cone-cone intersections

[Wesley Smith]

I was wondering if anyone knows how to perform cone-cone intersections
with GA. Let's say I have a circle C and a line L that passes through
the area of the circle and is thus neither perpendicular or parallel
to its axis. If I generate a circular rotation versor with
exp(-inf.C*k) where k is a scalar factor and apply it to the line L, L
will sweep out a double cone. Is there a way I can create an
equation for this cone and then use two equations of such cones to
perform an intersection with GA?

thanks!
wes

[Lanco]

Hi wes

Here is a link to a a generalization of
conformal GA by Zamora 2011
http://www.gaalop.de/wp-content/uploads/134-1061-Zamora.pdf
Cones are not worked out here,
but replacing c with ic and -1 with 0
in eq. [45-48] seems to give a
cone (an elliptical cone).
Intersections are as in eq. 61.

(Perwass, C. Geometric Algebra with
Applications in Engineering 2009, has
a chapter about conic space
of more technical nature.)

lanco

[Lanco]

By lanco
I have emailed Zamora with the text below,
and he has answered
"You are totally right!!!,
I will clarify that my apologies".

In your paper G6,3 GEOMETRIC ALGEBRA ...
is claimed, that the G6,3 model provides
a good framework to perform any
conformal transformation.
It seems only to work smoothly for quadric
surfaces with axis along directions e1,e2,e3.
The cause is that a rotation an angle strictly
between 0 and pi can not be performed by
a linear transformation of G6,3.