curve_intersections.json

[
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        "intersections": [
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        "curve1": "1",
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        "note": "The curves in this intersection are coincident: curve1([1/4, 1]) == curve24([0, 3/4]).",
        "intersections": [
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        "type": "tangent",
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        "curve2_params": [
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        "type": "standard",
        "intersections": [
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        "curve1_params": [
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        "curve2_params": [
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        "id": 26,
        "type": "no-intersection",
        "intersections": [[], []],
        "curve1_params": [],
        "curve2_params": []
    }, {
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        "id": 27,
        "type": "no-intersection",
        "intersections": [[], []],
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        "curve2_params": []
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        "curve1": "32",
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        "id": 28,
        "type": "standard",
        "intersections": [
            ["-3/64"],
            ["-1/4"]
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        "curve1_params": [
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        "curve2_params": [
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    }, {
        "curve1": "34",
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        "id": 29,
        "type": "standard",
        "intersections": [
            ["1/2"],
            ["21/8"]
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        "curve1_params": [
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        "curve2_params": [
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    }, {
        "curve1": "36",
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        "id": 30,
        "type": "standard",
        "intersections": [
            ["1/2"],
            ["-3/8"]
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        "curve1_params": [
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        "curve2_params": [
            "4/7"
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    }, {
        "curve1": "38",
        "curve2": "39",
        "id": 31,
        "type": "tangent",
        "intersections": [
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            [0]
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        "curve1_params": [
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        "curve2_params": [
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    }, {
        "curve1": "40",
        "curve2": "41",
        "id": 32,
        "type": "no-intersection",
        "intersections": [[], []],
        "curve1_params": [],
        "curve2_params": []
    }, {
        "curve1": "42",
        "curve2": "43",
        "id": 33,
        "type": "coincident",
        "note": "The curves in this intersection are coincident: curve42([1, 2]) == curve43([0, 1]). The algebraic curve that contains them both also has a self-crossing present on these curves.",
        "intersections": [
            ["-1/4", "-3/32"],
            ["11/8", "53/64"]
        ],
        "curve1_params": [
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        "curve1_polys": [
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        "curve2_polys": [
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    }, {
        "curve1": "44",
        "curve2": "45",
        "id": 34,
        "type": "coincident",
        "note": "The curves in this intersection are coincident: curve44([1, 2]) == curve45([0, 1]).",
        "intersections": [
            ["-75/256"],
            ["517/512"]
        ],
        "curve1_params": [
            1
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        "curve2_params": [
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    }, {
        "curve1": "46",
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        "id": 35,
        "type": "coincident",
        "note": "The curves in this intersection are coincident: curve46([1, 2]) == curve47([0, 1]).",
        "intersections": [
            ["-9/256"],
            ["583/512"]
        ],
        "curve1_params": [
            1
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        "curve2_params": [
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    }, {
        "curve1": "48",
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        "id": 36,
        "type": "standard",
        "intersections": [
            [1],
            [0]
        ],
        "curve1_params": [
            "1/2"
        ],
        "curve2_params": [
