curves.json

{
    "1": {
        "control_points": [
            [0, 0],
            ["1/2", 1],
            [1, 0]
        ],
        "implicitized": [
            [1, 0, 1],
            [-2, 1, 0],
            [2, 2, 0]
        ]
    },
    "2": {
        "control_points": [
            ["9/8", "1/2"],
            ["5/8", "-1/2"],
            ["1/8", "1/2"]
        ],
        "implicitized": [
            [25, 0, 0],
            [-32, 0, 1],
            [-80, 1, 0],
            [64, 2, 0]
        ]
    },
    "3": {
        "control_points": [
            [0, 0],
            ["3/2", 3],
            [3, 0]
        ],
        "implicitized": [
            [3, 0, 1],
            [-6, 1, 0],
            [2, 2, 0]
        ],
        "note": "This curve is #1 scaled up by a factor of 3."
    },
    "4": {
        "control_points": [
            [3, "3/2"],
            ["21/8", "-29/32"],
            ["-3/4", "39/16"]
        ],
        "implicitized": [
            [14301, 0, 0],
            [-9252, 0, 1],
            [-10233, 1, 0],
            [576, 0, 2],
            [2208, 1, 1],
            [2116, 2, 0]
        ]
    },
    "5": {
        "control_points": [
            [0, "3/4"],
            ["1/2", "-1/4"],
            [1, "3/4"]
        ],
        "implicitized": [
            [3, 0, 0],
            [-4, 0, 1],
            [-8, 1, 0],
            [8, 2, 0]
        ]
    },
    "6": {
        "control_points": [
            [0, 1],
            ["1/2", 0],
            [1, 1]
        ],
        "implicitized": [
            [1, 0, 0],
            [-1, 0, 1],
            [-2, 1, 0],
            [2, 2, 0]
        ]
    },
    "7": {
        "control_points": [
            [0, "17/64"],
            ["1/2", "15/64"],
            [1, "17/64"]
        ],
        "implicitized": [
            [17, 0, 0],
            [-64, 0, 1],
            [-4, 1, 0],
            [4, 2, 0]
        ]
    },
    "8": {
        "control_points": [
            [0, "3/8"],
            [1, "3/8"]
        ],
        "implicitized": [
            [3, 0, 0],
            [-8, 0, 1]
        ]
    },
    "9": {
        "control_points": [
            ["1/2", 0],
            ["1/2", "3/4"]
        ],
        "implicitized": [
            [1, 0, 0],
            [-2, 1, 0]
        ]
    },
    "10": {
        "control_points": [
            [0, 0],
            ["9/2", 9],
            [9, 0]
        ],
        "implicitized": [
            [9, 0, 1],
            [-18, 1, 0],
            [2, 2, 0]
        ],
        "note": "This curve is #1 scaled up by a factor of 9."
