X of NURBS

Intersections in between two Cubic-NURBS

Data sample of intersections

Here below are few pieces of NURBS intersection data for anyone who is intrested to do a double check.

Ex-01 degree 3 * degree 3

Blue NURBS -- Nonuniform, degree: 3, Ring-mode: OFF;

Control points: 11;

P00(233.00, 131.00), P01(288.00, 246.00), P02( 96.00, 373.00), P03(303.00, 327.00), P04(346.00, 478.00), P05(411.00, 335.00),

P06(572.00, 436.00), P07(542.00, 252.00), P08(631.00, 154.00), P09(467.00, 201.00), P10(453.00, 103.00)

Knots vector: 15;

{ 3.00, 3.00, 3.00, 3.00, 11.00, 20.00, 27.00, 33.00, 38.00, 45.00, 53.00, 53.00, 62.00, 65.00, 68.00 }

Control points of poly-Bezier: 25;

VALID (233.000000000000000, 131.000000000000000), (288.000000000000000, 246.000000000000000), (197.647058823529390, 305.764705882352930),

VALID (182.283737024221440, 330.189158016147640), (165.000000000000000, 357.666666666666690), (242.624999999999970, 340.416666666666630),

VALID (286.480823863636370, 367.616950757575750), (320.590909090909120, 388.772727272727250), (334.272727272727250, 436.818181818181810),

VALID (354.198523698523670, 429.048562548562590), (371.277777777777770, 422.388888888888910), (392.944444444444460, 374.722222222222170),

VALID (432.065656565656580, 371.419191919191920), (464.666666666666690, 368.666666666666630), (509.388888888888910, 396.722222222222230),

VALID (532.351851851851850, 393.921296296296300), (564.500000000000000, 390.000000000000000), (554.000000000000000, 325.600000000000020),

VALID (567.782222222222120, 269.911111111111150), (583.533333333333300, 206.266666666666680), (631.000000000000000, 154.000000000000000),

VALID (553.823529411764640, 176.117647058823540), (553.823529411764640, 176.117647058823540), (553.823529411764640, 176.117647058823540),

(553.823529411764640, 176.117647058823540)

Red NURBS -- Nonuniform, degree: 3, Ring-mode: ON;

Control points: 8;

P00(415.00, 115.00), P01(150.00, 229.00), P02(222.00, 424.00), P03(497.00, 411.00), P04(645.00, 283.00), P05(415.00, 115.00),

P06(150.00, 229.00), P07(222.00, 424.00)

Knots vector: 12;

{ 0.00, 0.00, 0.00, 3.00, 11.00, 20.00, 27.00, 27.00, 30.00, 38.00, 47.00, 54.00 }

Control points of poly-Bezier: 16;

VALID (293.110743801652920, 176.679752066115700), (160.800000000000010, 258.250000000000000), (189.600000000000020, 336.250000000000000),

VALID (247.984313725490210, 375.504901960784310), (313.666666666666690, 419.666666666666630), (416.791666666666630, 414.791666666666630),

VALID (508.736979166666630, 372.158854166666630), (580.250000000000000, 339.000000000000000), (645.000000000000000, 283.000000000000000),

VALID (484.000000000000000, 165.399999999999980), (484.000000000000000, 165.399999999999980), (484.000000000000000, 165.399999999999980),

VALID (484.000000000000000, 165.399999999999980), (415.000000000000000, 115.000000000000000), (342.727272727272750, 146.090909090909090),

(293.110743801652920, 176.679752066115700)

The intersection point(s) of above two NURBS: 8;

X00(250.433286023772210, 207.336361817501110), X01(200.079593922803380, 309.686327971267470), X02(219.376033286838090, 349.870979789034210),

X03(321.280897082626780, 403.260140472970990), X04(378.474111627186060, 405.984678769864500), X05(484.772987921869570, 382.322803214774690),

X06(558.044822693010930, 342.086898570587100), X07(572.590639154713470, 253.355903815887130)

