Mixing
When are cone geodesics planar
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I mentioned to my (high school) students today that the intersection of a plane and a cone gives a conic section.
One asked whether if you 'unroll the cone' the conic section becomes a straight line on the resulting circular sector.
I can find examples that show this is false in general, but are there instances where it is true?
More specifically, given the cone defined by $alpharho=z$ in cylindrical polars and the points $A=(rho_0,0,alpharho_0), B=(rho_0,phi_0,alpharho_0)$, is the geodesic from $A$ to $B$ ever a planar curve?
conic-sections plane-curves geodesic
asked Jan 21 at 19:35
DominicRDominicR
264
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add a comment |
$begingroup$
I mentioned to my (high school) students today that the intersection of a plane and a cone gives a conic section.
One asked whether if you 'unroll the cone' the conic section becomes a straight line on the resulting circular sector.
I can find examples that show this is false in general, but are there instances where it is true?
More specifically, given the cone defined by $alpharho=z$ in cylindrical polars and the points $A=(rho_0,0,alpharho_0), B=(rho_0,phi_0,alpharho_0)$, is the geodesic from $A$ to $B$ ever a planar curve?
conic-sections plane-curves geodesic
asked Jan 21 at 19:35
DominicRDominicR
264
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I suppose you are excluding degenerate conics where the plane of the section passes through the vertex of the cone in such a way that the conic section is a straight line in that plane. (This seems to be implied by your choice of notation.)
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– David K
Jan 21 at 19:40
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Indeed. (I guess I don't want the degenerate cases where the cone is flat or simply a line either.)
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– DominicR
Jan 21 at 19:58
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Try coming at it from the other direction: are there any line segments contained within a circular sector that are planar when “rolled up?” Any radius obviously qualifies, but those correspond to degenerate conics.
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– amd
Jan 21 at 21:12
add a comment |
$begingroup$
I mentioned to my (high school) students today that the intersection of a plane and a cone gives a conic section.
One asked whether if you 'unroll the cone' the conic section becomes a straight line on the resulting circular sector.
I can find examples that show this is false in general, but are there instances where it is true?
More specifically, given the cone defined by $alpharho=z$ in cylindrical polars and the points $A=(rho_0,0,alpharho_0), B=(rho_0,phi_0,alpharho_0)$, is the geodesic from $A$ to $B$ ever a planar curve?
conic-sections plane-curves geodesic
asked Jan 21 at 19:35
DominicRDominicR
264
$endgroup$
I mentioned to my (high school) students today that the intersection of a plane and a cone gives a conic section.
One asked whether if you 'unroll the cone' the conic section becomes a straight line on the resulting circular sector.
I can find examples that show this is false in general, but are there instances where it is true?
More specifically, given the cone defined by $alpharho=z$ in cylindrical polars and the points $A=(rho_0,0,alpharho_0), B=(rho_0,phi_0,alpharho_0)$, is the geodesic from $A$ to $B$ ever a planar curve?
conic-sections plane-curves geodesic
conic-sections plane-curves geodesic
asked Jan 21 at 19:35
DominicRDominicR
264
asked Jan 21 at 19:35
DominicRDominicR
264
asked Jan 21 at 19:35
DominicRDominicR
264
asked Jan 21 at 19:35
DominicRDominicR
264
asked Jan 21 at 19:35
DominicRDominicR
264
264
$begingroup$
I suppose you are excluding degenerate conics where the plane of the section passes through the vertex of the cone in such a way that the conic section is a straight line in that plane. (This seems to be implied by your choice of notation.)
$endgroup$
– David K
Jan 21 at 19:40
$begingroup$
Indeed. (I guess I don't want the degenerate cases where the cone is flat or simply a line either.)
$endgroup$
– DominicR
Jan 21 at 19:58
$begingroup$
Try coming at it from the other direction: are there any line segments contained within a circular sector that are planar when “rolled up?” Any radius obviously qualifies, but those correspond to degenerate conics.
$endgroup$
– amd
Jan 21 at 21:12
add a comment |
$begingroup$
I suppose you are excluding degenerate conics where the plane of the section passes through the vertex of the cone in such a way that the conic section is a straight line in that plane. (This seems to be implied by your choice of notation.)
$endgroup$
– David K
Jan 21 at 19:40
$begingroup$
Indeed. (I guess I don't want the degenerate cases where the cone is flat or simply a line either.)
$endgroup$
– DominicR
Jan 21 at 19:58
$begingroup$
Try coming at it from the other direction: are there any line segments contained within a circular sector that are planar when “rolled up?” Any radius obviously qualifies, but those correspond to degenerate conics.
$endgroup$
– amd
Jan 21 at 21:12
$begingroup$
I suppose you are excluding degenerate conics where the plane of the section passes through the vertex of the cone in such a way that the conic section is a straight line in that plane. (This seems to be implied by your choice of notation.)
$endgroup$
– David K
Jan 21 at 19:40
$begingroup$
I suppose you are excluding degenerate conics where the plane of the section passes through the vertex of the cone in such a way that the conic section is a straight line in that plane. (This seems to be implied by your choice of notation.)
$endgroup$
– David K
Jan 21 at 19:40
$begingroup$
Indeed. (I guess I don't want the degenerate cases where the cone is flat or simply a line either.)
$endgroup$
– DominicR
Jan 21 at 19:58
$begingroup$
Indeed. (I guess I don't want the degenerate cases where the cone is flat or simply a line either.)
$endgroup$
– DominicR
Jan 21 at 19:58
$begingroup$
Try coming at it from the other direction: are there any line segments contained within a circular sector that are planar when “rolled up?” Any radius obviously qualifies, but those correspond to degenerate conics.
$endgroup$
– amd
Jan 21 at 21:12
$begingroup$
Try coming at it from the other direction: are there any line segments contained within a circular sector that are planar when “rolled up?” Any radius obviously qualifies, but those correspond to degenerate conics.
$endgroup$
– amd
Jan 21 at 21:12
add a comment |
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$begingroup$
I suppose you are excluding degenerate conics where the plane of the section passes through the vertex of the cone in such a way that the conic section is a straight line in that plane. (This seems to be implied by your choice of notation.)
$endgroup$
– David K
Jan 21 at 19:40
$begingroup$
Indeed. (I guess I don't want the degenerate cases where the cone is flat or simply a line either.)
$endgroup$
– DominicR
Jan 21 at 19:58
$begingroup$
Try coming at it from the other direction: are there any line segments contained within a circular sector that are planar when “rolled up?” Any radius obviously qualifies, but those correspond to degenerate conics.
$endgroup$
– amd
Jan 21 at 21:12