How the All Screen Gain Calculator Works (Moved from Post #1)
I. Basic Description
The All Screen Gain Calculator can calculate the screen gain for both angular reflective and retro reflective front projection screens. The calculator will not work correctly for screens that diffuse light asymmetrically, such as ribbed or profiled screens. I wouldn't use it on the SilverStar, either. If in doubt about your screen type, please consult the manufacturer.
This explanation begins similar to the explanation of my earlier High Power Screen Gain Calculator. Those of you already familiar with that write-up may want to skip ahead a few paragraphs to Section II, labeled Differences Between Retro Reflective and Angular Reflective Screens.
First, I had to pick a 3-D coordinate system (x, y, z). While most any coordinate system would do, I wanted to pick one that was easy to visualize and use. I decided my x coordinate would represent horizontal distance parallel to the screen, and my y coordinate would represent vertical distance parallel to the screen (no big surprises here). I further decided to place the origin (zero value) of the x axis at the horizontal center of the screen, and the origin of the y axis at the floor of the room. My z coordinate represents distance perpendicular to the screen surface, and the origin of the z axis is at the screen surface.
Using the above coordinate system, a point on the screen 50 inches above the floor and at the horizontal center of the screen, for example, would be represented by the x, y, z coordinates (0, 50, 0). As a second example, a point defining a projector lens aligned horizontally with the screen center and placed 72 inches above the floor and 144 inches from the screen would be represented by (0, 72, 144). As a third example, a point defining a viewer's eyes positioned horizontally 24 inches to the right of the screen center at a height of 37 inches above the floor and 120 inches back from the screen would have the coordinates (24, 37, 120). Note that had I said to the left of the screen center, the coordinates would have been (-24, 37, 120). Well, you get the picture, dont you?
After accepting the data entered by a user, my calculator first defines the x, y, z coordinates of a Screen Point (a point on the screen at which the gain is to be evaluated), a PJ Lens Point (a point defining the front and center of the projector lens), and a Viewer Point (a point at which the viewer's eyes are located). The calculator then calculates a PJ Vector by subtracting the coordinates of the PJ Lens Point from those of the Screen Point. The PJ Vector has a direction parallel to a line from the PJ Lens Point to the Screen Point. The length of the PJ Vector is the distance along that line from the PJ Lens Point to the Screen Point. In a similar fashion, the calculator calculates a Viewer Vector by subtracting the coordinates of the Viewer Point from those of the Screen Point.
The angle (herein called Error Angle) between the PJ Vector (or a slightly modified version thereof called the Best View Vector) and the Viewer Vector is what determines the screen gain. Before the calculator can compute the Error Angle, it must create unit vectors (vectors of length = one) parallel to the PJ Vector and the Viewer Vector. This is begun by calculating the respective lengths of the PJ and Viewer Vectors (length = the square root of the sum of the squares of the elements of each vector). The calculator then divides the elements of each vector by the vector's own length, thereby deriving the PJ Unit Vector and the Viewer Unit Vector, which are unit vectors parallel to the PJ Vector and the Viewer Vector, respectively.
II. Differences Between Retro Reflective and Angular Reflective Screens
Here is where I describe the Best View Vector. This vector defines the viewing direction that will realize the highest gain from the screen at a particular Screen Point. The Best View Vector for a retro reflective screen is different from that for an angular reflective screen. For the retro reflective screen, the Best View Vector is identical to the PJ Vector, because the retro reflective screen is designed to send most of its reflected light right back in the direction of the projector.
So how do we determine the Best View Vector for the angular reflective screen? First, we must understand how the angular reflective screen reflects light. The angular reflective screen can be simplistically described as a big mirror immediately behind a translucent light-scattering layer. The mirror reflects light according to the familiar angle of reflection equals the angle of incidence rule, while the light-scattering layer scatters the reflected light so that it can be seen (with decreased brightness) at angles other than the precise angle of reflection. To realize the highest gain at a Screen Point, the Best View Vector for the Angular Reflective screen must point directly into the Screen Point and parallel to the angle of reflection. To calculate the Error Angle between the Best View Vector and the actual Viewer Vector, we will need to use their respective Unit Vectors.
Once we have determined the PJ Unit Vector for a Screen Point on the angular reflective screen, calculating the Best View Unit Vector is easy. Let the PJ Unit Vector at a Screen Point be represented in our x, y, and z dimensions as (X, Y, Z). The Angle of Reflectance Unit Vector is then (X, Y, -Z), because the mirror reflects the (perpendicular) z-axis component of the light without changing the x and y components. The Best View Unit Vector then is the reverse (negative) of the Angle of Reflectance Unit Vector, or (-X, -Y, Z). Thus to compute the Best View Unit Vector for the angular reflective screen, we simply multiply the x and y components of the PJ Unit Vector by -1, while leaving the z component intact!