The t value(s) of X point(s) of Blue NURBS: 8;

t00 = 5.003976470984580, t01 = 9.303052690478090, t02 = 16.005210294584195,

t03 = 23.010977225280026, t04 = 29.465897173177449, t05 = 35.451785784683210,

t06 = 41.943276377189356, t07 = 45.706631396041963

The t value(s) of X point(s) of Red NURBS: 8;

t00 = 4.010112519632701, t01 = 7.702805653314900, t02 = 9.515317924499612,

t03 = 13.891773968621063, t04 = 15.826893393025717, t05 = 19.225198830715613,

t06 = 21.765588465168268, t07 = 24.963410220273321

Ex02 degree 3 * degree 3

Blue NURBS -- Uniform, degree: 3, Ring-mode: ON;

Control points: 13;

P00(281.00, 288.00), P01(204.00, 464.00), P02(393.00, 340.00), P03(556.00, 482.00), P04(512.00, 301.00), P05(751.00, 206.00),

P06(474.00, 177.00), P07(427.00, 34.00), P08(330.00, 187.00), P09(107.00, 170.00), P10(281.00, 288.00), P11(204.00, 464.00),

P12(393.00, 340.00)

Knots vector: 17;

{ 0.00, 1.00, 2.00, 3.00, 4.00, 5.00, 6.00, 7.00, 8.00, 9.00, 10.00, 11.00, 12.00, 13.00, 14.00, 15.00, }

{ 1.00 }

Control points of poly-Bezier: 31;

VALID (248.333333333333340, 414.000000000000000), (267.000000000000000, 422.666666666666630), (330.000000000000060, 381.333333333333370),

VALID (388.666666666666740, 384.333333333333370), (447.333333333333370, 387.333333333333370), (501.666666666666690, 434.666666666666690),

VALID (521.500000000000000, 428.166666666666630), (541.333333333333260, 421.666666666666630), (526.666666666666740, 361.333333333333370),

VALID (559.166666666666740, 315.333333333333370), (591.666666666666740, 269.333333333333310), (671.333333333333370, 237.666666666666690),

VALID (665.000000000000000, 217.000000000000000), (658.666666666666630, 196.333333333333310), (566.333333333333370, 186.666666666666690),

VALID (512.333333333333370, 158.000000000000000), (458.333333333333370, 129.333333333333340), (442.666666666666690, 81.666666666666671),

VALID (418.666666666666630, 83.333333333333343), (394.666666666666630, 85.000000000000000), (362.333333333333370, 136.000000000000030),

VALID (309.000000000000000, 158.666666666666690), (255.666666666666690, 181.333333333333310), (181.333333333333340, 175.666666666666690),

VALID (173.166666666666690, 192.500000000000000), (165.000000000000000, 209.333333333333340), (223.000000000000000, 248.666666666666690),

VALID (239.166666666666660, 297.666666666666690), (255.333333333333310, 346.666666666666690), (229.666666666666690, 405.333333333333370),

(248.333333333333340, 414.000000000000000)

Red NURBS -- Nonuniform, degree: 3, Ring-mode: ON;

Control points: 8;

P00(136.00, 302.00), P01(351.00, 468.00), P02(619.00, 341.00), P03(615.00, 99.00), P04(262.00, 115.00), P05(136.00, 302.00),

P06(351.00, 468.00), P07(619.00, 341.00)

Knots vector: 12;

{ 0.00, 5.00, 14.00, 22.00, 31.00, 40.00, 42.00, 51.00, 59.00, 68.00, 77.00, 79.00 }

Control points of poly-Bezier: 16;

VALID (350.404977375565640, 419.190045248868780), (433.461538461538450, 428.923076923076910), (526.230769230769280, 384.961538461538450),

VALID (571.715384615384660, 308.530769230769240), (617.200000000000050, 232.100000000000020), (615.399999999999980, 123.200000000000020),

VALID (485.104545454545420, 109.290909090909080), (456.149999999999980, 106.199999999999990), (420.850000000000020, 107.800000000000010),

