# Cross section (?) of Great Pyramid from drone footage

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• neilparker62

#### neilparker62

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Is this a 'valid' cross-section ?
Following is a frame carefully chosen from this drone footage. At about 0:29 into the video the drone is directly in front of one of the four faces (if you know Cairo, you will know which one) and moving from left to right. At that point I have paused and 'frame-stepped' till the moment before the drone swings around the right edge of the pyramid. So my question is , does the picture present an accurate cross-sectional view / silhouette of the Great Pyramid ? One obvious concern is that the faint lines one can see (presumably representing one or other particular layer of stonework) on the pyramid face are not quite horizontal.

My plan is to super-impose lines along the pyramid edges and then measure the apex angle as well as the slope angle. And hopefully confirm that the base line along with pyramid edges do indeed accurately constitute an isosceles triangle. Further that if a perpendicular is dropped from the apex, the iso triangle will be divided into two equal Kepler (golden ratio) triangles.

if you know Cairo, you will know which one)
And what if we DON'T know Cairo? Why would you expect us to?

sophiecentaur
Sorry - I don't expect necessarily and the thought was just an aside since the question I am putting is 'face independent'. (other than that statistically one might like to have similar profiles of all 4 faces)

Draw a side view showing the height of the drone wrt the pyramid. Is it looking down ?

You will need to know the exact position of the drone relative to the face. An equilateral triangle will image equilateral only from directly normal to the centroid

What I was thinking is that if the drone is placed such that it has a 'side view' of the pyramid then the profile in the pic would be a fair reflection of the cross-section - something like the below. I do not expect it to reflect the exact triangular shape of the face itself. For that the drone would have to point it's camera such that its line of sight is exactly normal to the plane of the slanting face.

The specific frame which I chose is at a single instant in the video when only the 1 face is visible to meet the 'side-on' criterion. I am assuming the camera is pointing (more or less) horizontally but that could be completely wrong.

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Draw a side view showing the height of the drone wrt the pyramid. Is it looking down ?
I can't answer this I'm afraid. I guess to get the frame I am looking for, one would have to ask the drone operator to position the drone about half way up the pyramid and have the camera pointing horizontally such that it's line of sight is normal to whichever stonework layer is at that particular height.

What is it you are trying to do? I'm not clear on the POINT of this thread. Is it to determine if the pyramid is symmetrical ? If that it, I would think that has been determined many times over by various students of the pyramid.

The issue here is perspective. The nice thing about perspective (without aberration) is that straight lines always map onto straight lines. So if you could somehow locate the two lower back corners you would have a better fit to the center cross-section. Obviously also if the drone is very far away (∞) at base level the elevation you seek and the face will superpose.

neilparker62
What is it you are trying to do? I'm not clear on the POINT of this thread. Is it to determine if the pyramid is symmetrical ? If that it, I would think that has been determined many times over by various students of the pyramid.
Here is what I'm trying to do. Firstly obtain a 'real-time' measure of relevant angles. Secondly check the Kepler triangle theory. I've constructed the Kepler triangle based on GD being of unit length. It fits like a glove but the caveat is whether I have an authentic cross-section or not. The angles look good in respect of same. I wonder if a much more accurate determination could be made along similar lines.

Construction:

1. M is midpoint of CD which we are taking as the base of the triangle.
2. From M we draw a perpendicular to point I (just off diagram at the top) such that MI = GD
3. Construct J as midpoint of MI and arc MJ down to the horizontal at K.. Join KI which by Pythagoras thm is of length ##\sqrt{5}GD/2##.
4. With K as centre and radius set to KI arc down to base at D showing that ##MD=\phi GD##

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The issue here is perspective. The nice thing about perspective (without aberration) is that straight lines always map onto straight lines. So if you could somehow locate the two lower back corners you would have a better fit to the center cross-section. Obviously also if the drone is very far away (∞) at base level the elevation you seek and the face will superpose.
Yes indeed. I'm not worried about getting the whole triangle right down to base since the shape is the same all the way up. What I need to make sure of is that the drone picture accurately reflects the centre cross-section so we can check if it lines up with following:

I should indicate that whereas above shows sides ##\Phi,\sqrt{\Phi}## and 1, I have effectively constructed ##1,\sqrt{\phi}, \phi## which is a similar triangle. ##\Phi=\frac{1+\sqrt{5}}{2}## and ##\phi=\frac{\sqrt{5}-1}{2}##

Secondly check the Kepler triangle theory.
Why? Do you think that hasn't already been done? I mean, do you expect, with your crude measurement method, to do better than what has already been done or are you just interested in specifically doing it yourself? Do you expect a different result than what has already been done?

EDIT: I see I'm sounding very negative here. Not my intent, I just don't get why you want to do it.

weirdoguy
My hunch is a well-equipped drone could indeed do a very accurate aerial survey and establish important parameters such as the slope angle with a high degree of accuracy. Has it been done already (?) - seemingly not as far as I can ascertain. Most places I look seem to do nothing much more than quote Flinders Petrie's measurements from the last century. I think he did an excellent job but I'm sure modern technology has a lot more to offer. I can't even find any laser distance and/or angle measurements. You'd think that would be an obvious.

I suspect even my "crude measurement" could be considerably enhanced with high resolution graphics, more sophisticated statistics and techniques for edge detection.

I suspect even my "crude measurement" could be considerably enhanced with high resolution graphics, more sophisticated statistics and techniques for edge detection.
Good point.

Why? Do you think that hasn't already been done? I mean, do you expect, with your crude measurement method, to do better than what has already been done or are you just interested in specifically doing it yourself? Do you expect a different result than what has already been done?

EDIT: I see I'm sounding very negative here. Not my intent, I just don't get why you want to do it.
No worries.

Ok - can I perhaps rephrase the basic query ?

In principle, should it be possible to obtain an accurate cross-sectional view / silhouette outline of the Great Pyramid by placing a drone right in front of the pyramid holding absolutely steady with camera pointing in a direction normal to the pyramid's cross-section ? So that the view is similar to my graphic (only one face showing) but hopefully not tilted as appears to be the case based on some visible layers of masonry not being quite horizontal.

The reason I ask is because from what research I have done, the various determinations of slope angle are not particularly convincing and it's a key parameter. Accurate silhouette views would also give us a very graphic picture of just how good the builders were. And we will see that if we get such silhouette views in front of each face of the pyramid. Even in my very crude 'measurement' there is enough to show that there is no discernible deviation from 'straight and true' along the lines fitted to the 2 edges shown. And the measured angles are very close to what we expect from a 5/2 seked or rise/run of 14/11.

My hunch is a well-equipped drone could indeed do a very accurate aerial survey ...
I would point laser range finders along the edges, then place a drone with laser reflectors at the apex, where they all intersect.

neilparker62
With one image you will still need to know the relative position of the camera. For instance, if you know the camera is in the base plane of the pyramid, then you must know its distance from the two nearest corners. Or if it is face centered exactly you need to know its distance from the face.
One method to obtain this data is to include a known target in the image, and infer the absolute camera position from image of the target.

Another useful method is to use stereo views of the pyramid and locate the corners on two images . Better be good at projective geometry. Also you need to know the exact relative orientation of the two camera images.

If you really are trying to obtain precise measurements, a single image by itself will not suffice.
This kind of 3D location is done routinely for athletes and kinesthesiology using stereo imaging. I (my team) once designed and built such a system to locate coordinated medical sensors real time on a patient. For precise location I think laser scanning now is probably state of the art.
Good luck...sounds like fun !

neilparker62 and BvU
In principle, should it be possible to obtain an accurate cross-sectional view / silhouette outline of the Great Pyramid by placing a drone right in front of the pyramid holding absolutely steady with camera pointing in a direction normal to the pyramid's cross-section ?
If you mean normal to a horizontal cross section, the drone would need to be stationed directly above the apex of the pyramid. Or did you intend that the drone be placed normal to the altitude (vertical line from base to apex) of the pyramid?

