三角函数的查表法详解

很多算法中都会用到三角函数的计算，不管是在 IMU 的姿态解算中，还是FOC电机控制中做 Clarke 和 Park 变换时，都无法避免三角函数的运算，如果调用默认的C 语言函数，其效率会非常低，如果是在 M4 内核上，因为内核具备了浮点运算单元，速度还能稍快一些，但是对于 M0 内核的单片机，就不得不考虑更快速的查表法了。

为了数学和物理中的标准相同，我们在计算三角函数时通常采用弧度表示，因此，在编程的时候，所有的库函数都采用了相同的标准，也就是使用弧度。角度与弧度的转换关系为：

在程序中，我们可以利用宏定义来进行快速的的单位换算：

#define?_PI ?3.14159265358979f#define?_2PI 6.28318530717958f
const?float?r2d = (180.0f/_PI);const?float?d2r = (_PI/180.0f);
//使用方法 ?float?angle_degrees =?60.0; ? ? ? ??// 假设有一个60度的角float?angle_radians = angle_degrees * d2r;?// 转换为弧度

理解了角度与弧度的关系，接下来看一下表的点数，这个点数的大小决定了我们计算时的精度，三角函数是一个周期函数，我们只需要计算一个周期就可以，本质上我们是对于三角函数波形的一个周期内进行点位分割。我们先拿曲线拟合来举例子。

在曲线拟合的时候，我们会把曲线进行分段，然后在每一个小的分段里默认函数的对应关系为 y=kx 是线性的。只要我们分段足够多，也就是在线上标注的点越多，我们就能够越精确地找到 x 和 y 的对应关系。

换到三角函数上面也是一样的，拿 SIN 正弦函数来说

上图中，x 的范围设定为一个周期，也就是 0 → ?2π，对应到角度时 0 → 360度，如果我们在 x 轴上均匀的取 1024 个点，提前先计算好对应的y 值，那么有一个 x，我就能够根据数组的索引查询到对应的y，也就实现了一个 1024 点的sin(x)计算。

