What is the angle of perspective for circles in a view of The London Eye?

[Dalgoma]

Good Morning everybody. I hope that this thread is of interest.
I am a retired architect with an interest in Mathematics.
My picture shows a view of The London Eye.
We know that it views as an ellipse but the major axis (drawn), clearly is not at right angles to the axis of the wheel and if you "walk up" the centre of vision this angle will not change, so it bears no relation to the vanishing points of say, a bounding square.
I cannot find any information on this matter on line apart from the mundane methods of drawing an ellipse.
Is there any established method of calculating this angle when the angle of view, in both planes are known?

 

[mfb]

clearly is not at right angles to the axis of the wheel

Looks like a right angle to me.

The vanishing points of a bounding square will depend on the orientation of the square. You can always find one that agrees with the axes of the ellipse.

 

[.Scott]

Starting in 3D, draw a line directly from your eye to the center of the London Eye. That line will meet the London Eye at an angle. Now, project that line directly onto the London Eye. That will be your minor axis. Back to the 2D image: a right angle to that will be the Major axis you are drawing. The Major axis can change when you change position.

The "project that line directly onto the London Eye" operation may take some explanation: Take a coordinate system where the London Eye is on the Z=0 plane and the center of the London Eye is at 0,0,0. Now take the position of your eye (Xe,Ye,Ze) and zero the Z coordinate: (Xe,Ye,0). Now a line drawn from (Xe,Ye,0) to (0,0,0) will follow along the minor axis. Once again, go back to the 2D image to generate the major axis.

BTW: The intersection of the major and minor axis will not pass through the center of the London Eye.

Another point: We are assuming that we are "picturing" the London Eye by looking straight at it - or by photographing it with a camera pointing straight at it. If, instead, the focal plane is parallel to the London Eye, then the image will be circular no matter what the position of the camera.

 

[Dalgoma]

Looks like a right angle to me.

But if you have two wheels on an axle, both have different "tilt" angles. The can't both be at right angles to the axle.

 

[mfb]

Tilt of what relative to what?
The center of the London Eye is approximately the center of the image, so we can neglect camera effects. In the plane orthogonal to your view, going through the center of the Eye, one axis (the longer one of the wheel) is within this plane, while the other axis is orthogonal to the first axis - its projection will be orthogonal as well.

 

[Dalgoma]

Tilt of what relative to what?

The undoubted tilt of the axes of the elliptical image relative to the horizontal.
Yes, I see we are projecting the image of what we see on to a 2D plane orthogonal to the line of sight.
So the "tilt angle" should change if we pan the "camera" up, down or side to side.(that is, change the angle of the plane relative to the subject). But it doesn't.

 

[CWatters]

So the "tilt angle" should change if we pan the "camera" up, down or side to side.(that is, change the angle of the plane relative to the subject). But it doesn't.

If the camera is at the same/similar height as the axle then the major axis of the ellipse doesn't appear tilted...

I don't think your red line is on the major axis. I think it's more upright than you have drawn it.

 

[Dalgoma]

Thanks for all your replies.
Yes, maybe but it does tilt.
And yes, the major axis would be vertical if the camera was level with the hub - (and horizontal if directly in front).
I imagine viewing a flat circular lollipop being rotated by its stick viewed from, say below its centre. During the 360 degree rotation, the major axis would turn all angles, from horizontal to vertical when the lollipop was edge on. All the time the portion of stick encased in the lollipop would remain vertical and the same length.
BUT the area of lollipop either side of it would vary, because in perspective, one side for the most part will be closer to the observer and therefore appear bigger.
Also, except when edge on the image will always be an ellipse.
It has already been established that the intersection of the axes is not the centre of the wheel. So an imaginary axle cannot be co-incident with either of the axes.

 

[.Scott]

And yes, the major axis would be vertical if the camera was level with the hub - (and horizontal if directly in front).

We need to remember that we are assuming that the camera is pointing directly at the center of the London Eye. If we are standing off to the side, we can still get a circular image on our photo by pointing the camera directly towards the plane of the Eye. The Eye will not appear in the center of our photo, but it will appear as a circle.

Similarly, with the camera level with the hub, photos could be taken that show the major axis far from vertical.

With the camera at the bore line, any major axis orientation would be possible just by selecting a spot along the the circumference of the Eye and tilting the camera directly towards it.

 

[Dalgoma]

Thank you all.
I believe that the "tilt angle" is solely related to the location of the observer ie: x degrees left or right and y degrees above or below the centre of the disc. It has nothing to do with the distance from it or it's radius. And that angle is unique to that location.
What I will take away from your comments, gratefully received, is that is somehow related to the imaginary axle.
So - I will try to investigate this using the rules of perspective that I know, with the following assumptions:-
1 The disc is viewed with one eye. ie: a single viewpoint.
2 The 2D picture plane is orthogonal to the line of sight which is a line joining the viewpoint to the centre of the disc.
Hence, the image will be as taken with a pinhole camera, where straight lines in fact are straight on the image, with no lens distortion.
If anybody is still interested, I will let you know how I get on.

 

[Dalgoma]

. The Eye will not appear in the center of our photo, but it will appear as a circle.

Yes, Agreed. As drawing - With a viewpoint on the face of a spherical bubble, circles drawn on it, when viewed with a sight line through the centre of the bubble, will all have true circular images. Full circles though, are all outside our cone of vision but it seems that if you are standing the right distance back from the edge of a circular pond, looking horizontally, you would view the diametrically opposite bank as the arc of a true circle!