If you really are trying to obtain precise measurements, a single image by itself will not suffice.
If you are really trying to obtain precise measurements then you wouldn't use images at all, you would use a theodolite or similar surveying equipment. Which is what the surveyors that have measured the base length and apex height to greater than 1% accuracy have done (and adjusted for the original limestone facing stones).

Why do you think you can improve on this with measurements from images, even if you did have the necessary information about the position from which the image was taken?

hutchphd and phinds
If you mean normal to a horizontal cross section, the drone would need to be stationed directly above the apex of the pyramid. Or did you intend that the drone be placed normal to the altitude (vertical line from base to apex) of the pyramid?
Normal to the altitude. In simple terms imagine if somehow one could shine a huge torch on the pyramid and then capture the shadow on an imagined huge screen placed behind. In essence that's what I thought the drone pic would be and the measured slope angle I obtain is 52 degrees. Not very far out from the 51.8 or so which is the commonly accepted value.

Here I try the same idea on a small paper model of the pyramid I have made. Again it is a very crude pic - I can' t get the horizontal alignment right. Also I don't have any method of holding my cellphone vertical. All the same the measured angles are clearly enough cross-section angles - that is angles of the pyramid's sihouette and not of the face itself. So I think that's good enough as "proof of concept". The technical challenge is to implement it with perfect camera alignment. Then you should get a 'perfect silhouette'. Or am I misunderstanding the physics of projection / projective geometry ?

You are misunderstanding. The method you propose will only be exact if you are infinitely far away. The angle subtended depends upon both the actual spatial separation and the distance to the lens.

Is there sufficient attention given to the focal length of the drone camera, which will surely be very wide angle?
Doesn't that confound any attempted measurements?

In principle, should it be possible to obtain an accurate cross-sectional view / silhouette outline of the Great Pyramid by placing a drone right in front of the pyramid holding absolutely steady with camera pointing in a direction normal to the pyramid's cross-section ? So that the view is similar to my graphic (only one face showing) but hopefully not tilted as appears to be the case based on some visible layers of masonry not being quite horizontal.

The reason I ask is because from what research I have done, the various determinations of slope angle are not particularly convincing and it's a key parameter. Accurate silhouette views would also give us a very graphic picture of just how good the builders were. And we will see that if we get such silhouette views in front of each face of the pyramid. Even in my very crude 'measurement' there is enough to show that there is no discernible deviation from 'straight and true' along the lines fitted to the 2 edges shown. And the measured angles are very close to what we expect from a 5/2 seked or rise/run of 14/11.
I see the potential for a kind of 'horse cart before the chicken egg' problem here.

How can you place a drone in exactly the right spot - without using measurements of the pyramid to figure out where that spot is? By the time you've done that, you don't need the drone.

neilparker62
In principle you need to know only the exact spot where the camera is relative to the object. Plus you are trying to infer positions of parts of the object that are not actually directly in your image. The way you wish to proceed will not work, and several alternatives have been suggested. This is neither new science nor the best way to accurately determine these distances as has been pointed out.

You are misunderstanding. The method you propose will only be exact if you are infinitely far away. The angle subtended depends upon both the actual spatial separation and the distance to the lens.
Ok. So can we conclude that there's no way drone images can be used to obtain an accurate measurement of the pyramid's slope angle ? Or put another way we can't process drone images to obtain a 'true' silhouette or cross-sectional view of the pyramid ? I was hoping we might be able to improve on following from Flinders-Petrie:

Ok. So can we conclude that there's no way drone images can be used to obtain an accurate measurement of the pyramid's slope angle
Not what I said. Please read previous posts. Likely not the easiest way.

