3D2><SPAN style=3D"FONT-SIZE: 10pt; FONT-FAMILY: Arial">repeat =
360 [fd :r pd =0A=
fd 0 pu bk :r rt 1]</SPAN></FONT></SPAN></P>=0A=
<P class=3DMsoNormal><SPAN class=3DEmailStyle17><FONT face=3DArial =
color=3Dnavy =0A=
size=3D2><SPAN =0A=
style=3D"FONT-SIZE: 10pt; FONT-FAMILY: =
Arial">end</SPAN></FONT></SPAN></P>=0A=
<P class=3DMsoNormal><SPAN class=3DEmailStyle17><FONT face=3DArial =
color=3Dnavy =0A=
size=3D2><SPAN =0A=
style=3D"FONT-SIZE: 10pt; FONT-FAMILY: =
Arial"></SPAN></FONT></SPAN> </P>=0A=
<P class=3DMsoNormal><SPAN class=3DEmailStyle17><FONT face=3DArial =
color=3Dnavy =0A=
size=3D2><SPAN style=3D"FONT-SIZE: 10pt; FONT-FAMILY: Arial">If =
you’re not =0A=
familiar with the pd fd 0 trick, it creates a dot without displacing =
the =0A=
turtle. If the radius for this circle is too large, there are gaps =
between the =0A=
dots in the circumference. To make the dots closer together decrease =
the size =0A=
of the turn (e.g., rt 1 / 2)and increase the input to repeat =
accordingly =0A=
(repeat 360 * 2).</SPAN></FONT></SPAN></P>=0A=
<P class=3DMsoNormal><SPAN class=3DEmailStyle17><FONT face=3DArial =
color=3Dnavy =0A=
size=3D2><SPAN =0A=
style=3D"FONT-SIZE: 10pt; FONT-FAMILY: =
Arial"></SPAN></FONT></SPAN> </P>=0A=
<P class=3DMsoNormal><SPAN class=3DEmailStyle17><FONT face=3DArial =
color=3Dnavy =0A=
size=3D2><SPAN style=3D"FONT-SIZE: 10pt; FONT-FAMILY: Arial">That =
makes three (or =0A=
four, although the old Apple Logo method is a variation of =
Shawn’s method) =0A=
different ways to draw a circle. <SPAN> </SPAN>If you think about =
it, I’m =0A=
sure you could come up with even more.</SPAN></FONT></SPAN></P>=0A=
<P class=3DMsoNormal><SPAN class=3DEmailStyle17><FONT face=3DArial =
color=3Dnavy =0A=
size=3D2><SPAN =0A=
style=3D"FONT-SIZE: 10pt; FONT-FAMILY: =
Arial"><SPAN> =
&=
nbsp; &n=
bsp; &nb=
sp; &nbs=
p;  =
; =
=0A=
</SPAN></SPAN></FONT></SPAN></P>=0A=
<P class=3DMsoAutoSig><FONT face=3DArial color=3Dnavy size=3D2><SPAN =0A=
style=3D"FONT-SIZE: 10pt; COLOR: navy; FONT-FAMILY: =
Arial">Susan</SPAN></FONT><FONT =0A=
face=3DArial color=3Dnavy size=3D2><SPAN =0A=
style=3D"FONT-SIZE: 10pt; COLOR: navy; FONT-FAMILY: =
Arial"></SPAN></FONT></P>=0A=
<P class=3DMsoAutoSig><FONT face=3DArial color=3Dnavy size=3D2><SPAN =0A=
style=3D"FONT-SIZE: 10pt; COLOR: navy; FONT-FAMILY: =
Arial">-------------------------------</SPAN></FONT><FONT =0A=
face=3DArial color=3Dnavy><SPAN =0A=
style=3D"COLOR: navy; FONT-FAMILY: Arial"></SPAN></FONT></P>=0A=
<P class=3DMsoAutoSig><FONT face=3DArial color=3Dnavy size=3D2><SPAN =0A=
style=3D"FONT-SIZE: 10pt; COLOR: navy; FONT-FAMILY: Arial">Susan =0A=
Einhorn</SPAN></FONT><FONT face=3DArial color=3Dnavy size=3D2><SPAN =0A=