            "9/10"
        ]
    }, {
        "curve1": "50",
        "curve2": "54",
        "id": 37,
        "type": "standard",
        "intersections": [
            ["17/8"],
            ["23/8"]
        ],
        "curve1_params": [
            1
        ],
        "curve1_polys": [
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        ],
        "curve2_params": [
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        "curve2_polys": [
            [-12, 60, 1]
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    }, {
        "curve1": "51",
        "curve2": "54",
        "id": 38,
        "type": "standard",
        "intersections": [
            ["17/8"],
            ["23/8"]
        ],
        "curve1_params": [
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        ],
        "curve1_polys": [
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        "curve2_params": [
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        "curve2_polys": [
            [-12, 60, 1]
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    }, {
        "curve1": "52",
        "curve2": "54",
        "id": 39,
        "type": "standard",
        "intersections": [
            ["17/8"],
            ["23/8"]
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        "curve1_params": [
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        ],
        "curve1_polys": [
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        "curve2_params": [
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        "curve2_polys": [
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    }, {
        "curve1": "53",
        "curve2": "54",
        "id": 40,
        "type": "standard",
        "intersections": [
            ["17/8"],
            ["23/8"]
        ],
        "curve1_params": [
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        ],
        "curve1_polys": [
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        "curve2_params": [
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        "curve2_polys": [
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    }, {
        "curve1": "1",
        "curve2": "55",
        "id": 41,
        "type": "tangent",
        "intersections": [
            ["1/2"],
            ["1/2"]
        ],
        "curve1_params": [
            "1/2"
        ],
        "curve2_params": [
            "1/2"
        ]
    }, {
        "curve1": "56",
        "curve2": "57",
        "id": 42,
        "type": "tangent",
        "note": "The curves in this intersection are not only tangent, they also have the same curvature (-1/8) at this point. This corresponds to the parameters being a triple root of the functions `f_0(t) = Resultant_s(x_0(s) - x_1(t), y_0(s) - y_1(t)) = 256 (2 t - 9) (2 t - 1)^3` and `f_1(s) = Resultant_t(x_1(t) - x_0(s), y_1(t) - y_0(s)) = 256 (2 s + 3) (2 s - 1)^3`",
        "intersections": [
            [0],
            [0]
        ],
        "curve1_params": [
            "1/2"
        ],
        "curve2_params": [
            "1/2"
        ]
    }, {
        "curve1": "58",
        "curve2": "59",
        "id": 43,
        "type": "tangent",
        "note": "This intersection is uniquely challenging, it is both a point of tangency **and** an endpoint of both edges.",
        "intersections": [
            [1],
            [0]
        ],
        "curve1_params": [
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        ],
        "curve2_params": [
            0
        ]
    }, {
        "curve1": "60",
        "curve2": "59",
        "id": 44,
        "type": "tangent",
        "note": "This intersection is similar to #43, but the intersection occurs on the ``interior'' of the first curve, rather than at an endpoint.",
        "intersections": [
            [1],
            [0]
        ],
        "curve1_params": [
            "1/4"
        ],
        "curve2_params": [
            0
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    }, {
        "curve1": "61",
        "curve2": "62",
        "id": 45,
        "type": "tangent",
        "note": "The curves in this intersection are not only tangent, they also have the same curvature. This intersection is the same as #42, but the curves have been reparameterized so that the intersection occurs at (1/3, 1/3).",
        "intersections": [
            [0],
            [0]
        ],
        "curve1_params": [
            "1/3"
        ],
        "curve2_params": [
            "1/3"
        ]
    }, {
        "curve1": "63",
        "curve2": "64",
        "id": 46,
        "type": "tangent",
        "note": "These curves are tangent, but only up to machine precision. The x-value of the 2nd control point in curve #64 is constructed from an algebraic value that would make these curves **exactly** tangent.",
        "intersections": [
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        "curve1_polys": [
