The Sensitivity of a Spirit Level

The accuracy with which a platform can be leveled is a limit on the precision of the determination of one's latitude and longitude. I included some adjusting screws on my platform to help level it with a spirit level. So one might ask how well can one level a platform?


To check the level of something one places the level on a surface and notes the position of the bubble between the two pairs of parallel rings. One then then rotates the level lengthwise 180° along the same line and again notes the position of the bubble. If the bubble has moved from its former position then this is an indication that the surface is not level and is in need of adjustment. The double rings on each side of center are spaced 2mm apart. They can be used to determine the sensitivity of the spirit level. Raising one end of my 2ft spirit level 5/32th of an inch will move the bubble 2mm from the inner ring to the outer ring. This corresponds to a change in slope of 6.5mm per meter which is one measure of the sensitivity of the level. Another measure is the angular change which moves the bubble the same amount or 22.4 minutes of arc per 2mm shift in this case.

The precision that one can level something in a particular direction is less than the 2mm separation since one can position the bubble more accurately. The platform needs to be leveled in two directions about 90° apart. So it may be possible to measure angles to within about 5 minutes of arc.

Solstice Sun Alt-Azimuth

I've finally got some preliminary results for the Solstice observations last week. The high for the day was 105°F (41°C) and NIO had to shut down due to overheating after being exposed to direct sunlight for about an hour. The solution was to hang a towel over him and to provide some additional shade for the wrist wacth between photos. Since then NIO has been helping with the photo analysis by locating the positions of the corners of the 1 inch square grids as well as the grid coordinates of the Sun's position. The position of the camera was lowered closer to the plane of the graph paper and 14 megapixel images were captured. An example of one of the clipped images is shown below along with expanded views about NIO's coordinates for the corners. He used the intersecting line method to locate the corners but got confused when light from the Sun or outside the shaded region changed the appearance of the lines so some corrections were necessary.


The position of the Sun was found by determining the upper, lower, left and right bounds of the image of the Sun. The grey scale of the clips were "stretched" to make the grid lines and the Sun easier to see.


The grid coordinates of the Sun's image were used to find the Sun's altitude and azimuth as a function of the time. The height of the lens aperature was found by measurement to be 24.8 inches. A check against my longitude turned out to be worse than expected being almost 1 degree off. I set the wrist watch to UT just before starting the observations. Time was measured to the second so its precision was about 1 part in 3600. The μ coordinates were rather flat and so its errors were probably more significant. They may have thrown off the time of the peak slightly.

Supplemental (28 Jun 2011): The "platform" I used was warped being shaped somewhat like an inverted bowl. I noticed some curvature when leveling the platfrorm with a spirit level and tried to compensate for it. This is probably the major source of the error in the observed latitude and longitude since the measurements are relative to the platform and its grid system.

Rubber Bands and Plumb Lines

The mechanics problem of what happens when the rubber band is loaded with the plumb line surprisingly is easiest to solve using complex numbers. Suppose the plumb line with a weight of mass m hangs on a rubber band which is attached at two fixed points deparated by a horizontal distance D. To first approximation the force exerted by the rubber band is linear and proportional to the relative amount it is stretched or |Δz|/|z|. The proportionality constant is k.


The sum of the two complex numbers, z_1 and z_2, representing the two sides of the rubber band is D which is fixed and not dependent on the mass of the plumb line. Writing a second equation with primed z's for the weighted position and subtracting tells us that the sum of the changes is zero.



So the change of one side is the negative of the change in the other. The balance of forces requires that the sum of the forces exerted by the rubber bands equals minus the force of gravity which in complex notation is, i mg. Substituting -Δz_1 for Δz_2 and rearranging tells us that the plumb line will only drop vertically after it is loaded.

Watching what happens as the rubber band is loaded with the plumb line at various positions shows that its motion actually is vertical.

Supplemental: One can solve the problem in a similar manner using vectors but what seems to be odd about the problem is that the force is dependent on which end of the vector is attached to the common point. One would use the unit vectors i and j in place of 1 and the "imaginary" unit i respectively. The tension in the rubber band is not a vector quantity but the internal forces are balanced. The problem is difficult to work in terms of vector components.

Using a Plumb Line To Indicate The Lens Aperature Position

On the last set of local noon observations I had to estimate the position of the lens aperature after the fact. Today while setting things up I used a plumb line to help determine the aperature position (see image). The simplest arrangement I could come up with was placing a rubber band around the lens holder and attaching the plumb line with a bent paper clip. The plumb line had a slip knot to permit the adjustment of the string's length. One can tell if the paper clip is positioned in the middle of the aperature by sighting along it through the lens. One moves one's head sideways until the entire paper clip appears to be in the same direction and does the same moving forwards and backwards. If the paperclip is not centered in the aperature then one has to adjust its position on the rubber band to center it properly. The lens magnifies distance slightly so only small corrections are necessary. One should also sight along the rubber band on both sides of the lens holder to make sure that it is lined up with the string. The plumb line should swing freely and be slightly above the graph paper to reduce paralax. To stop the plumb line from swinging one can clamp it gently between two fingers to dampen out the oscillations and then release it by slowly moving one's fingers sideways and it will stay fixed at the position beneath the aperature. Sighting from the north and east allows one to determine the coordinates, (λ,μ), of the lens aperature position.


I did a set of solstice observations of the Sun's track about local noon today but it will take a little while to "reduce" the data. The image size used was 14 megapixels which resulted in the 1 inch grids being about twice their actual size in MSPaint. The finer lines on the grid turned out much clearer.