style=3D"FONT-SIZE: 10pt; COLOR: navy; FONT-FAMILY: =
Arial"></SPAN></FONT></P>=0A=
<P class=3DMsoAutoSig><FONT face=3DArial color=3Dnavy size=3D2><SPAN =0A=
style=3D"FONT-SIZE: 10pt; COLOR: navy; FONT-FAMILY: =
Arial">LCSI</SPAN></FONT><FONT =0A=
face=3DArial color=3Dnavy size=3D2><SPAN =0A=
style=3D"FONT-SIZE: 10pt; COLOR: navy; FONT-FAMILY: =
Arial"></SPAN></FONT></P>=0A=
<P class=3DMsoAutoSig><FONT face=3DArial color=3Dnavy size=3D2><SPAN =0A=
style=3D"FONT-SIZE: 10pt; COLOR: navy; FONT-FAMILY: =
Arial">www.MicroWorlds.com</SPAN></FONT><FONT =0A=
face=3DArial color=3Dnavy size=3D2><SPAN =0A=
style=3D"FONT-SIZE: 10pt; COLOR: navy; FONT-FAMILY: =
Arial"></SPAN></FONT></P>=0A=
<P class=3DMsoAutoSig><FONT face=3DArial color=3Dnavy size=3D2><SPAN =0A=
style=3D"FONT-SIZE: 10pt; COLOR: navy; FONT-FAMILY: =
Arial">susan@lcsi.ca</SPAN></FONT><FONT =0A=
face=3DArial color=3Dnavy size=3D2><SPAN =0A=
style=3D"FONT-SIZE: 10pt; COLOR: navy; FONT-FAMILY: =
Arial"></SPAN></FONT></P>=0A=
<P class=3DMsoNormal><SPAN class=3DEmailStyle17><FONT face=3DArial =
color=3Dnavy =0A=
size=3D2><SPAN =0A=
style=3D"FONT-SIZE: 10pt; FONT-FAMILY: =
Arial"></SPAN></FONT></SPAN> </P>=0A=
<P class=3DMsoNormal><SPAN class=3DEmailStyle17><FONT face=3DArial =
color=3Dnavy =0A=
size=3D2><SPAN =0A=
style=3D"FONT-SIZE: 10pt; FONT-FAMILY: =
Arial"><SPAN> =
&=
nbsp; &n=
bsp; &nb=
sp; &nbs=
p;  =
; =
&=
nbsp; &n=
bsp; =0A=
</SPAN></SPAN></FONT></SPAN></P>=0A=
<P class=3DMsoNormal style=3D"MARGIN-LEFT: 0.5in"><FONT face=3DTahoma =
color=3Dblack =0A=
size=3D2><SPAN =0A=
style=3D"FONT-SIZE: 10pt; COLOR: black; FONT-FAMILY: =
Tahoma">-----Original =0A=
Message-----<BR><B><SPAN style=3D"FONT-WEIGHT: bold">From:</SPAN></B> =0A=
mwforum-admin@lists.mathcats.com =0A=
[mailto:mwforum-admin@lists.mathcats.com]<B><SPAN =
style=3D"FONT-WEIGHT: bold">On =0A=
Behalf Of </SPAN></B>Logo Computer Systems Inc.<BR><B><SPAN =0A=
style=3D"FONT-WEIGHT: bold">Sent:</SPAN></B> Thursday, February 17, =
2005 10:34 =0A=
AM<BR><B><SPAN style=3D"FONT-WEIGHT: bold">To:</SPAN></B> =0A=
mwforum@lists.mathcats.com<BR><B><SPAN =0A=
style=3D"FONT-WEIGHT: bold">Subject:</SPAN></B> Re: [MWForum]Circle =
and arc =0A=
help</SPAN></FONT></P>=0A=
<P class=3DMsoNormal style=3D"MARGIN-LEFT: 0.5in"><FONT face=3D"Times =
New Roman" =0A=
size=3D3><SPAN style=3D"FONT-SIZE: 12pt"></SPAN></FONT> </P>=0A=
<P class=3DMsoNormal style=3D"MARGIN-LEFT: 1in; MARGIN-RIGHT: =
0.5in"><FONT =0A=
face=3D"Times New Roman" color=3Dblack size=3D3><SPAN =0A=
style=3D"FONT-SIZE: 12pt; COLOR: black">Hi!<BR>Could you program the =
circle with =0A=
a variable (or 2) to make it easy to make different sized circles? And =
how =0A=
