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Math Help - Cylinder vs planar projection - calculating the difference

  1. #1
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    Cylinder vs planar projection - calculating the difference

    I'm looking at two different computer graphics approaches. One of them uses a cylindrical 'virtual screen', the other uses a flat virtual screen. The problem is that while the first method, the cylindrical method, should be displayed on a cylindrical surface to look right, it is in fact mapped onto the flat computer screen. This entails some distortion. I'm trying to model this distortion.

    I've drawn a diagram. It's not as confusing as I've made it sound.
    Cylinder vs planar projection - calculating the difference-trig.jpg



    P is the projection centre. A B C are objects in the real world. The dotted lines are the light rays cast from A B C to the two projection surface: cylinder, and tangent

    For point A, there's no difference between the two projections.

    For point B, you can see that the ray P B intersects the two surfaces at two different places. On the cylinder, the arc is marked C. On the plane, the distance down the tangent is marked T.

    I can work out both. The value of C is:

    2*Pi*r * (angle / 360)

    Let's say the radius is 10.

    The value of T is:

    radius * Tan (angle)

    As the angle increases, so does the difference between C and T. This describes the distortion you'd expect when plotting the cylindrically derived coordinates onto the flat plane. It's clear that for small angles the distortion is very small, but that it gets exponentially bigger as the angle increases up to 90 degrees.

    I used excel to work out some values:

    angle difference
    1 3.72206E-05
    2 0.000297873
    3 0.001005936
    4 0.002386479
    5 0.004666219
    6 0.008074083
    7 0.012841779
    8 0.019204385
    9 0.027400961
    10 0.037675166
    11 0.05027592
    12 0.06545808
    13 0.083483156
    14 0.104620059
    15 0.129145897
    16 0.157346814
    17 0.189518881
    18 0.225969048
    19 0.267016164
    20 0.312992061
    21 0.364242734
    22 0.421129598
    23 0.484030853
    24 0.553342961
    25 0.629482248
    26 0.712886644
    27 0.804017581
    28 0.903362063
    29 1.011434936
    30 1.128781365
    31 1.255979569
    32 1.393643817
    33 1.542427741
    34 1.703027994
    35 1.8761883
    36 2.062703943
    37 2.263426764
    38 2.479270726
    39 2.711218123
    40 2.960326538
    41 3.227736634
    42 3.514680928
    43 3.822493663
    44 4.152621983
    45 4.506638569
    46 4.886256015
    47 5.293343193
    48 5.729943951
    49 6.198298549
    50 6.700868299
    51 7.240363998
    52 7.819778844
    53 8.442426679
    54 9.111986612
    55 9.832555281
    56 10.60870834
    57 11.44557309
    58 12.34891482
    59 13.3252397
    60 14.38191838
    61 15.52733514
    62 16.77106891
    63 18.12411461
    64 19.59915553
    65 21.21090104
    66 22.97650882
    67 24.91611711
    68 27.0535222
    69 29.4170495
    70 32.0406858
    71 34.96556928
    72 38.24197599
    73 41.93200756
    74 46.11328674
    75 50.88413124
    76 56.37094474
    77 62.73901936
    78 70.20873915
    79 79.08062204
    80 89.77538678
    81 102.9007312
    82 119.3681945
    83 140.6101861
    84 169.0140455
    85 208.8769712
    86 268.7933451
    87 368.816705
    88 569.1076344
    89 1170.468991
    90 3.42816E+17

    I don't know why it's given a value for angle = 90 because no ray can exist for such a scenario.

    Now - this is rather a long winded way to produce a function between angle and difference. Is there a tidier trigonometrical function to get what I want without needing a big long excel formua to subtract a side of a triangle from the arc of a circle. It just feels a bit long winded. Perhaps it will be obvious that what I want can be calculated much more simply. Thanks
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  2. #2
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    Maybe this could be helpful for you: Mercator projection - Wikipedia, the free encyclopedia
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