    },
    "11": {
        "control_points": [
            [0, 8],
            [6, 0]
        ],
        "implicitized": [
            [-24, 0, 0],
            [3, 0, 1],
            [4, 1, 0]
        ]
    },
    "13": {
        "control_points": [
            [0, 0],
            ["1/4", 2],
            ["1/2", -2],
            ["3/4", 2],
            [1, 0]
        ],
        "implicitized": [
            [-1, 0, 1],
            [8, 1, 0],
            [-36, 2, 0],
            [56, 3, 0],
            [-28, 4, 0]
        ]
    },
    "14": {
        "control_points": [
            [0, 0],
            ["3/8", "3/4"],
            ["3/4", "3/8"]
        ],
        "implicitized": [
            [1, 0, 1],
            [-2, 1, 0],
            [2, 2, 0]
        ]
    },
    "15": {
        "control_points": [
            ["1/4", "5/8"],
            ["5/8", "1/4"],
            [1, 1]
        ],
        "implicitized": [
            [1, 0, 0],
            [-1, 0, 1],
            [-2, 1, 0],
            [2, 2, 0]
        ]
    },
    "16": {
        "control_points": [
            ["1/4", "9/16"],
            ["5/8", "3/16"],
            [1, "15/16"]
        ],
        "implicitized": [
            [15, 0, 0],
            [-16, 0, 1],
            [-32, 1, 0],
            [32, 2, 0]
        ]
    },
    "17": {
        "control_points": [
            [11, 8],
            [7, 10],
            [3, 4]
        ],
        "implicitized": [
            [13, 0, 0],
            [8, 0, 1],
            [-18, 1, 0],
            [1, 2, 0]
        ]
    },
    "18": {
        "control_points": [
            [1, 0],
            ["3/2", -1],
            [2, 0]
        ],
        "implicitized": [
            [4, 0, 0],
            [-1, 0, 1],
            [-6, 1, 0],
            [2, 2, 0]
        ]
    },
    "19": {
        "control_points": [
            [2, 0],
            ["3/2", 1],
            [1, 0]
        ],
        "implicitized": [
            [-4, 0, 0],
            [-1, 0, 1],
            [6, 1, 0],
            [-2, 2, 0]
        ]
    },
    "20": {
        "control_points": [
            [1, 0],
            [-1, "1/4"],
            [1, "1/2"]
        ],
        "implicitized": [
            [1, 0, 0],
            [-8, 0, 1],
            [-1, 1, 0],
            [16, 0, 2]
        ]
    },
    "21": {
        "control_points": [
            ["-1/8", "-9/32"],
            ["1/2", "41/32"],
            ["9/8", "-9/32"]
        ],
        "implicitized": [
            [1, 0, 1],
            [-2, 1, 0],
            [2, 2, 0]
        ]
    },
    "22": {
        "control_points": [
            ["25/16", "-1/16"],
            ["-25/16", "1/4"],
            ["25/16", "9/16"]
        ],
        "implicitized": [
            [-1, 0, 0],
            [8, 0, 1],
            [1, 1, 0],
            [-16, 0, 2]
        ]
    },
    "23": {
        "control_points": [
            [3, "9/2"],
            [8, "9/2"]
        ],
        "implicitized": [
            [9, 0, 0],
            [-2, 0, 1]
        ],
        "note": "This curve is #8 scaled up by a factor of 5 and shifted by [3, 21/8]."
    },
    "24": {
        "control_points": [
            ["1/4", "3/8"],
            ["3/4", "7/8"],
            ["5/4", "-5/8"]
        ],
        "implicitized": [
            [1, 0, 1],
            [-2, 1, 0],
            [2, 2, 0]
        ],
        "note": "This is coincident with #1: curve1([1/4, 1]) == curve24([0, 3/4])."
    },
    "25": {
        "control_points": [
            [0, "1/2"],
            ["1/4", 1],
            ["3/4", "3/2"],
            [1, "1/2"]
        ],
        "implicitized": [
            [-17, 0, 0],
            [-348, 0, 1],
            [180, 1, 0],
            [732, 0, 2],
            [-1800, 1, 1],
            [1728, 2, 0],
            [64, 0, 3],
            [-576, 1, 2],
            [1728, 2, 1],
            [-1728, 3, 0]
        ],
        "note": "The algebraic curve that contains this curve has a self-crossing at [22/27, 17/18] when the parameters are (-1 +/- sqrt(33)) / 6."
    },
    "26": {
        "control_points": [
            ["3/8", 7],
            ["17/8", 8],
            ["31/8", 0],
            ["45/8", 1]
        ],
        "implicitized": [
            [6834, 0, 0],
            [-1029, 0, 1],
            [1398, 1, 0],
            [-1152, 2, 0],
            [128, 3, 0]
        ]
    },
    "27": {
        "control_points": [
            ["1/8", "1/4"],
            ["3/8", "3/4"],
            ["5/8", 0],
            ["7/8", "3/16"]
        ],
        "implicitized": [
            [395, 0, 0],
            [3456, 0, 1],
            [-13512, 1, 0],
            [29760, 2, 0],
            [-17920, 3, 0]
        ]
    },
    "28": {
        "control_points": [
            [0, 0],
            ["-1/2", "3/2"],
            [1, 1]
        ],
        "implicitized": [
            [-1, 0, 1],
            [-3, 1, 0],
            [1, 0, 2],
            [2, 1, 1],
            [1, 2, 0]
        ],
        "note": "This curve is #1 rotated about the origin by 45 degrees counterclockwise and scaled up by a factor of sqrt(2)."