三角函数的定点查表法（1024 点）

// 外部定义的正弦查找表 (0-1023 对应 0-2π)const?int16_t?SinTable[1024] =?{// ?0 ? ? 1 ? ? 2 ? ? 3 ? ? 4 ? ? 5 ? ? 6 ? ? 7 ? ? 8 ? ? 9 ? ? 10 ? ?11 ? ?12 ? ?13 ? ?14 ? ?15 ??? ??0, ? ?201, ?402, ?603, ?804, ?1005,?1206,?1407,?1608,?1809,?2009,?2210,?2410,?2611,?2811,?3012,? ??3212,?3412,?3612,?3811,?4011,?4210,?4410,?4609,?4808,?5007,?5205,?5404,?5602,?5800,?5998,?6195,? ??6393,?6590,?6786,?6983,?7179,?7375,?7571,?7767,?7962,?8157,?8351,?8545,?8739,?8933,?9126,?9319,? ??9512,?9704,?9896,?10087,10278,10469,10659,10849,11039,11228,11417,11605,11793,11980,12167,12353,? ??12539,12725,12910,13094,13279,13462,13645,13828,14010,14191,14372,14553,14732,14912,15090,15269,? ??15446,15623,15800,15976,16151,16325,16499,16673,16846,17018,17189,17360,17530,17700,17869,18037,? ??18204,18371,18537,18703,18868,19032,19195,19357,19519,19680,19841,20000,20159,20317,20475,20631,? ??20787,20942,21096,21250,21403,21554,21705,21856,22005,22154,22301,22448,22594,22739,22884,23027,? ??23170,23311,23452,23592,23731,23870,24007,24143,24279,24413,24547,24680,24811,24942,25072,25201,? ??25329,25456,25582,25708,25832,25955,26077,26198,26319,26438,26556,26674,26790,26905,27019,27133,? ??27245,27356,27466,27575,27683,27790,27896,28001,28105,28208,28310,28411,28510,28609,28706,28803,? ??28898,28992,29085,29177,29268,29358,29447,29534,29621,29706,29791,29874,29956,30037,30117,30195,? ??30273,30349,30424,30498,30571,30643,30714,30783,30852,30919,30985,31050,31113,31176,31237,31297,? ??31356,31414,31470,31526,31580,31633,31685,31736,31785,31833,31880,31926,31971,32014,32057,32098,? ??32137,32176,32213,32250,32285,32318,32351,32382,32412,32441,32469,32495,32521,32545,32567,32589,? ??32609,32628,32646,32663,32678,32692,32705,32717,32728,32737,32745,32752,32757,32761,32765,32766,? ??32767,32766,32765,32761,32757,32752,32745,32737,32728,32717,32705,32692,32678,32663,32646,32628,? ??32609,32589,32567,32545,32521,32495,32469,32441,32412,32382,32351,32318,32285,32250,32213,32176,? ??32137,32098,32057,32014,31971,31926,31880,31833,31785,31736,31685,31633,31580,31526,31470,31414,? ??31356,31297,31237,31176,31113,31050,30985,30919,30852,30783,30714,30643,30571,30498,30424,30349,? ??30273,30195,30117,30037,29956,29874,29791,29706,29621,29534,29447,29358,29268,29177,29085,28992,? ??28898,28803,28706,28609,28510,28411,28310,28208,28105,28001,27896,27790,27683,27575,27466,27356,? ??27245,27133,27019,26905,26790,26674,26556,26438,26319,26198,26077,25955,25832,25708,25582,25456,? ??25329,25201,25072,24942,24811,24680,24547,24413,24279,24143,24007,23870,23731,23592,23452,23311,? ??23170,23027,22884,22739,22594,22448,22301,22154,22005,21856,21705,21554,21403,21250,21096,20942,? ??20787,20631,20475,20317,20159,20000,19841,19680,19519,19357,19195,19032,18868,18703,18537,18371,? ??18204,18037,17869,17700,17530,17360,17189,17018,16846,16673,16499,16325,16151,15976,15800,15623,? ??15446,15269,15090,14912,14732,14553,14372,14191,14010,13828,13645,13462,13279,13094,12910,12725,? ??12539,12353,12167,11980,11793,11605,11417,11228,11039,10849,10659,10469,10278,10087,9896,?9704,? ??9512,?9319,?9126,?8933,?8739,?8545,?8351,?8157,?7962,?7767,?7571,?7375,?7179,?6983,?6786,?6590,? ??6393,?6195,?5998,?5800,?5602,?5404,?5205,?5007,?4808,?4609,?4410,?4210,?4011,?3811,?3612,?3412,? ??3212,?3012,?2811,?2611,?2410,?2210,?2009,?1809,?1608,?1407,?1206,?1005,?804, ?603, ?402, ?201,? ??0, ? ?-201,?-402,?-603,?-804,?-1005,-1206,-1407,-1608,-1809,-2009,-2210,-2410,-2611,-2811,-3012,? ??-3212,-3412,-3612,-3811,-4011,-4210,-4410,-4609,-4808,-5007,-5205,-5404,-5602,-5800,-5998,-6195,? ??-6393,-6590,-6786,-6983,?-7179,?-7375,?-7571,?-7767,?-7962,?-8157,?-8351,?-8545,?-8739,?-8933,?-9126,?-9319,? ??-9512,?-9704,?-9896,?-10087,-10278,-10469,-10659,-10849,-11039,-11228,-11417,-11605,-11793,-11980,-12167,-12353,? ??-12539,-12725,-12910,-13094,-13279,-13462,-13645,-13828,-14010,-14191,-14372,-14553,-14732,-14912,-15090,-15269,? ??-15446,-15623,-15800,-15976,-16151,-16325,-16499,-16673,-16846,-17018,-17189,-17360,-17530,-17700,-17869,-18037,? ??