I was hoping we might be able to improve on following from Flinders-Petrie:
Possibly, although I believe Flinders Petrie's work at Giza is generally well regarded. The Wikipedia entry cites Lehner, Mark; Hawass, Zahi (2017). Giza and the Pyramids: The Definitive History from which it presumably quotes the following measurements "Initially standing at 146.5 metres (481 feet), the Great Pyramid was ... The base was measured to be about 230.3 metres (755.6 ft) square". Assuming symmetry this equates to 51.83° or 51°50' (51°49'57" with spurious precision) for each face, in line with Petrie's measurements.

Edit: I have not seen Lehner and Hawass's book so I don't know if the 146.5m height is actually based on Petrie's measurement of angles anyway.

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In principle you need to know only the exact spot where the camera is relative to the object. Plus you are trying to infer positions of parts of the object that are not actually directly in your image. The way you wish to proceed will not work, and several alternatives have been suggested. This is neither new science nor the best way to accurately determine these distances as has been pointed out.
But finding that exact spot creates a "chicken-and-egg" situation per post #25. I am not sure about inferring positions. I have just marked points on the graphic which lie on the edges. Then joined them up. Admittedly I should do that for both edges - not just one and then 'infer' the other from a construction based on bisecting the base (which incorrectly assumes an isosceles triangle). I'm not claiming the base I've drawn is the actual base - it's just a horizontal line drawn from the lower of my chosen points. And I pointed out the problem that this line doesn't align with visible layers of masonry showing as faint lines on the graphic.

No worries if the way I propose doesn't work - that was the actually the question I put out in the first instance (is this a valid cross section ?).

I am not really concerned about distances - happy to trust Flinders Petrie et al on the base lengths. It's just a more accurate determination of the slope angle I'm after if that's possible. If along the way we can create accurate silhouette views/outlines in front of each of the 4 faces, that's a bonus. My idea was to create those views and then average the 4 slope angles thus obtained.

Why do you think you can improve on this with measurements from images, even if you did have the necessary information about the position from which the image was taken?
I had an idea which I put out for scrutiny and/or suggestions for improvement. If the answer is "no your idea doesn't work" - fair enough. If notwithstanding, other ideas for measuring the pyramid's slope angle come up so much the better and many thanks to all who have commented accordingly.

Ok. So can we conclude that there's no way drone images can be used to obtain an accurate measurement of the pyramid's slope angle ?
There are various methods to reconstruct the 3D shape from multiple images:
https://en.wikipedia.org/wiki/3D_reconstruction_from_multiple_images

But I don't think any of that will be more accurate than laser range finders or laser scanning.

This is getting a bit out of hand. What the OP desires to do is far simpler than a full scale image rendering in 3D as described in https://en.wikipedia.org/wiki/3D_reconstruction_from_multiple_images
In fact the requirement is to locate 5 points in space: the 5 vertices of the pyramid, using flat images. If the images are "perfect" this requires at very least 2 overhead pictures which contain all 5 vertices. It is easy to analyze this requirement by the information necessary to locate 5 points in space.

Consider the image of one pyramid vertex. That spot on the camera image can be thought of as defining a line (ray) emanating from the camera image plane through the lens center and out from the camera face. If the camera position and orientation are known that one image constrains the vertex location to a known line in space. The addition of a second (not colinear) camera image will similarly define a second ray and these rays intersect at the location of the pyramid vertex. In fact this problem is overdetermined: this is very useful for calibration or verification.
Caveats have also been made about optics. Unfortunately only a perfect pinhole camera will be distortion free and behave as described above. Any real world camera will have aberration and this is the fundamental limit to the accuracy of this technique. Multiple judiciously chosen images can reduce these problems. But you certainly need at least two images to do what you desire. As I recall the system I designed, using two modest cameras, would reliably 3D locate 10 sensors within 1mm at a distance of 1m in real time

OK, so, why not simply take measurements from Google Maps? Better yet, Google Earth - which has nuanced 3D flypast.

Both have high-infinite angles from which you can pick the most useful.

You need to know the exact position and orientation of the camera for the simplest techniques. And who knows what the preprocessing of the the images actually is. But maybe it would work.
Talk to the CIA image guys and gals...
I think for best accuracy a camera nearby is far preferable particularly for the depth perception.