In this posting I will consider the trigonometry of the R1. The considerations and formulas are independent from the actual lens/camera used. I will use my combo - Canon 15mm on a Canon EOS 5D - merely as an example. (So this is a generalized version of my previous posting regarding the calibration of the Canon 15mm on a Canon EOS 5D).
Before we start with trigonometry, we need to calibrate our lens at 0 degrees. I strongly recommend the method described by John Houghton at http://www.johnhpanos.com/epcalib.htm - 3. A DIRECT APPROACH - FOR SLRs ONLY. When you have calibrated the lens at 0 degrees, you will get an correction value - for later use we will call this value c.
The correction of the R1, when tilted, can be described by a simple trigonometrical formula. In Fig. 1 you can see a camera on the R1 tilted by 10 deg.
Let us have a closer look at the triangle in Fig. 2. The lens was calibrated at 0 degrees. So at 0 degrees the NPP (to be more precise - the Least Parallax Point) is located directly above the center of tilt/rotation, i.e. the point around which the lens is tilted. Here d is the vertical distance between the center of rotation and the NPP (In the vertical the NPP is located in the center of the lens).
When the lens is tilted counterclockwise by alpha degrees - as seen in Fig. 2 - then the NPP is dislocated from the vertical line and is no longer directly above the center of tilt. To compensate for this dislocation, one needs to move the lens by x to the right.
Both values d and alpha are know - d is measured and alpha is the value of tilt you chose at the R1. From those two values, x can be calculated:
x = d * tan(alpha)* (-1)
The factor -1 is used as the unit is moved counterclockwise. When using excel, make sure that alpha is converted to radian instead of degrees.
With this x you can now calculate the corrections at different tilt angles. Simply add x for each tilt angle to the correction value (c) that you have determined at 0 degrees.
For my lens I have determined the correction at tilt angle 0 degrees to c = 1.65 cm.
I will now calculate two versions of x:
x1 - For my lens I have determined d = 6.9 cm - from this I will calculate x1.
x2 - Nick Fan was so kind to send me the precise value of d for my lens as d = 7.226 cm - from this I will calculate x2.
alpha x1+c x2+c measured -15 3,499 3,586 -10 2,867 2,924 2,85 -7,5 2,558 2,601 2,55 -2,5 1,951 1,965 1,90 0 1,650 1,650 1,65 5 1,046 1,018 1,00 7,5 0,742 0,699 0,70 12,5 0,120 0,048
As you can see, for both cases, x1+c and x2+c, the calculated values differ less than 1 mm from the experimentally determined values.
Yes there is a difference in the values and I would like to quote a caveat, that Nick Fan has posted to me, when sending his value d:
"The height is 72.26mm (=7.226 cm). This is a theoretical value. It assumes the lens barrel is perfectly cylindrical, lens ring mount and lens are perfectly concentric etc. The NPP value derived from trigonometry should be used as a guide as starting point of NPP calibration."
Now with an error below 1 mm, the trigonometrical calculation is quite an acceptable starting point for the calibration. This also shows how precise a unit the R1 is! (And how nicely I have attached the lens ring to the my lens )
I have loaded a little excel sheet on my website http://www.freiburg-panorama.com/mat...ection.xls.zip
What you need to do in this sheet is:
- Determine the value for d by measurement
- Determine the value for c by calibrating your lens at 0 degrees.
- Enter your values for d and c - both values are lens specific
You will then get the correction values to start with at a given tilt angle in the column named x+c.