02-Jul-2023 08:59:27 circle_segment_test(): MATLAB/Octave version 5.2.0. Test circle_segment(). circle_segment_test01(): circle_segment_area_from_height() computes the area of a circle segment. R H Area 1.000000 1.000000 1.570796 1.000000 0.500000 0.614185 1.000000 0.250000 0.226656 1.000000 0.125000 0.081753 1.000000 0.062500 0.029185 1.000000 0.031250 0.010368 1.000000 0.015625 0.003674 1.000000 0.007812 0.001301 1.000000 0.003906 0.000460 1.000000 0.001953 0.000163 1.000000 0.000977 0.000058 circle_segment_test02(): gqcircsegm() computes a Gauss quadrature rule for a circle segment with circle of radius R and center (0,0), with segment angles in [-theta/2,+theta/2]. Tabulate some rules. Rule of precision 0 THETA = 6.28319 Grid order: 3 Sum(W) = 3.14159 X Y W 0.926632 0 0.258589 6.12323e-17 0 2.62442 -0.926632 0 0.258589 Rule of precision 1 THETA = 6.28319 Grid order: 4 Sum(W) = 3.14159 X Y W 0.970028 0 0.0703739 0.480049 0 1.50042 -0.480049 0 1.50042 -0.970028 0 0.0703739 Rule of precision 2 THETA = 6.28319 Grid order: 10 Sum(W) = 3.14159 X Y W 0.985634 -0.0975123 0.0119222 0.716222 -0.402917 0.360189 6.12323e-17 -0.57735 0.826575 -0.716222 -0.402917 0.360189 -0.985634 -0.0975123 0.0119222 0.985634 0.0975123 0.0119222 0.716222 0.402917 0.360189 6.12323e-17 0.57735 0.826575 -0.716222 0.402917 0.360189 -0.985634 0.0975123 0.0119222 Rule of precision 3 THETA = 6.28319 Grid order: 12 Sum(W) = 3.14159 X Y W 0.992298 -0.0715201 0.00473805 0.835113 -0.317588 0.174542 0.339731 -0.543011 0.606118 -0.339731 -0.543011 0.606118 -0.835113 -0.317588 0.174542 -0.992298 -0.0715201 0.00473805 0.992298 0.0715201 0.00473805 0.835113 0.317588 0.174542 0.339731 0.543011 0.606118 -0.339731 0.543011 0.606118 -0.835113 0.317588 0.174542 -0.992298 0.0715201 0.00473805 Rule of precision 4 THETA = 6.28319 Grid order: 21 Sum(W) = 3.14159 X Y W 0.995514 -0.0732842 0.00117913 0.898628 -0.339824 0.0492883 0.558451 -0.642557 0.218402 6.12323e-17 -0.774597 0.334925 -0.558451 -0.642557 0.218402 -0.898628 -0.339824 0.0492883 -0.995514 -0.0732842 0.00117913 0.995514 -4.26715e-18 0.0018866 0.898628 -1.97871e-17 0.0788612 0.558451 -3.74145e-17 0.349444 6.12323e-17 -4.51028e-17 0.53588 -0.558451 -3.74145e-17 0.349444 -0.898628 -1.97871e-17 0.0788612 -0.995514 -4.26715e-18 0.0018866 0.995514 0.0732842 0.00117913 0.898628 0.339824 0.0492883 0.558451 0.642557 0.218402 6.12323e-17 0.774597 0.334925 -0.558451 0.642557 0.218402 -0.898628 0.339824 0.0492883 -0.995514 0.0732842 0.00117913 Rule of precision 5 THETA = 6.28319 Grid order: 24 Sum(W) = 3.14159 X Y W 0.997217 -0.057753 0.000579528 0.93462 -0.275484 0.0264176 0.697838 -0.554809 0.137205 0.261542 -0.747634 0.27213 -0.261542 -0.747634 0.27213 -0.697838 -0.554809 0.137205 -0.93462 -0.275484 0.0264176 -0.997217 -0.057753 0.000579528 0.997217 -3.36281e-18 0.000927245 0.93462 -1.60408e-17 0.0422682 0.697838 -3.23052e-17 0.219528 0.261542 -4.35329e-17 0.435409 -0.261542 -4.35329e-17 0.435409 -0.697838 -3.23052e-17 0.219528 -0.93462 -1.60408e-17 0.0422682 -0.997217 -3.36281e-18 0.000927245 0.997217 0.057753 0.000579528 0.93462 0.275484 0.0264176 0.697838 0.554809 0.137205 0.261542 0.747634 0.27213 -0.261542 0.747634 0.27213 -0.697838 0.554809 0.137205 -0.93462 0.275484 0.0264176 -0.997217 0.057753 0.000579528 Rule of precision 0 THETA = 3.14159 Grid order: 3 Sum(W) = 1.5708 X Y W 0.98372 0 0.0285153 0.707107 0 0.687812 0.179706 0 0.85447 Rule of precision 1 THETA = 3.14159 Grid order: 4 Sum(W) = 1.5708 X Y W 0.993717 0 0.00699072 0.866597 0 0.252172 0.499009 0 0.760526 0.111919 0 0.551108 Rule of precision 2 THETA = 3.14159 Grid order: 10 Sum(W) = 1.5708 X Y W 0.997101 -0.0439271 0.00110751 0.932985 -0.207797 0.0485584 0.707107 -0.408248 0.219222 0.359915 -0.538659 0.326297 0.0760839 -0.575677 0.190213 0.997101 0.0439271 0.00110751 0.932985 0.207797 0.0485584 0.707107 0.408248 0.219222 0.359915 0.538659 0.326297 0.0760839 0.575677 0.190213 Rule of precision 3 THETA = 3.14159 Grid order: 12 Sum(W) = 1.5708 X Y W 0.998487 -0.0317441 0.000420179 0.963239 -0.155102 0.0205674 0.824494 -0.326706 0.115731 0.565871 -0.476022 0.245691 0.268645 -0.556126 0.264417 0.0549824 -0.576477 0.138571 0.998487 0.0317441 0.000420179 0.963239 0.155102 0.0205674 0.824494 0.326706 0.115731 0.565871 0.476022 0.245691 0.268645 0.556126 0.264417 0.0549824 0.576477 0.138571 Rule of precision 4 THETA = 3.14159 Grid order: 21 Sum(W) = 1.5708 X Y W 0.999136 -0.0321849 0.000101096 0.978334 -0.160366 0.00529996 0.890587 -0.352296 0.0340918 0.707107 -0.547723 0.0893122 0.454812 -0.689846 0.130719 0.207031 -0.757815 0.118352 0.0415505 -0.773928 0.0584563 0.999136 -1.87404e-18 0.000161753 0.978334 -9.33769e-18 0.00847994 0.890587 -2.05133e-17 0.0545469 0.707107 -3.18925e-17 0.1429 0.454812 -4.0168e-17 0.20915 0.207031 -4.41256e-17 0.189363 0.0415505 -4.50639e-17 0.09353 0.999136 0.0321849 0.000101096 0.978334 0.160366 0.00529996 0.890587 0.352296 0.0340918 0.707107 0.547723 0.0893122 0.454812 0.689846 0.130719 0.207031 0.757815 0.118352 0.0415505 0.773928 0.0584563 Rule of precision 5 THETA = 3.14159 Grid order: 24 Sum(W) = 1.5708 X Y W 0.999472 -0.0251643 4.84375e-05 0.986468 -0.126996 0.0026583 0.929017 -0.28663 0.0186652 0.799536 -0.465237 0.0559976 0.600619 -0.619318 0.0992311 0.370038 -0.719613 0.117649 0.163951 -0.764115 0.0962366 0.032487 -0.774188 0.0458463 0.999472 -1.46525e-18 7.75e-05 0.986468 -7.39466e-18 0.00425327 0.929017 -1.66898e-17 0.0298643 0.799536 -2.70896e-17 0.0895962 0.600619 -3.60613e-17 0.15877 0.370038 -4.19013e-17 0.188238 0.163951 -4.44925e-17 0.153979 0.032487 -4.5079e-17 0.0733541 0.999472 0.0251643 4.84375e-05 0.986468 0.126996 0.0026583 0.929017 0.28663 0.0186652 0.799536 0.465237 0.0559976 0.600619 0.619318 0.0992311 0.370038 0.719613 0.117649 0.163951 0.764115 0.0962366 0.032487 0.774188 0.0458463 Rule of precision 0 THETA = 1.5708 Grid order: 3 Sum(W) = 0.285398 X Y W 0.996045 0 0.0034545 0.92388 0 0.101863 0.767135 0 0.180081 Rule of precision 1 THETA = 1.5708 Grid order: 4 Sum(W) = 0.285398 X Y W 0.998493 0 0.000827221 0.966468 0 0.0336787 0.864972 0 0.12862 0.744848 0 0.122272 Rule of precision 2 THETA = 1.5708 Grid order: 10 Sum(W) = 0.285398 X Y W 0.99931 -0.0214398 0.000129186 0.983493 -0.104468 0.00615 0.92388 -0.220942 0.0325636 0.823382 -0.327639 0.0604919 0.732877 -0.392807 0.0433645 0.99931 0.0214398 0.000129186 0.983493 0.104468 0.00615 