New thing needed in GSP

[Steve Gray]

Several times now I have wanted to scale anisotropically, that is, by different factors in
X and Y. (The images have no circles.) I think this can be done by drawing vertical lines from every
point in the sketch downward to a common horizontal line, putting points on those intersections, and
scaling the upper points by some factor, then redrawing the sketch with the new scaled points. Even
with macros this is time-consuming, and a special tool would be appreciated.
I have no suggestion about handling circles, which obviously become ellipses, which might
open the larger issue of conics. You may be trying to avoid this but somehow I feel it will be
necessary in future versions.

Steve Gray

[Paul Kunkel]

Anisotropically is what it's called? That's what I've called the
sketch. See the link below. I don't quite get the construction you
described. If it works, you need to select the pre-image and the image,
and construct the locus. The tool in this file will construct the
image. You will need to construct the locus by yourself. If this is
something that is being done many times in the same sketch, let me suggest
that you edit the tool an make the scales assumed objects.

http://whistleralley.com/temporary/Anisotropic.gsp

Paul Kunkel
<whis...@whistleralley.com>
Email to this address must include the word "geometry" in the message.

[Nicholas Jackiw]

On Wed, 09 Feb 2005 23:30:27 +0800, Paul Kunkel wrote:
>At 12:31 PM 2/9/2005, Steve Gray wrote:
>> Several times now I have wanted to scale anisotropically,
that
>> is, by different factors in X and Y. (The images have no circles.)
I
>

> Anisotropically is what it's called? That's what I've called
the
>sketch. See the link below.>

><a
href="http://whistleralley.com/temporary/Anisotropic.gsp">http://whistleralley.com/temporary/Anisotropic.gsp</a>

The tool is Paul's sketch takes as given

1. Measurement x-scale
2. Measurement y-scale
3. Point dilation center
4. Point pre-image

and constructs the image point of pre-image (given#4) via the
anisotropic (non-uniform) scaling determined by givens #1-3. Being
able to work point-by-point is powerful if you want to construct and
measure the anisotropic geometry that results. For instance, since
these transforms are still affine, a segment goes to a segment under
this mapping. So you could build an Anisotropic Segment tool, by
wielding Paul's basic Anisotropic Point tool on the pre-image
segment's endpoints to construct the image segment's endpoints. Then
construct a Euclidean segment between those two image points. (Having
done this once, to define the tool: select the pre-image segment and
endpoints, the image segment and endpoints, and the center point and
two scale factors; then "Create New Tool.")

A more flexible extension of this idea is to construct an Anisotropic
Path tool. Since in Sketchpad, segments, rays, lines, axes, circles,
arcs, polygon perimeters, function plots and point loci are all paths,
such a tool can build the anisotropic image of a wide variety of
objects. The illustration in Paul's existing sketch has all you need
to define the tool: select the two scale factors and center point,
then select the green pre-image circle (your example path) and the red
image Anisotropic Circle to its left (your example transformed image
path). Create New Tool. Your new tool should have four givens as
above, but with a "Path Object" as pre-image (Given #4) rather than a
point. Turn the first three Givens into Assumed Objects so that you
have a tool with only one explicit Given, and you can click this tool
on any Sketchpad path object in order to construct its anisotropic
image. Since images defined this way are defined as point loci, and
since point loci are themselves paths, the tool can build anisotropic
images of any anisotropic image.

If I were introducing these ideas in a classroom, I'd stay away from
custom tools for a bit and start by plotting some points and some
familiar curves on the coordinate system. Then I'd change the
coordinate system (Graph menu | Grid Form) from Square to
Rectangular, and vary one axis' scaling while keeping the other one
constant. At the school level, there's a nice discussion to be had
about whether y = +/- sqrt(1-x^2) is still a circle when you've
stretched it into an apparent ellipse by creating an anisotropic
coordinate system. After establishing some intuition about how the
transformation works, I'd switch to trying to construct it -- and to
tools like these.

Nick Jackiw
Sketchpad Projects

ps. Thanks to all the people who pointed out to me by e-mail a simpler
way to create a random number that changes by push-button than I
described yesterday. Start with a Parameter, and make an Action Button
that animates that (random, once only) in a given domain. No points,
ratios, or range-shifting calculations needed. Ooops.