-18204,-18371,-18537,-18703,-18868,-19032,-19195,-19357,-19519,-19680,-19841,-20000,-20159,-20317,-20475,-20631,? ??-20787,-20942,-21096,-21250,-21403,-21554,-21705,-21856,-22005,-22154,-22301,-22448,-22594,-22739,-22884,-23027,? ??-23170,-23311,-23452,-23592,-23731,-23870,-24007,-24143,-24279,-24413,-24547,-24680,-24811,-24942,-25072,-25201,? ??-25329,-25456,-25582,-25708,-25832,-25955,-26077,-26198,-26319,-26438,-26556,-26674,-26790,-26905,-27019,-27133,? ??-27245,-27356,-27466,-27575,-27683,-27790,-27896,-28001,-28105,-28208,-28310,-28411,-28510,-28609,-28706,-28803,? ??-28898,-28992,-29085,-29177,-29268,-29358,-29447,-29534,-29621,-29706,-29791,-29874,-29956,-30037,-30117,-30195,? ??-30273,-30349,-30424,-30498,-30571,-30643,-30714,-30783,-30852,-30919,-30985,-31050,-31113,-31176,-31237,-31297,? ??-31356,-31414,-31470,-31526,-31580,-31633,-31685,-31736,-31785,-31833,-31880,-31926,-31971,-32014,-32057,-32098,? ??-32137,-32176,-32213,-32250,-32285,-32318,-32351,-32382,-32412,-32441,-32469,-32495,-32521,-32545,-32567,-32589,? ??-32609,-32628,-32646,-32663,-32678,-32692,-32705,-32717,-32728,-32737,-32745,-32752,-32757,-32761,-32765,-32766,? ??-32767,-32766,-32765,-32761,-32757,-32752,-32745,-32737,-32728,-32717,-32705,-32692,-32678,-32663,-32646,-32628,? ??-32609,-32589,-32567,-32545,-32521,-32495,-32469,-32441,-32412,-32382,-32351,-32318,-32285,-32250,-32213,-32176,? ??-32137,-32098,-32057,-32014,-31971,-31926,-31880,-31833,-31785,-31736,-31685,-31633,-31580,-31526,-31470,-31414,? ??-31356,-31297,-31237,-31176,-31113,-31050,-30985,-30919,-30852,-30783,-30714,-30643,-30571,-30498,-30424,-30349,? ??-30273,-30195,-30117,-30037,-29956,-29874,-29791,-29706,-29621,-29534,-29447,-29358,-29268,-29177,-29085,-28992,? ??-28898,-28803,-28706,-28609,-28510,-28411,-28310,-28208,-28105,-28001,-27896,-27790,-27683,-27575,-27466,-27356,? ??-27245,-27133,-27019,-26905,-26790,-26674,-26556,-26438,-26319,-26198,-26077,-25955,-25832,-25708,-25582,-25456,? ??-25329,-25201,-25072,-24942,-24811,-24680,-24547,-24413,-24279,-24143,-24007,-23870,-23731,-23592,-23452,-23312,? ??-23170,-23027,-22884,-22739,-22594,-22448,-22301,-22154,-22005,-21856,-21705,-21554,-21403,-21250,-21096,-20942,? ??-20787,-20631,-20475,-20317,-20159,-20000,-19841,-19680,-19519,-19357,-19195,-19032,-18868,-18703,-18537,-18371,? ??-18204,-18037,-17869,-17700,-17530,-17360,-17189,-17018,-16846,-16673,-16499,-16325,-16151,-15976,-15800,-15623,? ??-15446,-15269,-15090,-14912,-14732,-14553,-14372,-14191,-14010,-13828,-13645,-13462,-13279,-13094,-12910,-12725,? ??-12539,-12353,-12167,-11980,-11793,-11605,-11417,-11228,-11039,-10849,-10659,-10469,-10278,-10087,-9896,?-9704,? ??-9512,?-9319,?-9126,?-8933,?-8739,?-8545,?-8351,?-8157,?-7962,?-7767,?-7571,?-7375,?-7179,?-6983,?-6786,?-6590,? ??-6393,?-6195,?-5998,?-5800,?-5602,?-5404,?-5205,?-5007,?-4808,?-4609,?-4410,?-4211,?-4011,?-3811,?-3612,?-3412,? ??-3212,?-3012,?-2811,?-2611,?-2410,?-2210,?-2009,?-1809,?-1608,?-1407,?-1206,?-1005,?-804, ?-603, ??-402,?-201};
// 假设的联合体定义，用于将两个int16_t组合成一个int32_t，也就是将正弦值和余弦值放在一起typedef?union?{? ??int32_t?as_int32;? ??struct?? ? {? ? ? ??int16_t?sin_value;?// 高16位存放正弦值? ? ? ??int16_t?cos_value;?// 低16位存放余弦值? ? } as_words;}?sin_cos_result_t;
/**?* @brief 通过查表法快速计算角度的正弦值和余弦值。?* @param angle 输入角度，使用Q16格式（0x0000-0xFFFF 对应 0°-360°）?* @return 一个32位整数，高16位为sin(angle)，低16位为cos(angle)?*/int32_t?CalSinCos(int16_t?angle)?{? ??sin_cos_result_t?result;? ??uint16_t?angle_temp;? ??uint16_t?sin_index;? ??uint16_t?cos_index;
? ??// 1. 将输入角度视为无符号数（0~65535 对应 0~360度）? ? angle_temp = (uint16_t)angle;
? ??// 2. 将角度映射到查找表索引（0~1023）? ??// 右移6位相当于除以64 (65536/64=1024)? ? sin_index = angle_temp >>?6;
? ??// 3. 计算余弦索引：cos(θ) = sin(θ + 90°)? ??// 90度在1024点的表中对应256个索引的偏移? ??// 使用位与操作确保索引在0-1023范围内（相当于对1024取模）? ? cos_index = (sin_index +?256) &?0x03FF;
? ??// 4. 查表获取正弦和余弦值? ? result.as_words.sin_value = SinTable[sin_index];? ? result.as_words.cos_value = SinTable[cos_index];
? ??// 5. 返回组合后的结果? ??return?result.as_int32;}

