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Conic section Edge Flattened Curves
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Narasimham G.L.
science forum Guru Wannabe


Joined: 28 Apr 2005
Posts: 216

PostPosted: Sun Jul 16, 2006 10:19 pm    Post subject: Conic section Edge Flattened Curves Reply with quote

Planar curve( Conic sections) edges produced when a hollow cone is cut
by a plane are flattened out after cutting again along the cone's
generator.What equations describe such developed curves and do they
have a name ?
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Robert B. Israel
science forum Guru


Joined: 24 Mar 2005
Posts: 2151

PostPosted: Sun Jul 16, 2006 11:58 pm    Post subject: Re: Conic section Edge Flattened Curves Reply with quote

In article <1153088349.573264.218070@h48g2000cwc.googlegroups.com>,
Narasimham <mathma18@hotmail.com> wrote:
Quote:
Planar curve( Conic sections) edges produced when a hollow cone is cut
by a plane are flattened out after cutting again along the cone's
generator.What equations describe such developed curves and do they
have a name ?

I think you mean something like this. In spherical coordinates
(r, t, p) (with origin at the vertex and north pole along
the axis of the cone) where t is colatitude and p is
longitude the cone is t = c where 0 < c < pi/2 is constant.
The plane, assuming it doesn't pass through the vertex, is
r (a_1 sin(t) cos(p) + a_2 sin(t) sin(p) + a_3 cos(t)) = 1
for some constants a_1, a_2, a_3. Unrolling the cone maps
(r, c, p) to (r, (p - p_0) sin(c)) in polar coordinates on the
plane. Thus the resulting curve has equation

r (a_1 sin(c) cos(p_0 + theta/sin(c)) + a_2 sin(c) sin(p_0 +
theta/sin(c)) + a_3 cos(c)) = 1

in polar coordinates (r, theta).

Robert Israel israel@math.ubc.ca
Department of Mathematics http://www.math.ubc.ca/~israel
University of British Columbia Vancouver, BC, Canada
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Narasimham G.L.
science forum Guru Wannabe


Joined: 28 Apr 2005
Posts: 216

PostPosted: Tue Jul 18, 2006 1:16 am    Post subject: Re: Conic section Edge Flattened Curves Reply with quote

Robert Israel wrote:
Quote:
In article <1153088349.573264.218070@h48g2000cwc.googlegroups.com>,
Narasimham <mathma18@hotmail.com> wrote:

Planar curve( Conic sections) edges produced when a hollow cone is cut
by a plane are flattened out after cutting again along the cone's
generator.What equations describe such developed curves and do they
have a name ?

I think you mean something like this. In spherical coordinates
(r, t, p) (with origin at the vertex and north pole along
the axis of the cone) where t is colatitude and p is
longitude the cone is t = c where 0 < c < pi/2 is constant.
The plane, assuming it doesn't pass through the vertex, is
r (a_1 sin(t) cos(p) + a_2 sin(t) sin(p) + a_3 cos(t)) = 1
for some constants a_1, a_2, a_3. Unrolling the cone maps
(r, c, p) to (r, (p - p_0) sin(c)) in polar coordinates on the
plane. Thus the resulting curve has equation

r (a_1 sin(c) cos(p_0 + theta/sin(c)) + a_2 sin(c) sin(p_0 +
theta/sin(c)) + a_3 cos(c)) = 1

in polar coordinates (r, theta).

Robert Israel

Ok, can it now be reduced to a neat form involving only eccentricity
and minimum distance to vertex (for an even solution) ? with p_0 = 0 ,
a_i ' s absorbed by algebra & trig Smile. c is semi-vertical angle of
cone. Regards
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