0.92388 0.220942 0.0325636 0.823382 0.327639 0.0604919 0.732877 0.392807 0.0433645 Rule of precision 3 THETA = 1.5708 Grid order: 12 Sum(W) = 0.285398 X Y W 0.999642 -0.0154468 4.85453e-05 0.991066 -0.0770014 0.00252381 0.955505 -0.170305 0.0159225 0.884223 -0.26966 0.0399201 0.795097 -0.350153 0.0521886 0.725772 -0.39718 0.0320955 0.999642 0.0154468 4.85453e-05 0.991066 0.0770014 0.00252381 0.955505 0.170305 0.0159225 0.884223 0.26966 0.0399201 0.795097 0.350153 0.0521886 0.725772 0.39718 0.0320955 Rule of precision 4 THETA = 1.5708 Grid order: 21 Sum(W) = 0.285398 X Y W 0.999796 -0.0156284 1.16012e-05 0.994784 -0.0790097 0.000637003 0.972744 -0.179613 0.00446727 0.92388 -0.296425 0.013285 0.851798 -0.405788 0.0228016 0.775544 -0.488997 0.0244002 0.72123 -0.53656 0.0136746 0.999796 -9.09999e-19 1.85619e-05 0.994784 -4.60054e-18 0.0010192 0.972744 -1.04584e-17 0.00714763 0.92388 -1.72601e-17 0.0212561 0.851798 -2.3628e-17 0.0364825 0.775544 -2.84731e-17 0.0390403 0.72123 -3.12425e-17 0.0218793 0.999796 0.0156284 1.16012e-05 0.994784 0.0790097 0.000637003 0.972744 0.179613 0.00446727 0.92388 0.296425 0.013285 0.851798 0.405788 0.0228016 0.775544 0.488997 0.0244002 0.72123 0.53656 0.0136746 Rule of precision 5 THETA = 1.5708 Grid order: 24 Sum(W) = 0.285398 X Y W 0.999876 -0.0122004 5.53062e-06 0.996765 -0.0622557 0.000314896 0.982545 -0.144095 0.00236619 0.94891 -0.244421 0.00784097 0.894106 -0.346907 0.015795 0.826304 -0.436271 0.0216903 0.761651 -0.501929 0.0204689 0.718156 -0.539028 0.0107956 0.999876 -7.104e-19 8.84899e-06 0.996765 -3.62499e-18 0.000503833 0.982545 -8.39029e-18 0.00378591 0.94891 -1.4232e-17 0.0125456 0.894106 -2.01995e-17 0.0252719 0.826304 -2.5403e-17 0.0347045 0.761651 -2.92261e-17 0.0327502 0.718156 -3.13862e-17 0.0172729 0.999876 0.0122004 5.53062e-06 0.996765 0.0622557 0.000314896 0.982545 0.144095 0.00236619 0.94891 0.244421 0.00784097 0.894106 0.346907 0.015795 0.826304 0.436271 0.0216903 0.761651 0.501929 0.0204689 0.718156 0.539028 0.0107956 Rule of precision 0 THETA = 0.785398 Grid order: 3 Sum(W) = 0.0391457 X Y W 0.999018 0 0.000428449 0.980785 0 0.0132733 0.939925 0 0.0254439 Rule of precision 1 THETA = 0.785398 Grid order: 4 Sum(W) = 0.0391457 X Y W 0.999627 0 0.000102012 0.991606 0 0.0042782 0.965604 0 0.0173025 0.933986 0 0.017463 Rule of precision 2 THETA = 0.785398 Grid order: 10 Sum(W) = 0.0391457 X Y W 0.99983 -0.0106566 1.58762e-05 0.995889 -0.0522991 0.000771032 0.980785 -0.112635 0.0042464 0.954747 -0.171716 0.00831198 0.930786 -0.211059 0.00622756 0.99983 0.0106566 1.58762e-05 0.995889 0.0522991 0.000771032 0.980785 0.112635 0.0042464 0.954747 0.171716 0.00831198 0.930786 0.211059 0.00622756 Rule of precision 3 THETA = 0.785398 Grid order: 12 Sum(W) = 0.0391457 X Y W 0.999912 -0.00767229 5.9523e-06 0.997782 -0.0384299 0.000313962 0.988838 -0.086022 0.00203747 0.970585 -0.139002 0.00532004 0.947303 -0.184948 0.00727171 0.928883 -0.213835 0.00462371 0.999912 0.00767229 5.9523e-06 0.997782 0.0384299 0.000313962 0.988838 0.086022 0.00203747 0.970585 0.139002 0.00532004 0.947303 0.184948 0.00727171 0.928883 0.213835 0.00462371 Rule of precision 4 THETA = 0.785398 Grid order: 21 Sum(W) = 