VALID (389.556698564593260, 112.688038277511960), (248.736842105263180, 134.684210526315780), (189.052631578947370, 223.263157894736820),

VALID (200.366396761133610, 295.368182900690610), (210.423076923076910, 359.461538461538450), (276.576923076923090, 410.538461538461550),

(350.404977375565640, 419.190045248868780)

The intersection point(s) of above two NURBS: 10;

X00(295.036805127753840, 404.272861499322350), X01(457.460021383564250, 404.026093750103090), X02(544.397441299525890, 344.916148996594870),

X03(580.710298782180870, 291.698964779687000), X04(594.816900660440300, 187.229436438413700), X05(452.836443958424640, 107.709201748263440),

X06(371.680527382515890, 115.921196026572930), X07(248.911203138980340, 174.250785454474510), X08(204.583425911182870, 241.901443675604040),

X09(242.845902511902070, 370.603896727491470)

The t value(s) of X point(s) of Blue NURBS: 10;

t00 = 3.443174606235227, t01 = 4.415992109206632, t02 = 5.794503490192864,

t03 = 6.181369263376602, t04 = 7.562455283551381, t05 = 8.517401670062238,

t06 = 9.529247426502009, t07 = 10.350631030634428, t08 = 11.579208904844538,

t09 = 12.497302874893402

The t value(s) of X point(s) of Red NURBS: 10;

t00 = 56.912398055510806, t01 = 25.835011278203773, t02 = 29.454932188081585,

t03 = 31.642978566762807, t04 = 35.431504448229290, t05 = 40.699537734634625,

t06 = 42.390579118830090, t07 = 46.003228968779389, t08 = 48.845877791822339,

t09 = 54.559735379918614

Ex03 degree 3 * degree 3

Blue NURBS -- Nonuniform, degree: 3, Ring-mode: ON;

Control points: 13;

P00(281.00, 288.00), P01(204.00, 464.00), P02(393.00, 340.00), P03(556.00, 482.00), P04(512.00, 301.00), P05(751.00, 206.00),

P06(474.00, 177.00), P07(427.00, 34.00), P08(330.00, 187.00), P09(107.00, 170.00), P10(281.00, 288.00), P11(204.00, 464.00),

P12(393.00, 340.00)

Knots vector: 17;

{ 0.00, 0.00, 8.00, 9.00, 15.00, 18.00, 18.00, 20.00, 29.00, 37.00, 42.00, 42.00, 50.00, 51.00, 57.00, 60.00, }

{ 0.00 }

Control points of poly-Bezier: 31;

VALID (233.100000000000020, 401.885714285714300), (222.899999999999980, 451.600000000000020), (336.299999999999950, 377.199999999999930),

VALID (446.544444444444480, 415.511111111111060), (501.666666666666690, 434.666666666666690), (556.000000000000000, 482.000000000000000),

VALID (529.600000000000020, 373.399999999999980), (529.600000000000020, 373.399999999999980), (529.600000000000020, 373.399999999999980),

VALID (529.600000000000020, 373.399999999999980), (512.000000000000000, 301.000000000000000), (555.454545454545500, 283.727272727272750),

VALID (585.706829056111360, 269.040017398869110), (721.842105263157920, 202.947368421052630), (590.631578947368440, 189.210526315789480),

VALID (518.706304531381870, 151.775541795665620), (454.772727272727250, 118.500000000000000), (437.681818181818130, 66.500000000000000),

VALID (394.374663797740710, 104.440828402366860), (367.307692307692320, 128.153846153846160), (330.000000000000000, 187.000000000000000),

VALID (244.230769230769250, 180.461538461538450), (244.230769230769250, 180.461538461538450), (244.230769230769250, 180.461538461538450),

VALID (244.230769230769250, 180.461538461538450), (107.000000000000000, 170.000000000000000), (261.666666666666690, 274.888888888888910),

VALID (242.348148148148140, 369.980246913580290), (239.933333333333340, 381.866666666666670), (234.800000000000010, 393.599999999999970),

(233.100000000000020, 401.885714285714300)