    },
    "29": {
        "control_points": [
            [-1, 1],
            ["1/2", "1/2"],
            [0, 2]
        ],
        "implicitized": [
            [2, 0, 0],
            [-3, 0, 1],
            [-1, 1, 0],
            [1, 0, 2],
            [2, 1, 1],
            [1, 2, 0]
        ],
        "note": "This curve is #6 rotated about the origin by 45 degrees counterclockwise and scaled up by a factor of sqrt(2)."
    },
    "30": {
        "control_points": [
            ["1/2", "1/2"],
            ["-1/4", "5/4"]
        ],
        "implicitized": [
            [1, 0, 0],
            [-1, 0, 1],
            [-1, 1, 0]
        ],
        "note": "This curve is #9 rotated about the origin by 45 degrees counterclockwise and scaled up by a factor of sqrt(2)."
    },
    "31": {
        "control_points": [
            ["1/4", "67/4"],
            ["49/4", "3/4"]
        ],
        "implicitized": [
            [-205, 0, 0],
            [12, 0, 1],
            [16, 1, 0]
        ],
        "note": "This curve is #11 scaled up by a factor of 2 and shifted by [1/4, 3/4]."
    },
    "32": {
        "control_points": [
            ["-1/4", "-1/4"],
            ["3/16", "-1/4"],
            ["5/8", "-1/4"]
        ],
        "implicitized": [
            [1, 0, 0],
            [4, 0, 1]
        ],
        "note": "This is a degree-elevated version of the line segment y = -1/4 for -1/4 <= x <= 5/8."
    },
    "33": {
        "control_points": [
            ["-1/8", 0],
            ["1/16", "-1/2"],
            [0, -1]
        ],
        "implicitized": [
            [-1, 0, 0],
            [-3, 0, 1],
            [-8, 1, 0],
            [-2, 0, 2]
        ]
    },
    "34": {
        "control_points": [
            ["1/2", "15/8"],
            ["1/2", "21/8"]
        ],
        "implicitized": [
            [1, 0, 0],
            [-2, 1, 0]
        ]
    },
    "35": {
        "control_points": [
            ["5/8", "21/8"],
            ["-5/8", "21/8"]
        ],
        "implicitized": [
            [21, 0, 0],
            [-8, 0, 1]
        ]
    },
    "36": {
        "control_points": [
            ["1/2", "-3/8"],
            [0, "-3/4"]
        ],
        "implicitized": [
            [3, 0, 0],
            [4, 0, 1],
            [-3, 1, 0]
        ]
    },
    "37": {
        "control_points": [
            ["-1/2", "-7/8"],
            ["5/4", 0]
        ],
        "implicitized": [
            [-5, 0, 0],
            [-8, 0, 1],
            [4, 1, 0]
        ]
    },
    "38": {
        "control_points": [
            ["1/2", "1/8"],
            ["5/4", "-1/4"],
            [2, "1/2"]
        ],
        "implicitized": [
            [1, 0, 0],
            [-2, 0, 1],
            [-2, 1, 0],
            [1, 2, 0]
        ]
    },
    "39": {
        "control_points": [
            ["1/2", "-1/8"],
            [1, "1/8"],
            ["3/2", "-1/8"]
        ],
        "implicitized": [
            [-1, 0, 0],
            [-2, 0, 1],
            [2, 1, 0],
            [-1, 2, 0]
        ]
    },
    "40": {
        "control_points": [
            ["-0x1.1f2347525aff1p-2", "-0x1.b10f43da717d8p-3"],
            ["-0x1.3912a66009422p-2", "-0x1.263d62f131d03p-4"]
        ],
        "note": "This curve and #41 are from a real-world failure, so the values do not have a rational or algebraic representation."