0.0391457 X Y W 0.99995 -0.00775859 1.42018e-06 0.998708 -0.0393594 7.88433e-05 0.993193 -0.090227 0.000564763 0.980785 -0.151116 0.00173302 0.962166 -0.211049 0.00309 0.942131 -0.259679 0.00343197 0.927666 -0.289242 0.00197379 0.99995 -4.51763e-19 2.27228e-06 0.998708 -2.2918e-18 0.000126149 0.993193 -5.25369e-18 0.000903621 0.980785 -8.79912e-18 0.00277284 0.962166 -1.22888e-17 0.00494399 0.942131 -1.51205e-17 0.00549115 0.927666 -1.68419e-17 0.00315806 0.99995 0.00775859 1.42018e-06 0.998708 0.0393594 7.88433e-05 0.993193 0.090227 0.000564763 0.980785 0.151116 0.00173302 0.962166 0.211049 0.00309 0.942131 0.259679 0.00343197 0.927666 0.289242 0.00197379 Rule of precision 5 THETA = 0.785398 Grid order: 24 Sum(W) = 0.0391457 X Y W 0.999969 -0.00605461 6.76235e-07 0.9992 -0.0309758 3.88389e-05 0.995654 -0.0721345 0.000296699 0.987167 -0.123697 0.0010077 0.973135 -0.178341 0.00209468 0.955502 -0.228494 0.002977 0.938444 -0.26757 0.002898 0.926843 -0.290823 0.0015602 0.999969 -3.52544e-19 1.08198e-06 0.9992 -1.80364e-18 6.21422e-05 0.995654 -4.20021e-18 0.000474719 0.987167 -7.20254e-18 0.00161233 0.973135 -1.03843e-17 0.00335148 0.955502 -1.33046e-17 0.0047632 0.938444 -1.55799e-17 0.00463681 0.926843 -1.69339e-17 0.00249633 0.999969 0.00605461 6.76235e-07 0.9992 0.0309758 3.88389e-05 0.995654 0.0721345 0.000296699 0.987167 0.123697 0.0010077 0.973135 0.178341 0.00209468 0.955502 0.228494 0.002977 0.938444 0.26757 0.002898 0.926843 0.290823 0.0015602 circle_segment_test03(): gqcircsegm() computes a Gauss quadrature rule for a circle segment with circle of radius R and center (0,0), with segment angles in [-theta/2,+theta/2]. Plot some rules. Graphics saved as "rule_p9_t3.14159.png". Graphics saved as "rule_p9_t1.5708.png". Graphics saved as "rule_p9_t0.785398.png". Graphics saved as "rule_p9_t0.392699.png". circle_segment_test04(): gqcircsegm() computes a Gauss quadrature rule for a circle segment with circle of radius R and center (0,0), with segment angles in [-theta/2,+theta/2]. Radius R = 1 Angle Theta = 0.785398 Estimate the integral of (x+y+2)^k using precision P rule. P: 1 3 5 7 9 11 K 0 0.965604.5 0.988838.5 0.995654.5 0.998013.5 0.998976.5 0.999422.5 1 3.853949.5 3.953072.5 3.981791.5 3.991688.5 3.995718.5 3.997585.5 2 15.382010.5 15.803171.5 15.923856.5 15.965291.5 15.982134.5 15.989929.5 3 61.393192.5 63.176242.5 63.682202.5 63.855327.5 63.925586.5 63.958070.5 4 245.034547.5 252.559280.5 254.675921.5 255.397954.5 255.690539.5 255.825699.5 5 977.990031.5 1009.654705.5 1018.492190.5 1021.498412.5 1022.714939.5 1023.276474.5 6 3903.386325.5 4036.290512.5 4073.122954.5 4085.620062.5 4090.670891.5 4093.000619.5 7 15579.325263.5 16135.854182.5 16289.109289.5 16340.986037.5 16361.928139.5 16371.581376.5 8 62180.720904.5 64506.206726.5 65142.909864.5 65357.967847.5 65444.690994.5 65484.641145.5 9 248177.760390.5 257876.072706.5 260517.541520.5 261407.968369.5 261766.678286.5 261931.827322.5 10 990535.327620.5 1030909.617060.5 1041853.819268.5 1045536.270135.5 1047018.372606.5 1047700.360943.5 11 3953457.528673.5 4121261.144528.5 4166550.069489.5 4181762.702149.5 4187880.137203.5 4190693.653162.5 circle_segment_test05(): for circle