-1

With my recently bought LED stab lamp the following "experiment" was done. Put the lamp on a table and watched the circle on the ceiling of the room. Then I positioned myself in the room in a certain distance and saw then the ellipse on the ceiling. Took a photo by the way. The question I am unable to calculate: what is the formula for the ellipse that one sees. It reminds me to a brochure of HP where maybe a similar problem was solved in the "computer graphics" formulas. Both pure 3D but maybe also 4D matrix calculations (?) are welcome. Of course any "elementary" of solving.

Addendum 20171003:

Want to direct your attention to two texts of David Eberly dealing with a maybe more than similar problem and delivering the math that might be helpful when occupying oneself with the above question. The titles are

David Eberly-Parallel Projection of an Ellipse

David Eberly-Perspective Projection of an Ellipse 20150310

You may visit the homepage of Dave.

amWhy
  • 209,954
Wolfgang Tintemann
  • 755
  • 2
    Can you work out what information you will need to compute the answer? Hint: some measurements will be required. – Wildcard Sep 20 '17 at 14:04
  • 1
    ... for example the distance to the wall... but not only. Moreover, if you are looking for a "formula" $(x/a)^2+(y/b)^2=1$ for this ellipse, you need axes on your wall... – Jean Marie Sep 20 '17 at 14:14
  • 1
    I think you should begin to work on a vertical plane giving a 2D simpler problem. – Jean Marie Sep 20 '17 at 14:27
  • 2
    These two questions which I posted recently might be relevant: 2351662, 2346896. – Hypergeometricx Sep 20 '17 at 14:44
  • Circle not on the wall but perpendicular upward on the ceiling. Ah one needs the distance of the light source to the ceiling or from the floor ? Then – Wolfgang Tintemann Sep 20 '17 at 17:43
  • Circle not on the wall but perpendicular upward on the ceiling. Ah one needs the distance of the light source to the ceiling or from the floor ? Then the distance of my eye or camera lense above the floor and distance of my feet from LED lamp perpendicular axis floor to ceiling. And the angle of my arms holding the camera ? – Wolfgang Tintemann Sep 20 '17 at 17:50
  • The video in 2346896 shows the experiment on the floor. So there should be a general formula for the ceiling situation when triple space coordinates are given for position of LED source of light (a,b,c) and my feet position (x,y,z) on floor or better the camera lense position (u,v,w) ? – Wolfgang Tintemann Sep 20 '17 at 18:04
  • Then this tilt angle also say alpha. – Wolfgang Tintemann Sep 20 '17 at 18:04
  • And coordinates on the ceiling (m,n) – Wolfgang Tintemann Sep 20 '17 at 18:06
  • What about the diameter of the projected circle which is dependent of the LED lamp construction. – Wolfgang Tintemann Sep 20 '17 at 18:08
  • Lets say diameter d when I look to the ceiling when my position on the floor is x=a and y=b the floor coordinates of the LED lamp. – Wolfgang Tintemann Sep 21 '17 at 10:15

2 Answers2

2

Just a hint, maybe of some use.

The point source emits a cone of light, and the intersection with a plane perpendicular to the cone axis is indeed a circumference.

Imagine you are just above the spot, you somehow manage (transparent ceiling?) to observe it straight below your head, with our eyes at $A$ in the sketch below: you will see a circumference, the blod black line representing a diameter of it.

enter image description here

Now you start moving along a direction of your choice without varying your distance from the ceiling.

The sketch above is taken on a plane perpendicular to the ceiling, and containing the line along which you are moving. The bold black line represents the side view of the circle. The perceived size of one axis is related to the angle subtended by the bold black line and your eyes.

Of course another such angle can be characterised, subtended by your eyes and the diameter of the circumference that goes through the middle of the black line, perpendicular to the plane used for the sketch.

The two subtended angles define the perceived ellipse.

An aedonist
  • 2,568
  • After some rethought of the many comments I made above I had an idea that may be the one that you describe but I dont grasp. I thought this way. I see the circle with diameter d. But this is an image built in the brain by the background cells of the eye. Or in case of camera on the light catching chip. – Wolfgang Tintemann Sep 21 '17 at 08:23
  • Now I also thought that the circle seems to be the result of a light cone with its vertex (?) quasi above and behind the ceiling. – Wolfgang Tintemann Sep 21 '17 at 08:26
  • The ellipse I see or the camera takes as image or photo is then the cone projection of the circle seen on the ceiling. Oh I am puzzled. Must repeat the experiment. Moment please. – Wolfgang Tintemann Sep 21 '17 at 08:29
  • Now. If I stand in line with the lamp I see a circle. Cone of parallel rays a cylinder (?). If I move I see diffrent shaped ellipses and somehow one can imagine the ellipse being the intersection of the ceiling plane and an appropriate Cone. Isnt it ? – Wolfgang Tintemann Sep 21 '17 at 08:38
  • Yes it has indeed a lot to do with cones and conic section. Have you had a look at https://math.stackexchange.com/questions/2351662/origin-centred-elliptical-spotlight-with-conical-light-source-of-fixed-apertur, it contains very detailed discussions you will find useful (in a comment above, Hypergeometric points at another useful question as well) – An aedonist Sep 21 '17 at 09:14
  • If the ceiling would be a mirror what would the camera image be like ? – Wolfgang Tintemann Sep 21 '17 at 10:25
1

Assume a coordinate system attached to the camera, with the $z$ axis on the optical axis. Assuming a focal distance $f$, the projection equations for any point in space are

$$x=f\frac XZ,\\y=f\frac YZ.$$

Now let the center of the circle be C, and two orthogonal vectors or length equal to the radius in the plane of the circle be $U, V$ (in the camera frame). The parametric equation of the circle is

$$X=C_x+U_x\cos\theta+V_x\sin\theta,\\ Y=C_y+U_y\cos\theta+V_y\sin\theta,\\ Z=C_z+U_z\cos\theta+V_z\sin\theta.$$

Plug this in the projection equations, and you have a parametric equation of the ellipse.

  • Is Cx and Cy equal to zero in this case ? – Wolfgang Tintemann Sep 21 '17 at 10:05
  • @WolfgangTintemann: no, why should they ? –  Sep 21 '17 at 11:08
  • Misunderstanding on my side. As above already described I thought in lets say room coordinates. Lamp lense fixed at (a,b,c) and camera lense fixed at (u,v,w) as an example. Circle diameter d or radius r. You have an coordinate system at camera lense (X,Y,Z) which called camera frame. Z axis perpendicular to (X,Y) plane. Experimentally one would go to the lamp and mark center C on the ceiling above (?). If one walks back to the fixed camera one would take the image of the ellipse. Where does the marked center C point lie on the ellipse ? – Wolfgang Tintemann Sep 21 '17 at 15:08
  • As I checked visually the center C is the same on circle and on the ellipses one sees irrespective position (u,v,w). – Wolfgang Tintemann Sep 21 '17 at 19:14
  • same position (u,v,w) – Wolfgang Tintemann Sep 22 '17 at 16:19