[help-3dldf] Re: Polyhedra and ellipses

[Laurence Finston]

On Tue, 7 Dec 2004, Martijn van Manen wrote:

>
> I'll try to answer some of your questions in another
> installment. What I would like to say though is that the
> number of fun and exciting polyhedra is rather endless.
> I'll make some selection.

I've made a start on the Archimedean polyhedra with the truncated
octahedron.  They ought to keep me busy for a while.

> Ofcourse there are "smart" methods
> of finding the vertices and the edges and so on.
> Those mathematicians in the nineteenth century produced
> formula after formula. There are endless and sometimes
> pointless calculations.

This sounds promising.

> What you should maybe figure out is how to construct a nice
> data structure for them, so that afterwards you can traverse it
> and find intersections with lines and planes. The people in
> computational geometry love the "doubly connected edge list".
> You might want to implement it.

Once I have a basic understanding of the methods I'll be able to figure
out a data structure to use.

> For the tangencies of two ellipses, or two quadrics, I'll
> give you some methods too.

Thanks.  I plan to implement classes for parabolae and hyperbolae, so I
may implement a base class `Conic_Section'.  If
appropriate, I'll implement a function for finding intersections of
`Conic_Sections' and overload it with specializations for two
`Ellipses', an `Ellipse' and a `Parabola', etc.  If `Quadric' is a better
name, I'll use it;  I don't know the its definition yet.

Laurence