segment with a given radius r, circle_segment_area_from_height() computes the area a, given the height. circle_segment_height_from_area() computes height H, given the area. Check that these functions are inverses of each other using random values of R, A, and H. R H => A => H2 1.245157 2.151451 4.473100 2.151451 2.579155 4.892919 20.490429 4.892919 1.744315 0.851534 1.806679 0.851534 0.523962 0.510657 0.417299 0.510657 4.149542 6.532125 45.672532 6.532125 R A => H => A2 4.548168 11.740885 2.150562 11.740885 4.386661 24.350291 3.714236 24.350292 1.594196 0.206956 0.198745 0.206956 3.582195 32.584521 5.397931 32.584521 3.958018 44.593084 6.729770 44.593085 circle_segment_test06(): circle_segment_sample_from_height() samples a circle segment. Plot 100 points from several segments. Graphics saved as "sample_t3.14159.png". Graphics saved as "sample_t1.5708.png". Graphics saved as "sample_t0.785398.png". Graphics saved as "sample_t0.392699.png". circle_segment_test07(): for circle segment with a given radius r, circle_segment_angle_from_height() computes the angle theta, given the height. circle_segment_height_from_angle() computes height H, given the angle. Check that these functions are inverses of each other using random values of R, T, and H. R H => T => H2 2.876996 2.696507 3.016039 2.696507 0.571479 0.334622 2.286888 0.334622 0.730698 1.053788 4.057616 1.053788 0.210532 0.158764 2.644718 0.158764 0.170507 0.164487 3.070969 0.164487 R T => H => T2 3.857612 3.104687 3.786433 3.104687 3.145218 6.182296 6.286435 6.182296 2.446393 5.592013 4.748148 5.592013 2.728998 1.322039 0.574818 1.322039 1.933598 4.068935 2.798371 4.068935 circle_segment_test08(): circle_segment_contains_point() reports whether a circle segment contains a point. Pick a circle segment at random. Compute 1000 sample points in the surrounding box. Compare the area of the segment to the percentage of points contained in the circle segment. N Omega1 Omega2 Area Estimate 1000 2.46477 3.46598 0.0795434 0.084 1000 1.281 4.13277 1.28299 1.308 1000 1.28491 5.70899 2.69141 2.676 1000 5.97394 10.0182 2.41466 2.372 1000 5.46367 9.20746 2.15512 2.156 circle_segment_test09(): circle_segment_area_from_chord() and circle_segment_centroid_from_chord() evaluate the area and centroid of a circle segment, given r, c and p1:p2. circle_segment_area_from_sample() and circle_segment_centroid_from_sample() give us monte carlo estimates. gqcircsegm() can estimate these values by quadrature. Start easy, with R = 1, C = (0,0), and Theta centered. Area CentroidX CentroidY 0.0391457 0.954429 0 0.0391457 0.954429 -0 0.0395841 0.954285 0.00102525 circle_segment_test10(): gqcircsegm() computes a Gauss quadrature rule for a circle segment with circle of radius R and center (0,0), with segment angles in [-theta/2,+theta/2]. Radius R = 1 Angle Theta = 0.785398 Estimate the integral of (x+y+2)^k using QUAD2D. K 0 0.039146.5 1 0.115653.5 2 0.342862.5 3 1.019859.5 4 3.043618.5 5 9.112436.5 6 27.367568.5 7 82.443510.5 8 249.089055.5 9 754.725272.5 10 2293.066015.5 11 6985.452528.5 circle_segment_test11(): circle_segment_rotation_from_chord() is given the endpoints of a chord, and is asked to determine the angle of the central radius vector. We make a table of all pairs of angles that are multiples of pi/12, determine the corresponding chord endpoints, and compute the rotation angle, also printed as a multiple of pi/12. 