The calculator next calculates the dot product of the (appropriately determined for screen type) Best View Unit Vector and the Viewer Unit Vector. (The dot product = the sum of the products of the corresponding elements of each of the unit vectors). The Unit Vector Dot Product is a scalar (a real number), which conveniently happens to be equal to the cosine of the Error Angle between the two unit vectors. (See the tutorial linked below.) The calculator then calculates the Error Angle as the arc cosine of the Unit Vector Dot Product. The above described calculations are performed in parallel three times to obtain values for Screen Points at screen left, screen center, and screen right.
The calculator uses a mix of linear and exponential interpolation to estimate the screen gain corresponding to the Error Angle at each Screen Point. Between on-axis (zero Error Angle) and the manufacturer-recommended maximum viewing angle, the gain is assumed to vary linearly. At angles greater than the maximum viewing angle, the gain is assumed to decay exponentially towards a value equal to the minimum gain for large off-axis angles. While this approach is somewhat less accurate than using a table look-up of screen gain versus error angle data for specific make and model screens, it is a more practical approach to designing a generic calculator that is usable for a wide range of front projection screens for which basic gain parameters are available. The accuracy is deemed more than sufficient to make comparisons of projector and viewer positions, as well as screen type, for home theater performance optimization. Those who want to check the calculator-estimated gains should feel free to use the Calculated Error Angle values (which are highly accurate) to look up the corresponding gain values on a screen gain chart for their particular screen(s) of interest.
III. Tips and Comments
1. Be sure to specify the correct type of screen. Glass bead and High Power screens are retro reflective; most other gain screens are angular reflective. Should you specify the wrong screen type, the calculated results will be erroneous. If uncertain, consult the screen manufacturer.
2. The Screen Left and Screen Right gains are calculated for the far edges of the screen. If it makes you feel better, you can have the calculator compute these gains at points removed somewhat from the screen edges.
To do this, just specify a smaller-than-actual horizontal width of viewing area. For example, if your screen's actual viewing area is 87 inches wide, and you want to calculate the Screen Left and Screen Right gains at 6 inches inside the screen edges, just specify a viewing area width of 87-12=75 inches. (The Screen Center gain will not be affected.)
3. Projector position is extremely important for the retro reflective screen. Ideally, the projector lens height should be near viewer eye level to get maximum screen gain. If other factors force a higher projector position, try to place it as low as possible to avoid losing much of the potential screen gain. It is perhaps less well known that the distance from the projector lens to the retro reflective screen is also important for gain uniformity across the screen in regard to off-center viewing positions. All on-center positions will see excellent gain uniformity with the retro reflective screen. You can add two equally-off-center positions to that happy list by placing the projector just a few inches behind the row of off-center seats. The off-center seats will, of course, experience lower gain than the center seats, but the gain advantageously will not change much from one side of the screen to the other. Use this calculator to experiment with various projector positions to see what I mean.
4. Projector height is important for the angular reflective screen, as well--if you are interested in making centerscreen your brightest area, top-to-bottom.. For example, when the projector is at too high an angle, the top half of the angular reflective screen will be brighter than the bottom half. If, for an on-center viewer, the calculator computes Screen Center gain at less than your specified on-axis centerscreen gain value, then the projector is not at the optimum vertical angle for that viewer, and the projector should be adjusted upward or downward. A greater projector distance (i.e., throw) from the angular reflective screen will generally produce better gain uniformity in all directions. This has to be traded off with reduced overall illumination (due to the slower zoom lens setting required for a greater throw). You can use the calculator to explore gain uniformity with different projector positions.
5. If you should happen to run the same input parameters in this calculator and in my older High Power Screen Gain Calculator, you will get somewhat different screen gain results (but should get the same Computed Error Angles). The reasons are twofold: (a) The older calculator is based on earlier gain specs that are a little different from today's published specs for the High Power screen; and (b) the older calculator uses a table-look-up method, which can produce slightly more accurate screen gain values than the method used in the new calculator, as discussed above.
6. If any users see anything that they think may be questionable about the calculations and assumptions used in this calculator, please let me know either in the open forum or by PM. I do not want to spread misinformation. Also, if any of you are fortunate enough to own a one-degree spot photometer, I would be delighted to have you measure a real-world set-up and see how closely the measurements compare with what the calculator predicts.
7. Readers seeking more in-depth information on the calculation techniques described above are referred to Chapters 0 through 10 of an excellent Vector Math tutorial from Central Connecticut State University, which is linked below.http://chortle.ccsu.edu/VectorLessons/vectorIndex.html