Prove that a straight line on a Lambert conformal conic is a great circle
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Prove that a straight line on a Lambert conformal conic is a great circle
During a heavy groundschool session recently somebody challenged my assertion that a straight line on a Lambert conformal conic projection was a great circle and asked me to prove it. I said I'd get back to them but at the moment I can't see how to prove it. Help!
HFD
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Simply, on a normal chart the longitude lines are vertical. On a conical projection map the longitude lines taper towards the poles. If you could wrap a conical map with a line drawn on it around a sphere then it would approximate a great circle line.
Saying that however, willing to be shot down in flames.
Saying that however, willing to be shot down in flames.
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It's O'level maths. You need to draw a globe and then explain how the conical projection map is made and how that relates to a globe and then you can use geometry to mathematically prove that it is a great circle line. If he still doesn't get it then he needs to learn more maths or just believe you.
AP3456G Part 1 Section 3 Chapter 2. This actually talks about conversion angle, the difference between a straight line and a great circle and explains why it can be ignored on a Lamberts Conformal.
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All I can offer is that every text book on Nav says that it is!
The chart was specifically designed to minimise the errors associated with plotting radio bearings. Scale Deviation has been designed not to exceed 1% and is achieved by two factors:
The ratio of the spacing of the SPs to the Latitude range of the chart does not exceed 1:6
The maximum spacing of the SPs does not exceed 14 degrees
If these parameters are applied, the SD will not exceed 1% and any variation between a straight line on the chart and a great circle will not exceed 0.75 degrees and can therefore be considered constant. Not exactly O level maths!
(Ref Maps and Charts - RAF College of Air Warfare) probably the same source as the Bristol notes.
The ratio of the spacing of the SPs to the Latitude range of the chart does not exceed 1:6
The maximum spacing of the SPs does not exceed 14 degrees
If these parameters are applied, the SD will not exceed 1% and any variation between a straight line on the chart and a great circle will not exceed 0.75 degrees and can therefore be considered constant. Not exactly O level maths!
(Ref Maps and Charts - RAF College of Air Warfare) probably the same source as the Bristol notes.
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As Whopity says plus:
A conformal conic projection is not exactly a conic projection, i.e. the projection of a sphere onto a cone. The conformal condition (i.e. the same scale locally in any direction) requires the solution of a partial differential equation, which is definitely beyond O-level maths.
It is not possible to model the surface of a sphere exactly onto any flat surface (which includes cones because you can unwrap them). So arcs of arbitrary Great Circles will not, in general, be straight lines in any chart projection, though they can be "straight enough" within the confines of the chart, as Whopity said.
A conformal conic projection is not exactly a conic projection, i.e. the projection of a sphere onto a cone. The conformal condition (i.e. the same scale locally in any direction) requires the solution of a partial differential equation, which is definitely beyond O-level maths.
It is not possible to model the surface of a sphere exactly onto any flat surface (which includes cones because you can unwrap them). So arcs of arbitrary Great Circles will not, in general, be straight lines in any chart projection, though they can be "straight enough" within the confines of the chart, as Whopity said.
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Thanks everyone.
The person who asked for proof has now accepted the statement (and also that a pilot doesn't need the proof) but it's raised my interest and it would be good to get to the bottom of it. Does anyone want to volunteer the maths? (I have A level maths and first degree level applied engineering maths but it seems a long time ago
)
HFD
who used to eat partial differentials, path integrals, grad, div, curl, poynting, stoke's, green's and assorted other delicacies for breakfast - but now prefers Alpen
The person who asked for proof has now accepted the statement (and also that a pilot doesn't need the proof) but it's raised my interest and it would be good to get to the bottom of it. Does anyone want to volunteer the maths? (I have A level maths and first degree level applied engineering maths but it seems a long time ago
)HFD
who used to eat partial differentials, path integrals, grad, div, curl, poynting, stoke's, green's and assorted other delicacies for breakfast - but now prefers Alpen
Thread Starter
O levels were exams that were around when A level results had value, when all (most) degree courses were demanding, when people went to University for the education rather than the parties, when around 5% of school leavers went to University, and when most fresh grads had a choice of jobs because of the foregoing. I was tempted to add something about grants or the 11+ filter but thought I'd probably upset enough people already.