Red NURBS -- Nonuniform, degree: 3, Ring-mode: ON;

Control points: 8;

P00(136.00, 302.00), P01(351.00, 468.00), P02(619.00, 341.00), P03(615.00, 99.00), P04(262.00, 115.00), P05(136.00, 302.00),

P06(351.00, 468.00), P07(619.00, 341.00)

Knots vector: 12;

{ 0.00, 0.00, 8.00, 9.00, 15.00, 18.00, 18.00, 26.00, 27.00, 33.00, 36.00, 36.00 }

Control points of poly-Bezier: 16;

VALID (281.114285714285760, 409.271428571428600), (377.800000000000010, 455.299999999999950), (538.599999999999910, 379.100000000000020),

VALID (590.422222222222220, 246.144444444444470), (616.333333333333370, 179.666666666666690), (615.000000000000000, 99.000000000000000),

VALID (518.727272727272750, 103.363636363636360), (518.727272727272750, 103.363636363636360), (518.727272727272750, 103.363636363636360),

VALID (518.727272727272750, 103.363636363636360), (262.000000000000000, 115.000000000000000), (150.000000000000000, 281.222222222222230),

VALID (239.481481481481470, 378.387654320987620), (250.666666666666660, 390.533333333333300), (265.000000000000000, 401.600000000000020),

(281.114285714285760, 409.271428571428600)

The intersection point(s) of above two NURBS: 10;

X00(291.327818927225850, 413.595925710885920), X01(421.027881081683400, 408.547387273355070), X02(578.319918989318580, 272.598376631177930),

X03(525.536931292176750, 342.578251604597880), X04(604.472094685767390, 186.655295767634950), X05(458.977417447159890, 109.152073615787400),

X06(365.743835213757390, 133.015507926772700), X07(281.322211213222430, 179.074181966050360), X08(215.338890519318990, 265.634873804699960),

X09(240.064376944079300, 379.016567691431020)

The t value(s) of X point(s) of Blue NURBS: 10;

t00 = 11.917708910781101, t01 = 14.537057320468932, t02 = 19.841863124837591,

t03 = 18.321049833609408, t04 = 25.748545666054216, t05 = 32.257318689222195,

t06 = 38.521481402692288, t07 = 41.211972819755935, t08 = 47.005220497106748,

t09 = 50.255796460501841

The t value(s) of X point(s) of Red NURBS: 10;

t00 = 9.206697395119475, t01 = 11.469771187321090, t02 = 14.589776043290964,

t03 = 13.329623345709681, t04 = 15.877440182812212, t05 = 18.650793777494613,

t06 = 19.832886153555414, t07 = 21.249606801224136, t08 = 23.314146228122386,

t09 = 26.017287155115664

Ex04 degree 3 * degree 4

Blue NURBS -- Nonuniform, degree: 3, Ring-mode: ON;

Control points: 13;

P00(281.00, 288.00), P01(204.00, 464.00), P02(393.00, 340.00), P03(556.00, 482.00), P04(512.00, 301.00), P05(751.00, 206.00),

P06(474.00, 177.00), P07(427.00, 34.00), P08(330.00, 187.00), P09(107.00, 170.00), P10(281.00, 288.00), P11(204.00, 464.00),

P12(393.00, 340.00)

Knots vector: 17;

{ 0.00, 2.00, 4.00, 13.00, 18.00, 20.00, 25.00, 33.00, 40.00, 43.00, 45.00, 49.00, 51.00, 60.00, 65.00, 67.00, }

{ 2.00 }

Control points of poly-Bezier: 31;

VALID (280.937500000000000, 399.517857142857110), (310.312500000000000, 394.250000000000000), (369.375000000000000, 355.500000000000000),

VALID (434.761904761904760, 386.690476190476150), (460.916666666666690, 399.166666666666630), (488.083333333333310, 422.833333333333310),

VALID (505.811904761904710, 432.842857142857160), (550.133333333333330, 457.866666666666670), (535.466666666666700, 397.533333333333300),