    },
    "41": {
        "control_points": [
            ["-0x1.25a8a28e98475p-2", "-0x1.692bd46818fe8p-3"],
            ["-0x1.7daf3df0a049ep-2", "-0x1.1361ffff031f1p-2"]
        ],
        "note": "This curve and #40 are from a real-world failure, so the values do not have a rational or algebraic representation."
    },
    "42": {
        "control_points": [
            [0, 2],
            ["-1/2", 1],
            ["-1/4", "3/4"],
            ["-3/32", "53/64"]
        ],
        "implicitized": [
            [16, 0, 0],
            [-40, 0, 1],
            [16, 1, 0],
            [32, 0, 2],
            [-24, 1, 1],
            [20, 2, 0],
            [-8, 0, 3],
            [12, 1, 2],
            [-6, 2, 1],
            [1, 3, 0]
        ],
        "note": "The algebraic curve that contains this curve has a self-crossing at [-1/4, 11/8] when the parameters are (3 +/- sqrt(5)) / 3."
    },
    "43": {
        "control_points": [
            ["-3/32", "53/64"],
            ["1/16", "29/32"],
            ["1/8", "21/16"],
            ["-3/4", "13/8"]
        ],
        "implicitized": [
            [16, 0, 0],
            [-40, 0, 1],
            [16, 1, 0],
            [32, 0, 2],
            [-24, 1, 1],
            [20, 2, 0],
            [-8, 0, 3],
            [12, 1, 2],
            [-6, 2, 1],
            [1, 3, 0]
        ],
        "note": "This is coincident with #42: curve43([0, 1]) == curve42([1, 2]), and their shared (parent) algebraic curve has a self-crossing."
    },
    "44": {
        "control_points": [
            [0, 2],
            ["-1/4", "3/2"],
            ["-5/16", "19/16"],
            ["-75/256", "517/512"]
        ],
        "implicitized": [
            [16, 0, 0],
            [-40, 0, 1],
            [16, 1, 0],
            [32, 0, 2],
            [-24, 1, 1],
            [20, 2, 0],
            [-8, 0, 3],
            [12, 1, 2],
            [-6, 2, 1],
            [1, 3, 0]
        ],
        "note": "This is coincident with #42: curve44([0, 1]) == curve42([0, 1/2]), and their shared (parent) algebraic curve has a self-crossing."
    },
    "45": {
        "control_points": [
            ["-75/256", "517/512"],
            ["-35/128", "213/256"],
            ["-11/64", "101/128"],
            ["-3/32", "53/64"]
        ],
        "implicitized": [
            [16, 0, 0],
            [-40, 0, 1],
            [16, 1, 0],
            [32, 0, 2],
            [-24, 1, 1],
            [20, 2, 0],
            [-8, 0, 3],
            [12, 1, 2],
            [-6, 2, 1],
            [1, 3, 0]
        ],
        "note": "This is coincident with #42: curve45([0, 1]) == curve42([1/2, 1]), and their shared (parent) algebraic curve has a self-crossing."
    },
    "46": {
        "control_points": [
            ["-3/32", "53/64"],
            ["-1/64", "111/128"],
            ["5/128", "253/256"],
            ["-9/256", "583/512"]
        ],
        "implicitized": [
            [16, 0, 0],
            [-40, 0, 1],
            [16, 1, 0],
            [32, 0, 2],
            [-24, 1, 1],
            [20, 2, 0],
            [-8, 0, 3],
            [12, 1, 2],
            [-6, 2, 1],
            [1, 3, 0]
        ],
        "note": "This is coincident with #42: curve46([0, 1]) == curve42([1, 3/2]), and their shared (parent) algebraic curve has a self-crossing."
