Type-free Circles and Ellipses [T]

> >The entire discussion about whether a circle is an ellipse hinges on
 how
 you
> >define circle and ellipse.
> >
> >If you take the classical definition of an ellipse, from Euclidean
 geometry,
> >defined based on the sum of the distances from each point on the ellipse to
> >the two foci, then a circle is NOT an ellipse. A circle does not have two
> >foci.
>
>
> Unfortunately, not everyone would agree. I think a circle has two coincident
> foci, and an ellipse with coincident foci is a circle.
>
> For instance, if one stretches an ellipse such that the length of its major
> semiaxis equals the length of its minor semiaxis, the foci will shift such
> that they occupy the same point. Does the ellipse suddenly cease to be an
> ellipse when this happens? In what way does the resulting shape differ from
> a circle?
>
>
>
>

A Topmind view of shapes:

I defined an ellipse as "a rectangle with a 100 percent smoothing factor", and a circle as "a square with a 100 percent smoothing factor". This is a "has-a" viewpoint. A rectangle has-a zero percent smoothing and an ellipse has-a 100 percent.

You can get all kinds of nice hybrids
("tweeners") that way. Possibilities
open up if you move away from is-a
thinking.

However, I don't think it is practical
to define an explicit case for circle
when an ellipse can satisfy that.

An even more generic approach to shapes
is a bunch of "segments" where the segments can be curves or strait lines. In that
approach you don't really need even an
"ellipse type", because it can be made
out of four curved segments.

Add the continious smoothing to this
mix, and you can get just about any shape. If fact, the smoothing may be able to
replace curves by having a "bleed factor". the bleed factor may work better if
we define nodes instead of segments.
This needs a bit more exploration.

Thus, one does not even need "types"
for shapes. A shape is just a variable
number of segments (or nodes), in which each segment has indepedent attributes.

-T- Received on Sun Aug 26 2001 - 04:51:15 CEST