about the arc?</SPAN></FONT><FONT color=3Dblack><SPAN =0A=
style=3D"COLOR: black"></SPAN></FONT></P>=0A=
<P class=3DMsoNormal style=3D"MARGIN-LEFT: 1in; MARGIN-RIGHT: =
0.5in"><FONT =0A=
face=3D"Times New Roman" color=3Dblack size=3D3><SPAN =0A=
style=3D"FONT-SIZE: 12pt; COLOR: black">Would the spell checker =
from =0A=
MSWord work with Pro or EX on Mac OS 10)?</SPAN></FONT><FONT =0A=
color=3Dblack><SPAN style=3D"COLOR: black"></SPAN></FONT></P>=0A=
<P class=3DMsoNormal style=3D"MARGIN-LEFT: 1in; MARGIN-RIGHT: =
0.5in"><FONT =0A=
face=3D"Times New Roman" color=3Dblack size=3D3><SPAN =0A=
style=3D"FONT-SIZE: 12pt; COLOR: black"></SPAN></FONT><FONT =
color=3Dblack><SPAN =0A=
style=3D"COLOR: black"></SPAN></FONT> </P>=0A=
<P class=3DMsoNormal style=3D"MARGIN-LEFT: 0.5in"><FONT face=3D"Times =
New Roman" =0A=
color=3Dblack size=3D3><SPAN style=3D"FONT-SIZE: 12pt; COLOR: =
black">Hi =0A=
Kate,</SPAN></FONT><FONT color=3Dblack><SPAN =0A=
style=3D"COLOR: black"></SPAN></FONT></P>=0A=
<P class=3DMsoNormal style=3D"MARGIN-LEFT: 0.5in"><FONT face=3D"Times =
New Roman" =0A=
color=3Dblack size=3D3><SPAN =0A=
style=3D"FONT-SIZE: 12pt; COLOR: black"></SPAN></FONT><FONT =
color=3Dblack><SPAN =0A=
style=3D"COLOR: black"></SPAN></FONT> </P>=0A=
<P class=3DMsoNormal style=3D"MARGIN-LEFT: 0.5in"><FONT face=3D"Times =
New Roman" =0A=
color=3Dblack size=3D3><SPAN style=3D"FONT-SIZE: 12pt; COLOR: black">I =
used the =0A=
following procedure in a -tools-.txt file and was able to create =
different =0A=
sized circles.</SPAN></FONT><FONT color=3Dblack><SPAN =0A=
style=3D"COLOR: black"></SPAN></FONT></P>=0A=
<P class=3DMsoNormal style=3D"MARGIN-LEFT: 0.5in"><FONT face=3D"Times =
New Roman" =0A=
color=3Dblack size=3D3><SPAN =0A=
style=3D"FONT-SIZE: 12pt; COLOR: black"></SPAN></FONT><FONT =
color=3Dblack><SPAN =0A=
style=3D"COLOR: black"></SPAN></FONT> </P>=0A=
<P class=3DMsoNormal style=3D"MARGIN-LEFT: 0.5in"><FONT face=3DTimes =
color=3Dblack =0A=
size=3D5><SPAN style=3D"FONT-SIZE: 18pt; COLOR: black; FONT-FAMILY: =
Times">to cir =0A=
:r<BR>repeat 360 [fd 2 * pi * :r / 360 rt 1]</SPAN></FONT><FONT =0A=
color=3Dblack><SPAN style=3D"COLOR: black"></SPAN></FONT></P>=0A=
<P class=3DMsoNormal style=3D"MARGIN-LEFT: 0.5in"><FONT face=3DTimes =
color=3Dblack =0A=
size=3D5><SPAN =0A=
style=3D"FONT-SIZE: 18pt; COLOR: black; FONT-FAMILY: =
Times">end</SPAN></FONT><FONT =0A=
color=3Dblack><SPAN style=3D"COLOR: black"></SPAN></FONT></P>=0A=
<P class=3DMsoNormal style=3D"MARGIN-LEFT: 0.5in"><FONT face=3D"Times =
New Roman" =0A=
color=3Dblack size=3D3><SPAN style=3D"FONT-SIZE: 12pt; COLOR: =
black">Unfortunately, =0A=
the spell checker features are only included with Windows versions of =
Pro and =0A=