    },
    "47": {
        "control_points": [
            ["-9/256", "583/512"],
            ["-7/64", "165/128"],
            ["-5/16", "47/32"],
            ["-3/4", "13/8"]
        ],
        "implicitized": [
            [16, 0, 0],
            [-40, 0, 1],
            [16, 1, 0],
            [32, 0, 2],
            [-24, 1, 1],
            [20, 2, 0],
            [-8, 0, 3],
            [12, 1, 2],
            [-6, 2, 1],
            [1, 3, 0]
        ],
        "note": "This is coincident with #42: curve47([0, 1]) == curve42([3/2, 2]), and their shared (parent) algebraic curve has a self-crossing."
    },
    "48": {
        "control_points": [
            [1, -3],
            [1, 3]
        ],
        "implicitized": [
            [1, 0, 0],
            [-1, 1, 0]
        ]
    },
    "49": {
        "control_points": [
            [10, 0],
            [0, 0]
        ],
        "implicitized": [
            [1, 0, 1]
        ]
    },
    "50": {
        "control_points": [
            ["9/4", "97/32"],
            ["17/8", "23/8"]
        ],
        "implicitized": [
            [-7, 0, 0],
            [32, 0, 1],
            [-40, 1, 0]
        ]
    },
    "51": {
        "control_points": [
            ["17/8", "23/8"],
            ["37/16", "91/32"]
        ],
        "implicitized": [
            [155, 0, 0],
            [-48, 0, 1],
            [-8, 1, 0]
        ]
    },
    "52": {
        "control_points": [
            ["9/4", "97/32"],
            ["35/16", "189/64"],
            ["17/8", "23/8"]
        ],
        "implicitized": [
            [-7, 0, 0],
            [32, 0, 1],
            [-40, 1, 0]
        ],
        "note": "This is a degree-elevated version of curve #50."
    },
    "53": {
        "control_points": [
            ["17/8", "23/8"],
            ["71/32", "183/64"],
            ["37/16", "91/32"]
        ],
        "implicitized": [
            [155, 0, 0],
            [-48, 0, 1],
            [-8, 1, 0]
        ],
        "note": "This is a degree-elevated version of curve #51."
    },
    "54": {
        "control_points": [
            ["269/128", "23/8"],
            ["553/256", "23/8"],
            ["1137/512", "23/8"]
        ],
        "implicitized": [
            [-23, 0, 0],
            [8, 0, 1]
        ],
        "note": "This curve has a ``bad'' parameterization. It is a line, but the curve appears quadratic: `x(s) = ((s + 30)^2 + 176) / 512, y(s) = 23 / 8`. If `553/256 = 2212/1024` were changed to `2213/1024` then the curve would present as a line: `x(s) = (61 s + 1076) / 512, y(s) = 23 / 8`. There are no self-intersections or cusps on the interval [0, 1], but the curve is still ``bad'' since it has a ``false'' degree of 2 (rather than the actual degree of 1). The quadratic curve behaves badly around `s = -30` because it will lead to duplicate values of `x`, for example `x(-31) = x(-29)` and in general `x(-30 + d) = x(-30 - d)`."
    },
    "55": {
        "control_points": [
            [0, "1/2"],
            [1, "1/2"]
        ],
        "implicitized": [
            [1, 0, 0],
            [-2, 0, 1]
        ]
    },
    "56": {
        "control_points": [
            [12, 4],
            [-4, -4],
            [-4, 4]
        ],
        "implicitized": [
            [-16, 0, 1],
            [1, 0, 2],
            [-2, 1, 1],
            [1, 2, 0]
        ]
    },
    "57": {
        "control_points": [
            [6, 1],
            [-2, -1],
            [-2, 1]
        ],
        "implicitized": [
            [-16, 0, 1],
            [4, 0, 2],
            [-4, 1, 1],
            [1, 2, 0]
        ]
    }
}