0.0 1.0 2.0 3.0 4.0 5.0 6.0 7.0 8.0 9.0 10.0 11.0 12.0 0.0: 6.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0 5.5 6.0 1.0: 6.5 7.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0 5.5 6.0 6.5 2.0: 7.0 7.5 8.0 2.5 3.0 3.5 4.0 4.5 5.0 5.5 6.0 6.5 7.0 3.0: 7.5 8.0 8.5 9.0 3.5 4.0 4.5 5.0 5.5 6.0 6.5 7.0 7.5 4.0: 8.0 8.5 9.0 9.5 10.0 4.5 5.0 5.5 6.0 6.5 7.0 7.5 8.0 5.0: 8.5 9.0 9.5 10.0 10.5 11.0 5.5 6.0 6.5 7.0 7.5 8.0 8.5 6.0: 9.0 9.5 10.0 10.5 11.0 11.5 0.0 6.5 7.0 7.5 8.0 8.5 9.0 7.0: -2.5 -2.0 -1.5 -1.0 -0.5 0.0 0.5 1.0 -4.5 -4.0 -3.5 -3.0 -2.5 8.0: -2.0 -1.5 -1.0 -0.5 -0.0 0.5 1.0 1.5 2.0 -3.5 -3.0 -2.5 -2.0 9.0: -1.5 -1.0 -0.5 -0.0 0.5 1.0 1.5 2.0 2.5 3.0 -2.5 -2.0 -1.5 10.0: -1.0 -0.5 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 -1.5 -1.0 11.0: -0.5 -0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0 -0.5 12.0: -0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0 5.5 6.0 circle_segment_test12(): circle_segment_rule_from_chord() computes a quadrature rule for a circle segment of radius R, center C, chord P1:P2, algebraic precision P. Plot some such rules. Created graphics image "test12_rule1.png". Graphics saved as "test12_rule2.png". circle_segment_test13(): gauss() computes the points and weights for a Gauss quadrature rule, given the ALPHA and BETA recursion coefficients. LEGENDRE RULE Point Weight -0.973907 0.0666713 -0.865063 0.149451 -0.67941 0.219086 -0.433395 0.269267 -0.148874 0.295524 0.148874 0.295524 0.433395 0.269267 0.67941 0.219086 0.865063 0.149451 0.973907 0.0666713 HERMITE RULE Point Weight -3.43616 7.64043e-06 -2.53273 0.00134365 -1.75668 0.0338744 -1.03661 0.240139 -0.342901 0.610863 0.342901 0.610863 1.03661 0.240139 1.75668 0.0338744 2.53273 0.00134365 3.43616 7.64043e-06 LAGUERRE RULE Point Weight 0.137793 0.308441 0.729455 0.40112 1.80834 0.218068 3.40143 0.0620875 5.5525 0.00950152 8.33015 0.000753008 11.8438 2.82592e-05 16.2793 4.24931e-07 21.9966 1.83956e-09 29.9207 9.91183e-13 circle_segment_test14(): r_jacobi() computes recursion coefficients ALPHA and BETA for a Jacobi weight w(x)=(1-x)^A * (1+x)^B. Legendre A = 0, B = 0 Alpha Beta 0 2 0 0.333333 0 0.266667 0 0.257143 0 0.253968 0 0.252525 0 0.251748 0 0.251282 0 0.25098 0 0.250774 Chebyshev Type 1 A = -0.5, B = -0.5 Alpha Beta 0 3.14159 0 0.5 0 0.25 0 0.25 0 0.25 0 0.25 0 0.25 0 0.25 0 0.25 0 0.25 Chebyshev Type 2 A = 0.5, B = 0.5 Alpha Beta 0 1.5708 0 0.25 0 0.25 0 0.25 0 0.25 0 0.25 0 0.25 0 0.25 0 0.25 0 0.25 General Jacobi weight A = 0.5, B = 1.5 Alpha Beta 0.25 1.5708 0.0833333 0.1875 0.0416667 0.222222 0.025 0.234375 0.0166667 0.24 0.0119048 0.243056 0.00892857 0.244898 0.00694444 0.246094 0.00555556 0.246914 0.00454545 0.2475 circle_segment_test17(): Test gqcircsect(). Created graphics image "test16_acute.png". Created graphics image "test16_obtuse.png". circle_segment_test17(): Test gqcircsegm(). Created graphics image "test17.png". circle_segment_test(): Normal end of execution. 02-Jul-2023 09:00:03