Re: LOGO-L> Re: Total Turtle Trip

[George Mills <mills@softronix.com>]

Mike Doyle wrote:
> 
> In-Reply-To: <memo.275746@cix.compulink.co.uk>
> Total Turtle Trip
> 
> The Turtle has two capabilities:
> 
> To move in a positive or negative direction.
> To turn in a clockwise or anticlockwise direction.
> 
> The commands are, respectively:
> Forward (back) and
> Right (left)
> 
> The movement is expressed in arbitrary units.
> The turn is expressed as degrees of arc.
> 
> It follows that Turtle geometry implicitly relies upon a (conventional)
> understanding of a circle, i.e. as an arc described by a rotating point.
> That is to say that the Turtle has inbuilt the 'knowledge' developed
> through using a compass to draw and divide circles.
> 
> The turtle, however, cannot move in a circular arc. It can only move by
> 'step and turn', leading to procedures of the form:
> 
> To shape :sides
> Repeat :sides (forward 1 right 360/:sides)
> End
> 

I agree with your statements and it's some of these reasons I include
an ARC function in MSWLogo that does not use the poly method. You can
draw lines exactly tangent and other arcs exactly tangent. Exactly
is only as exact as 64bit floating point allows and the resolution of
the
display device.

ARC comes in 2 flavors, one called ARC which assumes the turtle
is at the center and lets the ARC sweep around it. ARC2 assumes
the turtle is ON the arc.

ARC is written using trig functions and SETXY. Each point is exactly
on the ARC (exact as 64bit floating point allows). I have posted
ARC written in Logo many moons ago it's pretty simple.

ARC is actually written from ELLIPSEARC which allows you to specify
two Radius and where to start the sweep and how long to sweep.
>From ELLIPSEARC you can create simple procedures for ELLIPSE, ARC,
CIRCLE and ELLIPSEARC2, ELLIPSE2, ARC2 and CIRCLE2 (the last 4
drawing from the edge).

Since the number crunching is built it allows you to draw ARCs
extremely fast. ARC also understand the resolution of your screen
and does not waste vectors on small ARCs and generates more vectors
on large arcs so that it always looks smooth.

ARC is still an approximation along the line segments.
But has the characteristic that if you say ARC2 90 100
you will be exactly at the same place as SETXY 100 100.
That is each vertex is exact (within 64bit floating point
accuracy).

Using the poly method the only point on the actual ARC is the first
one. However if you use enough points in the POLY method you
can approach the 64bit floating point error but then the floating
point errors themselves will prevent you from ever actually reaching
that limit.

> Which will draw a regular polygon.
> 
> The expression defines a generic polygon and is thereby a general
> definition of a polygon as a sequence of steps and turns. It follows that
> it is not possible to draw a tangent to the shape which will touch it only
> at a point. Either the tangent will touch a corner or it will touch a
> side. In the latter case the contact area will be one unit.
> 
> If we think of doing Turtle geometry using a school geometry set, we see
> that the instruments we would use would be ruler and protractor. The
> protractor is derived from a circular arc, which the compass produced. On
> the computer the protractor is 'hidden' (a pity because the kids then have
> to 'discover' the metric) and only the ruler, as steps along a (straight)
> number line, is available.
> 
> The 'circle' described in Mindstorms and Turtle Geometry is simply the
> perceptual illusion of a circle created when the Turtle draws a many-sided
> polygon on the computer screen. Papert's assertion that such a shape might
> help children to develop notions of the differential calculus are somewhat
> dubious. More clearly, this perceptual circle demonstrates the nature of
> digital representation - a sufficient number of discrete samples will
> produce the same sensation as the continuous original.
> 
> To summarise:
> All turtle shapes are of a step/turn form.
> The procedure definition specify this.
> A circle is a continuous arc centred on a point.
> It is not possible to pass from step/turn to continuity - the definitions
> (procedures) remain distinctly different in their form. Hence, a circle
> may not be described in Turtle geometry. (Yet Turtle geometry relies on
> Euclid's compass for its turn metric and Descartes' coordinates for its
> positioning on the screen.)
> 
> I refer, of course, to the language (Logo procedures) and not to the
> capabilities of the mind to imagine so small a step that it becomes a
> point. Unfortunately, this mental process leads to a 'circle' of zero
> circumference - otherwise known as a point.
> 
> The Turtle Circle is a nice conceit. I find the thought of teaching Turtle
> geometry to children, with the presumption that it has mathematical
> validity, extremely worrying. If, however, you can fault my thesis, I will
> be happy to revise my opinion.
> 
> Michael Doyle (aka Micheal O Duill)
> 37 Bright Street
> SKIPTON BD23 1QQ UK
> Tel/fax: +44 (0)1756 794601
> Email: mikedoyle@cix.co.uk
> 
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-- 
===============================================================
George Mills (mills@softronix.com)
http://www.softronix.com/
The www page contains some very powerful educational software.
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