Re: 5 point conics; constructing its tangent

Xah Lee wrote :
On Feb 8, 8:41 pm, Virgil <Vir...@xxxxxxx> wrote:
In article <080220082303217382%cbarron...@xxxxxxxxxxxx>,
Carl Barron <cbarron...@xxxxxxxxxxxx> wrote:

In article
<582ad14a-5ba2-4fea-b111-ed9c8335c...@xxxxxxxxxxxxxxxxxxxxxxxxxxx>, Xah
Lee <x...@xxxxxxxxxx> wrote:

recently i read, that given a circle, one cannot construct a its
center.

What sort of construction rules prohibits that construction?

the construction is using straight edge only, i read here:
http://www.cut-the-knot.org/impossible/straightedge.shtml

i didn't read the proof anyway... but then it is possible to construct
anything a straightedge and compass can do, given a circle and using
straight edge...

All points constructible with compass and straightedge are
constructible with straightedge alone, given a circle
*with its center* (quite easy),
or given two intersecting circles (I don't know how),
or given two concentric circles.

But otherwise (if you don't know the center) it is impossible.

And also, all points constructible with compass and straightedge are
constructible with compass alone.

A recent thread here asked to construct tangents to circle
from a given point with straightedge alone (quite easy),
*not* knowing its center.

anyway, i'm interested in constructing conic's focus, using
straightedge and compass.


As Carl Barron wrote, it is "easy" analitically (with equations)
but this doesn't answer the question : how to do practically
with compass and straightedge.

The key to do that is projective geometry.
First of all we have to construct the axes of the conic section.
And even before that, the center (if it is not a parabola).
Then knowing axes, elementary methods give the focii.

I had this problem a couple years ago, asked, and received valuable
answers.
<http://www.math.niu.edu/~rusin/known-math/95/ellips.ctr>

Rather complicated.

My own problem was to construct the intersection points of a given
line with the conic section defined through 5 points.
That is in <http://chephip.free.fr/pbg_en/jsp145.html>

May be this will help you.
(I thought I had the focii construction also on my web site, but it
is indeed in the todo list)

Regards.

--
Philippe C., mail : chephip+news@xxxxxxx
site : http://chephip.free.fr/ (recreational mathematics)


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