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Definition of constant-curvature curve embedded on an Ellipsoid of revolution

How to define small circle on an ellipsoidA question about a curve on the surface of a spheresemi ellipsoid and cylinder parametrize the curveSurface of revolution with constant planar ellipse intersectionsEgg curve constant width.Can every curve be subdivided equichordally?Use scale in projection to solve for curvatureIs a surface of revolution from sine curve space packing?Solid angle definition from an ellipsoid surfaceConstant scalar curvature with positive Ricci curvatureHow to define small circle on an ellipsoid

0

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$begingroup$

I am interested in identifying a type of curve so I can do literature review on it.

What is the name of a curve embedded on an ellipsoid of revolution in which the curvature of the embedded curve is constant? Or where might I find such a curve described or analyzed?

Background:
On the plane, one of the characteristics of an arc (segment of a circle) is that curvature does not change as one traverses the length of the arc. In fields such as surveying, many roadway and railway curves are implemented as arcs. The radiuses of these are usually small relative to the size of the earth, so it is fine that planar arc segments are used for this. Errors are adjusted for at survey-project boundaries.

But for for long alignments (say a river, a mountain range, or an ice crevice), if planar curves are used, the small errors accumulate. It would be desirable to embed the alignment on the ellipsoid of the Earth. Similar issues present when approximating craters on any ellipsoidal body such as the Moon.

I am interested in finding the mathematical definition of a curve embedded on the ellipsoid with a center at any location in which the curve is defined as having a constant Degree of Curvature in the embedded context.

In other words, I want to take this one characteristic of planar circles and arc segments, constant curvature, and figure out what this is on the ellipsoid. Just the name of it, and a reference to its description if anyone happens to know it.

This questions is a follow-on to How to define small circle on an ellipsoid, which I now see was not formed as well as it should have been.

geometry circle noneuclidean-geometry geodesy

share|cite|improve this question

asked 2 days ago

philologonphilologon

15817

$endgroup$

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  $begingroup$
  Do you really need this for an ellipsoid? The Earth might be close enough to a sphere for your purposes. (Just curious.)
  $endgroup$
  – Ethan Bolker
  2 days ago
  

  
  
  
  


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  @EthanBolker, yes. If a sphere were close enough for my purposes I would still be designing roads to make other people rich.
  $endgroup$
  – philologon
  2 days ago
  

  
  
  
  


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  $begingroup$
  You have to use the so-called "Darboux frame" with the two curvatures which are $k_g$, the geodesic curvature and $k_n$, the normal curvature ; see for example web.cs.iastate.edu/~cs577/handouts/surface-curvature.pdf
  $endgroup$
  – Jean Marie
  2 days ago
  

  
  
  
  


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  $begingroup$
  Your issue could be considered in terms of "oblate spheroidal coordinates" en.wikipedia.org/wiki/Oblate_spheroidal_coordinates
  $endgroup$
  – Jean Marie
  yesterday
  
  

  
  
  
  


• 
  
  
  

  
  1
  

  
  
  
  
  
  

  
  $begingroup$
  I think the answer is not easy. You can find here some ideas, even if in that case constant geodesic curvature was required.
  $endgroup$
  – Aretino
  yesterday
  

  
  
  
  

 | 
show 4 more comments

0

0

$begingroup$

I am interested in identifying a type of curve so I can do literature review on it.

What is the name of a curve embedded on an ellipsoid of revolution in which the curvature of the embedded curve is constant? Or where might I find such a curve described or analyzed?

Background:
On the plane, one of the characteristics of an arc (segment of a circle) is that curvature does not change as one traverses the length of the arc. In fields such as surveying, many roadway and railway curves are implemented as arcs. The radiuses of these are usually small relative to the size of the earth, so it is fine that planar arc segments are used for this. Errors are adjusted for at survey-project boundaries.

But for for long alignments (say a river, a mountain range, or an ice crevice), if planar curves are used, the small errors accumulate. It would be desirable to embed the alignment on the ellipsoid of the Earth. Similar issues present when approximating craters on any ellipsoidal body such as the Moon.

I am interested in finding the mathematical definition of a curve embedded on the ellipsoid with a center at any location in which the curve is defined as having a constant Degree of Curvature in the embedded context.

In other words, I want to take this one characteristic of planar circles and arc segments, constant curvature, and figure out what this is on the ellipsoid. Just the name of it, and a reference to its description if anyone happens to know it.

This questions is a follow-on to How to define small circle on an ellipsoid, which I now see was not formed as well as it should have been.

geometry circle noneuclidean-geometry geodesy

share|cite|improve this question

asked 2 days ago

philologonphilologon

15817

$endgroup$

• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  Do you really need this for an ellipsoid? The Earth might be close enough to a sphere for your purposes. (Just curious.)
  $endgroup$
  – Ethan Bolker
  2 days ago
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  @EthanBolker, yes. If a sphere were close enough for my purposes I would still be designing roads to make other people rich.
  $endgroup$
  – philologon
  2 days ago
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  You have to use the so-called "Darboux frame" with the two curvatures which are $k_g$, the geodesic curvature and $k_n$, the normal curvature ; see for example web.cs.iastate.edu/~cs577/handouts/surface-curvature.pdf
  $endgroup$
  – Jean Marie
  2 days ago
  

  
  
  
  


• 
  
  
  

  
  1
  

  
  
  
  
  
  

  
  $begingroup$
  Your issue could be considered in terms of "oblate spheroidal coordinates" en.wikipedia.org/wiki/Oblate_spheroidal_coordinates
  $endgroup$
  – Jean Marie
  yesterday
  
  

  
  
  
  


• 
  
  
  

  
  1
  

  
  
  
  
  
  

  
  $begingroup$
  I think the answer is not easy. You can find here some ideas, even if in that case constant geodesic curvature was required.
  $endgroup$
  – Aretino
  yesterday
  

  
  
  
  

 | 
show 4 more comments

$begingroup$

I am interested in identifying a type of curve so I can do literature review on it.

What is the name of a curve embedded on an ellipsoid of revolution in which the curvature of the embedded curve is constant? Or where might I find such a curve described or analyzed?

Background:
On the plane, one of the characteristics of an arc (segment of a circle) is that curvature does not change as one traverses the length of the arc. In fields such as surveying, many roadway and railway curves are implemented as arcs. The radiuses of these are usually small relative to the size of the earth, so it is fine that planar arc segments are used for this. Errors are adjusted for at survey-project boundaries.

But for for long alignments (say a river, a mountain range, or an ice crevice), if planar curves are used, the small errors accumulate. It would be desirable to embed the alignment on the ellipsoid of the Earth. Similar issues present when approximating craters on any ellipsoidal body such as the Moon.

I am interested in finding the mathematical definition of a curve embedded on the ellipsoid with a center at any location in which the curve is defined as having a constant Degree of Curvature in the embedded context.

In other words, I want to take this one characteristic of planar circles and arc segments, constant curvature, and figure out what this is on the ellipsoid. Just the name of it, and a reference to its description if anyone happens to know it.

This questions is a follow-on to How to define small circle on an ellipsoid, which I now see was not formed as well as it should have been.

geometry circle noneuclidean-geometry geodesy

share|cite|improve this question

asked 2 days ago

philologonphilologon

15817

$endgroup$

share|cite|improve this question

asked 2 days ago

philologonphilologon

15817

share|cite|improve this question

asked 2 days ago

philologonphilologon

15817

asked 2 days ago

philologonphilologon

15817

asked 2 days ago

philologonphilologon

15817