2D Interpolation

To obtain the position of the Sun from an image we only have its pixel coordinates to work with and those of other points in the image. The grid coordinates of the corners of a 1 inch square which contains the Sun are known and their pixel coordinates are easily obtained. For a small section of the image we can use a simpler equation since the curvature of the lines can be ignored. This is analogous to using the method used in projective drawings. The situation is as follows. We want to find the coordinates, (λ,μ), of the point, x.


The values of λ or μ change by one unit as move from the end of one side to the other. We can draw a line through the point from one side to the other where λ has the same value and similarly for μ. The pixel coordinates corners can be represented by a set of column vectors, x_k, and the ends of horizontal line through x are ξ_1 and ξ_2. Any point on the line between these endpoints can be represented by a linear function of λ. Likewise for any point on the vertical line through x as a function of μ.


Both equations result in the same function of x in terms of λ and μ since the vector difference for the λμ term is the same in each case. This equation can be solved by a simple predictor-corrector method for λ and μ. The last term is a small correction so by ignoring it we can obtain a first estimate for λ and μ. Replacing λ and μ by λ + Δλ and μ + Δμ respectively we get the corrections which satisfies the original equations.


The terms of the matrix above those of the two column vectors indicated and so consists of four separate terms. The use of vectors was intended to simplify the notation. Adding the corrections to the estimates gives improved estimates of the grid coordinates of the point. Only one iteration was used to convert the pixel coordinates to the grid coordinates.

Observing Local Noon

The Summer Solstice is fast approaching and so some details on observing local noon might be a suitable topic for this blog. I used the setup below to capture images of the Sun projected onto a plywood board with graph paper affixed to it to serve as a coordinate grid. The observations were made about local noon on 27 May 2011.


The coordinates, (λ,μ), of the Sun were found by doing a 2D interpolation of an image's pixel coordinates for the Sun relative to the corner coordinates of the 1 inch grid which contained the Sun's image. This was easier than finding a formula to convert pixel coordinates to grid coordinates for each image. λ increases along the line of sight and μ increases to the left both from the bottom right corner. The Sun followed the track indicated in the next image. The large red dot at the bottom indicates the point below the aperature of the lens which is needed to determine a minimum distance corresponding to the Sun's maximum altitude.


Plotting the distance of the Sun from the point below the aperature of the lens versus time results in a curve with a fairly well defined minimum. Estimating the time of the minimum and doing a polynomial fit about this time gives a simple polynomial. The curve shown is that of a 6 degree polynomial fit.


The time of local noon is determined by the best fit and comparing this time with that of the Greenwich local noon allows one to determine the hour angle of one's position. I'll try to give more details on this procedure in future blogs.

Are We In Need of a Reality Check?

It may seem that we are expending an excessive amount of time on studying a piece of paper but what we are interested in is the coordinate transformation which takes us from the plane of the paper to the plane of the camera image. If one compares the lines in the image with a straight edge there is a noticable curvature. This seems to run counter to the rules used in perspective drawing. Parallel lines there remain straight and converge on vanishing points. It is likely that this is actually a lower order of approximation. One sometimes encounters a discrepancy between appearance and reality. Parallel lines do not actually converge in the real world since their separation remains constant. But if we consider two parallel lines one on either side of us they will appear to converge in both directions that they move away from us. If we turn 90° the two lines angle appears to be 90° to this direction. We find ourselves in a bit of a quandary and so we need to reexamine our initial assumptions.

This is a position that idealists often get themselves into. But their assumptions do not get past the naive impressionist. The real world is his source of knowledge. He takes that as it is. False assumptions can make the world appear surreal and perhaps we should take this as an indication that we should be a little more critical in our thinking.

Some related Wikipedia articles on what we have been doing:

Ruled surface

Curvature

Geodesic curvature

Perspective (graphical)

Homography

Real Projective Plane

History of manifolds and varieties


Supplemental: There may be some trickery involved in perspective drawings but they are still an effective illusion.

Zen Seems To Have Worked

I corrected the positions for all the grid points using the best intersection point for each pair of lines. The average deviation of these points from the positions from the equation fit was nearly halved.


A more centrally located origin was also chosen and as a result the formulas required some additional modification.

Supplemental (3 Jun 2011): I haven't come to a final decision on "Zen" yet. Each intersection is determined by all the points in its two lines. But although the deviation of the improved points is about a half the measured positions it hasn't been shown that the procedure results in more accurate functions.

Improved Estimates for the Grid Points

We started with measured positions for the positions of the intersections of the grid lines. Using an assumed intersection point for the origin made it easier to fit the curves through it. But the intersection of the two axes also has an error associated with it which a search of the neighborhood of the origin showed. This gave us an improved estimate of the position for the origin. One could repeat the process for all pairs of lines for the grid to get improved estimates of these pixel points in the image. Chosing a pair of axes for the grid was an arbitrary decision. We could have used any of the intersections for the origin.

It's likely that the improved estimates of the grid points will result in a lower variance for the fitted transformation function. We can use this improved fit as our equivalent of Ptolemy's Governing Faculty. There seems to be a number of converging approaches in the sciences. The psychologists study evoked potentials. The engineering approach is fuzzy logic. For the mystic there is Zen. We might consider this to be the rational version.

We can't rule out that it is possible to go beyond the actual measurements. Just what our limit in precision is remains to be seen. Redundant data gives better results.