HFD
(tin hat firmly attached and egress plan intact)
HFD
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If a great circle is defined as the shortest route between 2 points on a globe whose characteristic is that every line of latitude it crosses is the same (and the centre of the arc is the centre of the earth)you have your answer. Draw any straight line between any 2 points on a Lambert chart and the angle it crosses each line of latitude is the same. Or has my memory of Nav faded that much?
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The GC as a straight line is only an approximation and only holds over arcs of about 20-30 degrees before it will noticeably deviate.
As an example; the equator is a great circle but will appear as an arc on a Lambert chart unless the standard parallels are at equal lattitudes N & S of the equator (in which case it is a cylindrical rather than conic projection).
In pure geometry the great circle is a plane that intersects the centre of the earth. If the area of interest is near the standard parallels of the Lambert, then it will be orthogonal to the cone in that region and hence appear close to a straight line. But if you look at the antipodean point of the great circle, it will intersect the cone at an oblique angle and the locus will be a distinct curve.
In simpler terms, it only holds true if you remain close to the standard parallels.
As an example; the equator is a great circle but will appear as an arc on a Lambert chart unless the standard parallels are at equal lattitudes N & S of the equator (in which case it is a cylindrical rather than conic projection).
In pure geometry the great circle is a plane that intersects the centre of the earth. If the area of interest is near the standard parallels of the Lambert, then it will be orthogonal to the cone in that region and hence appear close to a straight line. But if you look at the antipodean point of the great circle, it will intersect the cone at an oblique angle and the locus will be a distinct curve.
In simpler terms, it only holds true if you remain close to the standard parallels.
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I make no claims to detailed navigational knowledge but.....
Surely another reason that the relationship between GC and line on a chart can only be an approximation is that the projection is based on a sphere whilst the earth is not a perfect sphere but rather an oblate spheroid.
It's taken 35 years but I knew that little fact would come in useful sometime.
Surely another reason that the relationship between GC and line on a chart can only be an approximation is that the projection is based on a sphere whilst the earth is not a perfect sphere but rather an oblate spheroid.
It's taken 35 years but I knew that little fact would come in useful sometime.
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When we talk about "Great Circles", we are working under the approximative assumption that the Earth is a sphere. Furthermore, the Lambert Conic projection is not unique. It consists of a plane defined by two secant lines (standard parallel circles) and an origin (those latitude parallels can of course also be ellipsoids in case of the "true" Earth, but then the concept of "Great Circle" becomes problematic; Unless by "Great Circle" we mean "Geodesic", i.e. the curve with the shortest distance between two points on the surface).
Carthographers will typically choose at least one the standard parallels to be close to the region of interest. They also apply additional corrections to comply with other (often legal) constraints. As such, Mark1's comment is spot on : its ok as long as you are close to an intersection parallel. Other projections have similar properties, i.e. that locally straight lines represent shortest distances on a map (e.g. stereographic projections near the pole).
The projection that hfd has in mind is a gnomonic projection. It is used by astronomers when plotting meteoric trajectories (straight lines in the sky become straight lines on a gnomonic stellar map). You seldom see them in an aviation context, because the projection distorts the "surfaces" of the earth quite dramatically, so that people don't "recognize" country contours any more.
Carthographers will typically choose at least one the standard parallels to be close to the region of interest. They also apply additional corrections to comply with other (often legal) constraints. As such, Mark1's comment is spot on : its ok as long as you are close to an intersection parallel. Other projections have similar properties, i.e. that locally straight lines represent shortest distances on a map (e.g. stereographic projections near the pole).
The projection that hfd has in mind is a gnomonic projection. It is used by astronomers when plotting meteoric trajectories (straight lines in the sky become straight lines on a gnomonic stellar map). You seldom see them in an aviation context, because the projection distorts the "surfaces" of the earth quite dramatically, so that people don't "recognize" country contours any more.
Last edited by proudprivate; 9th Sep 2011 at 02:19. Reason: removed irrelevant bits