VALID (549.421794871794870, 351.270512820512810), (571.750000000000000, 277.250000000000000), (667.350000000000020, 239.250000000000000),

VALID (646.304074074074150, 214.642592592592560), (627.888888888888910, 193.111111111111090), (520.166666666666630, 181.833333333333310),

VALID (468.658333333333300, 120.058333333333340), (446.583333333333310, 93.583333333333329), (434.833333333333370, 57.833333333333336),

VALID (410.733333333333350, 74.133333333333340), (394.666666666666630, 85.000000000000000), (373.111111111111090, 119.000000000000010),

VALID (340.157407407407390, 140.250000000000030), (274.250000000000000, 182.750000000000000), (162.750000000000000, 174.250000000000000),

VALID (156.516666666666680, 192.394444444444450), (153.400000000000010, 201.466666666666670), (176.600000000000020, 217.199999999999990),

VALID (193.831818181818190, 234.072727272727260), (271.375000000000000, 310.000000000000000), (228.062500000000000, 409.000000000000000),

(280.937500000000000, 399.517857142857110)

Red NURBS -- Nonuniform, degree: 4, Ring-mode: ON;

Control points: 9;

P00(136.00, 302.00), P01(351.00, 468.00), P02(619.00, 341.00), P03(615.00, 99.00), P04(262.00, 115.00), P05(136.00, 302.00),

P06(351.00, 468.00), P07(619.00, 341.00), P08(615.00, 99.00)

Knots vector: 14;

{ 0.00, 2.00, 4.00, 13.00, 18.00, 20.00, 25.00, 27.00, 36.00, 41.00, 43.00, 48.00, 50.00, 59.00 }

Control points of poly-Bezier: 21;

VALID (548.594671201814090, 347.376417233560120), (566.293650793650840, 329.682539682539700), (580.944444444444460, 308.222222222222230), (591.246031746031690, 283.015873015873010),

VALID (595.987024439405330, 257.441672965482500), (607.839506172839490, 193.506172839506180), (584.938271604938340, 127.271604938271590), (508.355379188712560, 111.522045855379190),

VALID (431.625842466616290, 119.143061854371380), (400.934027777777770, 122.191468253968250), (370.218750000000000, 128.979166666666660), (342.906250000000000, 137.770833333333310),

VALID (319.193497474747520, 148.216540404040390), (212.486111111111110, 195.222222222222230), (178.673611111111110, 275.722222222222230), (235.704861111111140, 357.829365079365060),

VALID (327.840702947845780, 390.815334467120120), (379.027281746031750, 409.140873015873010), (441.048611111111090, 412.305555555555600), (504.347222222222230, 391.611111111111140),

(548.594671201813980, 347.376417233560120)

The intersection point(s) of above two NURBS: 10;

X00(318.478725353640810, 387.219480185612720), X01(455.004446022656620, 398.139741541786410), X02(551.891131141422650, 343.989521524040130),

X03(585.553892002903350, 292.250416085278570), X04(592.228630835697120, 190.383854136649750), X05(466.692832944589900, 117.644639704141550),

X06(341.487919764120310, 139.381100436151030), X07(293.473643319878420, 161.160981772891550), X08(219.059474286005810, 265.592312861674830),

X09(243.781689037343710, 332.822028672840930)

The t value(s) of X point(s) of Blue NURBS: 10;

t00 = 14.635644241775895, t01 = 18.515388019947384, t02 = 25.266598564546086,

t03 = 27.474636420870088, t04 = 35.744529584471046, t05 = 40.090280123948091,

t06 = 44.972957019656242, t07 = 45.848963847690570, t08 = 52.208651090930253,

t09 = 54.805725173228851

The t value(s) of X point(s) of Red NURBS: 10;

t00 = 35.767861543989511, t01 = 38.797696655775439, t02 = 18.094289465980594,

t03 = 19.304062581307008, t04 = 21.366443259351893, t05 = 24.423391930600939,

t06 = 26.552363402770517, t07 = 27.581155205197490, t08 = 31.191216140753415,

t09 = 33.398572536323528