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            [2718441, -10007712, 35492904, -89404128, 118148976, -141239040, 219477760, -215449600, 92550400, -14592000,  2304000],
            [2718441, -10007712, 35492904, -89404128, 118148976, -141239040, 219477760, -215449600, 92550400, -14592000,  2304000]
        ],
        "curve2_params": [
            "0x1.1aa1c972f3091p-1",
            "0x1.c681757f704adp-1",
            "0x1.0ebde6aa68c39p-7"
        ],
        "curve2_polys": [
            [9, -35, -65, 165, 25, 1],
            [40, -4280, -66551, -170260, 491504, -404620, 228094, -51020, 10024, -540, 9],
            [40, -4280, -66551, -170260, 491504, -404620, 228094, -51020, 10024, -540, 9]
        ]
    }, {
        "curve1": "65",
        "curve2": "66",
        "id": 47,
        "type": "tangent",
        "note": "These curves are **exactly** tangent at the apex of curve #66.",
        "intersections": [
            ["1/4", "0x1.49e81d355d4adp-1", "0x1.3b0bf165515aap+0"],
            ["3/4", "0x1.d55399c5397a4p-1", "-0x1.22a9cce29cbd2p+0"]
        ],
        "curve1_params": [
            "3/8",
            "0x1.24f40e9aaea56p-1",
            "0x1.bb0bf165515aap-1"
        ],
        "curve1_polys": [
            [3, -8],
            [507, -1472, 1024],
            [507, -1472, 1024]
        ],
        "curve2_params": [
            "3/4",
            "0x1.d617e2caa2b53p-1",
            "0x1.53d03a6aba959p-2"
        ],
        "curve2_polys": [
            [3, -4],
            [39, -160, 128],
            [39, -160, 128]
        ]
    }, {
        "curve1": "67",
        "curve2": "68",
        "id": 48,
        "type": "standard",
        "note": "These curves (actually lines) are **very** close to identical, but only intersect once.",
        "intersections": [
            ["1/2"],
            ["1/2"]
        ],
        "curve1_params": [
            "1/2"
        ],
        "curve2_params": [
            "1/2"
        ]
    }, {
        "curve1": "69",
        "curve2": "70",
        "id": 49,
        "type": "standard",
        "note": "These curves are **very** close to identical, but only intersect once.",
        "intersections": [
            ["0x1.d413cccfe7799p-1"],
            ["1/2"]
        ],
        "curve1_params": [
            "0x1.6a09e667f3bcdp-1"
        ],
        "curve1_polys": [
            [-1, 0, 2]
        ],
        "curve2_params": [
            "0x1.6a09e667f3bcdp-1"
        ],
        "curve2_polys": [
            [-1, 0, 2]
        ]
    }, {
        "curve1": "71",
        "curve2": "72",
        "id": 50,
        "type": "standard",
        "note": "These curves are **very** close to identical and close to linear, and they intersect twice.",
        "intersections": [
            ["0x1.dbb45f3a33413p-2", "0x1.a7a1e6d182739p-2"],
            ["-0x1.4d03a7a539fe4p-1", "-0x1.717f0642f61dfp-1"]
        ],
        "curve1_params": [
            "0x1.3c07c30226a3cp-2",
            "0x1.7d5b269eb3fecp-1"
        ],
        "curve2_params": [
            "0x1.30f6f2bdde113p-2",
            "0x1.8700df9eac93bp-1"
        ]
    }, {
        "curve1": "73",
        "curve2": "74",
        "id": 51,
        "type": "standard",
        "note": "This is a problematic edge-edge intersection that occurs in surface-surface intersection case #49. This intersection is from a real-world case, so the intersection values do not have a rational or algebraic representation. However, the intersection was computed in high-precision.",
        "intersections": [
            ["0x1.6900a8d9a4a7ep-1"],
            ["-0x1.3d611a06f255fp-1"]
        ],
        "curve1_params": [
            "0x1.42968093ddf88p-2"
        ],
        "curve2_params": [
            "0x1.529acd85b333ep-7"
        ]
    }, {
        "curve1": "75",
        "curve2": "76",
        "id": 52,
        "type": "standard",
        "note": "This is a problematic edge-edge intersection that occurs in surface-surface intersection case #50. This intersection is from a real-world case, so the intersection values do not have a rational or algebraic representation. However, the intersection was computed in high-precision.",
        "intersections": [
            ["0x1.bf8ad4846373bp-1"],
            ["0x1.637d99fa68cacp-3"]
        ],
        "curve1_params": [
            "0x1.7ce08ef2b6425p-4"
        ],
        "curve2_params": [
            "0x1.350363e3ce847p-4"
        ]
    }, {
        "curve1": "77",
        "curve2": "78",
        "id": 53,
        "type": "standard",
        "note": "This is a problematic edge-edge intersection that occurs in surface-surface intersection case #51. This intersection is from a real-world case, so the intersection values do not have a rational or algebraic representation. However, the intersection was computed in high-precision.",
        "intersections": [
            ["0x1.537baa83c36dap-3"],
            ["-0x1.f5949a0ad50f6p-1"]
        ],
        "curve1_params": [
            "0x1.52cfc17cd81b6p-2"
        ],
        "curve2_params": [
            "0x1.c1f44f818b884p-2"
        ]
    }
]