EX. I used to use an OS 9 application called, SpellTools which =
is =0A=
available at no charge for download from the following address: =0A=
=
http://mac.softpedia.com/progDownload/SpellTools-Download-1097.html =
=0A=
It's a universal spell checker that works with all of your text =
oriented =0A=
applications. I have never tried it under OS =
X.</SPAN></FONT><FONT =0A=
color=3Dblack><SPAN style=3D"COLOR: black"></SPAN></FONT></P>=0A=
<P class=3DMsoNormal style=3D"MARGIN-LEFT: 0.5in"><FONT face=3D"Times =
New Roman" =0A=
color=3Dblack size=3D3><SPAN =0A=
style=3D"FONT-SIZE: 12pt; COLOR: black"></SPAN></FONT><FONT =
color=3Dblack><SPAN =0A=
style=3D"COLOR: black"></SPAN></FONT> </P>=0A=
<P class=3DMsoNormal style=3D"MARGIN-LEFT: 0.5in"><FONT face=3D"Times =
New Roman" =0A=
color=3Dblack size=3D3><SPAN =0A=
style=3D"FONT-SIZE: 12pt; COLOR: black">Regards.</SPAN></FONT><FONT =0A=
color=3Dblack><SPAN style=3D"COLOR: black"></SPAN></FONT></P>=0A=
<P class=3DMsoNormal style=3D"MARGIN-LEFT: 0.5in"><FONT face=3D"Times =
New Roman" =0A=
color=3Dblack size=3D3><SPAN =0A=
style=3D"FONT-SIZE: 12pt; COLOR: black"></SPAN></FONT><FONT =
color=3Dblack><SPAN =0A=
style=3D"COLOR: black"></SPAN></FONT> </P>=0A=
<P class=3DMsoNormal style=3D"MARGIN-LEFT: 0.5in"><FONT face=3D"Times =
New Roman" =0A=
color=3Dblack size=3D3><SPAN =0A=
style=3D"FONT-SIZE: 12pt; COLOR: black">Shawn</SPAN></FONT><FONT =0A=
color=3Dblack><SPAN style=3D"COLOR: black"></SPAN></FONT></P>=0A=
<P class=3DMsoNormal style=3D"MARGIN-LEFT: 0.5in"><FONT face=3D"Times =
New Roman" =0A=
color=3Dblack size=3D3><SPAN =0A=
style=3D"FONT-SIZE: 12pt; COLOR: black"></SPAN></FONT><FONT =
color=3Dblack><SPAN =0A=
style=3D"COLOR: black"></SPAN></FONT> </P><PRE =
style=3D"MARGIN-LEFT: 0.5in"><FONT face=3D"Courier New" color=3Dblack =
size=3D2><SPAN style=3D"FONT-SIZE: 10pt; COLOR: black">-- =
</SPAN></FONT><FONT color=3Dblack><SPAN style=3D"COLOR: =
black"></SPAN></FONT></PRE>=0A=
<P class=3DMsoNormal style=3D"MARGIN-LEFT: 0.5in"><FONT face=3D"Times =
New Roman" =0A=
color=3Dblack size=3D3><SPAN =0A=
style=3D"FONT-SIZE: 12pt; COLOR: black"></SPAN></FONT><FONT =
color=3Dblack><SPAN =0A=
style=3D"COLOR: black"></SPAN></FONT> </P>=0A=
<P class=3DMsoNormal style=3D"MARGIN-LEFT: 0.5in"><FONT face=3DGeneva =
color=3Dblack =0A=
size=3D2><SPAN style=3D"FONT-SIZE: 10pt; COLOR: black; FONT-FAMILY: =
Geneva">Logo =0A=
Computer Systems Inc.<BR>1 Westmount Square, Suite =
300<BR>Montreal,QC =0A=
Canada<BR>H3Z 2P9<BR>info@lcsi.ca<BR>www.lcsi.ca</SPAN></FONT><FONT =0A=
color=3Dblack><SPAN =0A=
style=3D"COLOR: =
black"></SPAN></FONT></P></DIV></BLOCKQUOTE></DIV></BODY></HTML>=0A=
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