[Blindmath] A 3d to 2d description resource?

Tue Jan 31 17:38:51 UTC 2012

Hi Pranav,

Here is another way to think about it. I suspect that at some point in your
life you used modeling clay to create 3d models of various objects. For
example, you probably rolled the clay on the table top with your hand to
create long skinny cylindrical shapes like snakes.

Assume that you have some modeling clay that has a very special property.
That property is that if it is touched by any object, it will leave a black
mark on the object where the object came in contact with the clay.

Assume that you can create a perfect cylinder with the clay and stand it up
on its end like a drinking glass.

You take a very thin but stiff sheet of metal with one sharp edge (a really
big knife) and you very carefully press the knife down on the top of the
cylinder cutting it into two half cylinders. Then you carefully remove the
clay from each side of the knife and examine the mark left on the knife by
the clay. If you did it carefully enough, that mark will be a rectangle.
More importantly, it will be the projection of the 3D cylinder onto the 2D
plane represented by the knife.

The width of the rectangle will be equal to the diameter of the cylinder.
The height of the rectangle will be equal to the height of the cylinder.

Now suppose you do the same thing, but this time you slice the cylinder
closer to the edge. You would still get a rectangle, but in this case, the
width of the rectangle would be equal to the length of the chord of the
circle at the point where you made the slice.

Now for another approach. The sun is sufficiently far from the earth that
for all practical purposes, the rays from the sun are parallel when they
reach the earth. That is necessary for this experiment to work properly.

Take your cylinder, or your drinking glass and carefully glue it to a piece
of white cardboard. Take the contraption out into the sunlight and prop it
up at an angle so that the direction of the sun's rays are perpendicular to
the surface of the cardboard. Then ask a sighted person to use a black
marking pen and carefully draw a line along the edge of the shadow that the
cylinder casts onto the cardboard.

Then remove the cylinder from the cardboard and view it using sonification.
You will find that to the extent that the marking was accurate, the black
mark forms a rectangle on the white cardboard. This is the projection of
the outer extremity (Whapples) of the 3D cylinder onto the 2D cardboard.

And yes, I was required to take a course in engineering drawing about 150
years ago when I received my engineering degree. And it was valuable in
understanding this sort of thing. However, a course with the title
 Analytic Geometry was even more valuable.That is where I learned to think
about the intersections of 3D objects, such as the intersection of a beer
can that is partly submerged in the sand at the beach with only the upper
portion of the can sticking up out of the sand at an angle. In this case,
the flat surface of the sand effectively slices the beer can at an angle
producing an elliptical shape at the intersection. Once again, we have the
projection of a 3D object (the beer can) on a 2D plane (the surface of the
sand).

Dick Baldwin

On Tue, Jan 31, 2012 at 10:36 AM, Michael Whapples <mwhapples at aim.com>wrote:

> Regarding the thing about the cylinder looking like a rectangle from the
> side. Try thinking of it by imagining viewing it from the side, what are
> the most extreme points of what would be visible? What shape do these
> create? A valid question might be, but why doesn't the bit coming towards
> me alter things? I have two answers to this. Firstly the simple one is to
> think of it as Richard described about how people see in 2D, when using two
> eyes you get an image from two slightly different angles and a clever bit
> of processing can say what 3D shape would create the difference between the
> two images. If you only have a single detector (eg. a single camera with a
> single lens) then the bulge in the cylinder cannot be detected as the
> distance cannot be identified. The more complicated answer, well some
> drawings you may encounter may show one side of the rectangle (representing
> the side of the cylinder) with one side darker than the other, this
> probably is to show where the shadow is forming if the light were coming
> from one side. If light were coming from all directions then no such
> shading should be drawn as no shadow would form.
>
>
> Michael Whapples
>
> -----Original Message----- From: Pranav Lal
> Sent: Tuesday, January 31, 2012 4:08 PM
> To: john.gardner at orst.edu ; 'Blind Math list for those interested in
> mathematics'
> Subject: Re: [Blindmath] A 3d to 2d description resource?
>
>
> Hi John,
> <snip
> If anybody would like to organize an archive of images and tutorials that
> could explain the principles of 3d to 2d projections understandable by
> blind
> people, it would be a good subject for the www.Access2Science.com web
> site.
> PL] I would be happy to do this. I was thinking about a wiki where we could
> start adding descriptions and linking to relevant SVG files.
>
> <snip This seems to be one of the most difficult
> things for most congenitively blind people to understand.  As a person who
> was sighted for much of my life, I understand it intuitively but have never
> tried to explain it to someone who didn't understand it.  The idea of
> projection should not be too difficult.  But projection is only a tiny part
> of drawing images.
> PL] I am congenitively blind so can relate to this. I was discussing this
> with my civil engineer dad and he mentioned that a cylindrical glass would
> look like a rectangle when seen in 2d. This is totally beyond me right now.
> Engineers learn all this in a subject called engineering drawing. Has
> anyone
> on this list studied it?
>
> Pranav
>
>
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-- 
Richard G. Baldwin (Dick Baldwin)
Home of Baldwin's on-line Java Tutorials
http://www.DickBaldwin.com

Professor of Computer Information Technology
Austin Community College
(512) 223-4758
mailto:Baldwin at DickBaldwin.com
http://www.austincc.edu/baldwin/