General Concepts:
The results created from the InspectorG software are to be interpreted as an optical distortion map, not a physical shape calculation. The surface plots do look like a surface layout but they are really a depiction of the local curvature of the reflective surface. Optical distortion is directly related to the local curvature of a reflective surface and this can be determined from the surface shape through two differentiations producing the local gradient(slope) and finally the curvature. The reverse is theoretically possible by twice integrating the local curvature to produce a surface elevation. In practice, this is quite difficult because of error amplification from the digital integrations. In some specific cases, such as when the curvature closely approximates a sinusoidal shape this integration can produce a reasonable estimate of the elevation deviation as measured by the traditional straight edge and feeler gauge method.
This program is designed to measure the reflected lens power of a flat sample of tempered glass by digitally capturing the reflected image of a lighted grid board with a readily available digital camera. It is preferable that the glass is supported nearly vertical to eliminate any gravitational bending and situated approximately parallel to the grid board at a measured distance away. The camera is located in the plane of the grid board and the grid lines appear as a half-size rectangular array in both the vertical and horizontal directions on the surface of the glass. The distance between the glass and the grid board must be chosen as a compromise between low resolution of the resulting quality measure from close spacing and overly distorted reflections from large spacings. Too much distortion makes it difficult for the program to properly resolve the location of each grid. The case of transmitted optical distortion is very similar with one very important difference, the magnitude of light ray deviation is much less in this case.
Lens Optics:
The measure of optical distortion in this program is in millidiopters. This is the familiar unit that is used to specify the strength of eyeglasses. Reflection from a mirrored surface is very similar to the transmitted effect in a glass lens. The Physics definition of diopter is 1/the focal length measured in meters, where focal length is the distance from the mirror that a parallel beam will meet at a point. Another related feature is the radius of curvature, which for a given mirror (reflection from a glass surface) the radius of curvature is 2 times the focal length. Thus 2/radius of curvature is also a measure in diopters if we measure the radius in meters. Because the magnification reflected from a flat sample is very low, the millidiopter (1/1000 of a diopter) is the appropriate unit for quantifying roll wave distortion. For example if the radius is 20 meters at a location on the glass the "optical power" would be 100 millidiopters and using the GANA accepted roll wave equation the depth measured would be about .0015 inches if the roller diameter was 3 inches. Perhaps the easiest way to understand focal length is to visualize parallel light rays coming to the glass surface and note that they would be redirected from the curved surface and all would cross at a distance from the glass.
If the glass at the point we are evaluating is locally spherical we have an exact reading of optical power in millidiopters. Actually, any point will be distorted in a more complex way that will magnify the length of an arrow differently in each direction. The approach of InspectorG is to treat the area change of a small square area as the average magnification at that point. This magnification is then related to the average curvature and consequently to the diopter value through the physical lens law. This result is termed "Optical Power" in the software. In the common case of roller conveyence the curvature varies along the direction of motion, but the cross magnification is very nearly unity. In this case the square becomes rectangular shaped and we calculate the results as millidiopters of rollwave power by calculating the linear magnification in the flow direction only. InspectorG calculates both sets of numerical data and plots of both are available to the user, dependent on the application. In almost every case these two are not the same. The glass shape will usually have roller waves and it will almost always have some cross draw shape.
mDp Sign:
This can be a confusing issue because it changes the appearance of an optical distortion plot greatly but it may not be clear when it should be positive or negative. By definition, the focal length of a convex mirror is negative, and the focal lenth of a concave mirror is positive. An object appears smaller when viewed in a convex mirror and larger in a concave mirror. This relationship is more complicated when the objects are a distance from the mirror near the focal length but in the case of the reflection from a nearly flat mirror the focal length is very long and this holds true. From this, we conclude that the milliDiopter value in the vicinity of a local peak is negative, remember the focal length is negative, and the magnification is less than one.
In a misguided effort to make the plot look more like the physical shape of the sample, the plot in version one of InspectorG was inverted. For any with access to this software, this can be seen by comparing the values on the plot with those in the tabular window. This is corrected in the second version. A little thought will show that the actual sign of the distortion is not really that important since a positive curvature on one side of the sample is a similar negative value when looked at from the other side. This has been tested and is true within any thickness variation of the sample.
Gravity:
The orientation of the setup is chosen to be a vertical plane for both the glass sample and the gridboard to minimize the distortion from gravity, which can produce very noticeable bow especially on thin samples.
Gridline Location:
Finding the position of the gridlines in the images is one of the more demanding operations in the analysis performed by the InspectorG software. The proceedure used is to examine a pixel scan perpendicular to the line and do a curve fit to a Gaussian curve. This provides a series of points along a line that is then fit to a multisegment cubic spline function. This allows a complete mathematical specification of the complete grid array that is accurate within approximately one tenth of a pixel. These curve definitions can be individually examined in the software diagnostic section.