上面的代码实现了一个 1024 点位的三角函数查表，可以快速的计算出一个角度对应的 sin 和 cos 值。

上面的算法最大的缺点是 1024 个点占用的存储空间太大了，就一个表占用了我们 1K * 2 字节，接下来，我们利用三角函数的特性来把这个表压缩一下。

压缩版 256 点三角函数查表

没错，这里的公式就是一个口诀，奇变偶不变，符号看象限。

sin(90°-α) = cosα

sin(90°+α) = cosα

cos(90°-α) = sinα

cos(90°+α) = -sinα

sin(180°-α) = sinα

sin(180°+α) = -sinα

cos(180°-α) = -cosα

cos(180°+α) = -cosα

sin(270°-α) = -cosα

sin(270°+α) = -cosα

cos(270°-α) = -sinα

cos(270°+α) = sinα

推导太麻，我们从图形上来推测吧。

我们将一个周期的正弦波分成 4 个象限，其中红色为 sin，绿色为 cos

第一象限 (U0_90)：角度θ本身就在0-90度之间。

hSin直接查表 sin(θ)，值为正。记录为查表 +[0 - 256]

hCos通过 cos(θ) 值时正值。波形上正好与 sin 镜像，因此可以理解为反着顺序查表，记录为+[256-0]

这里使用0xFF - (u8)(hindex)就对应 (90° - θ)这个角度在表中的索引。

第二象限 (U90_180)：

正弦：首先值为正，波形上看，正好和第一象限对称，倒序查表即可，+[256 - 0]。

余弦：首先值时负，波形上看，将它加负号后反转到 X 轴上方，正好和第一象限 sin 相同，-[0 - 256]。

第三象限 (U180_270)：

正弦：首先值为负，波形上看，将它加负号反转到 X 轴上方，正好和第一象限 sin 相同，-[0 - 256]。

余弦：首先值为负，波形上看，将它加负号反转到 X 轴上方，正好和第一象限的 cos 相同，-[256 - 0]。

第四象限 (U270_360)：

正弦：首先值为负，波形上看，将它加负号反转到 X 轴上方，正好和第二象限 sin 相同，-[256 - 0]。

余弦：首先值为正，波形上看，它和第一象限的 sin 值一样，所以正序查表，+[0 - 256]

接下来，直接按照不同象限来列式子即可。

#define?SIN_MASK	 0x0300#define?U0_90 ? ?	 0x0200#define?U90_180 ?	 0x0300#define?U180_270 	 0x0000#define?U270_360 	 0x0100
//外部定义的正弦查找表（0°-90°），256个点，Q15格式const?int16_t?hSin_Cos_Table[256] =?{? ??0x0000,0x00C9,0x0192,0x025B,0x0324,0x03ED,0x04B6,0x057F,? ??0x0648,0x0711,0x07D9,0x08A2,0x096A,0x0A33,0x0AFB,0x0BC4,? ??0x0C8C,0x0D54,0x0E1C,0x0EE3,0x0FAB,0x1072,0x113A,0x1201,? ??0x12C8,0x138F,0x1455,0x151C,0x15E2,0x16A8,0x176E,0x1833,? ??0x18F9,0x19BE,0x1A82,0x1B47,0x1C0B,0x1CCF,0x1D93,0x1E57,? ??0x1F1A,0x1FDD,0x209F,0x2161,0x2223,0x22E5,0x23A6,0x2467,? ??0x2528,0x25E8,0x26A8,0x2767,0x2826,0x28E5,0x29A3,0x2A61,? ??0x2B1F,0x2BDC,0x2C99,0x2D55,0x2E11,0x2ECC,0x2F87,0x3041,? ??0x30FB,0x31B5,0x326E,0x3326,0x33DF,0x3496,0x354D,0x3604,? ??0x36BA,0x376F,0x3824,0x38D9,0x398C,0x3A40,0x3AF2,0x3BA5,? ??0x3C56,0x3D07,0x3DB8,0x3E68,0x3F17,0x3FC5,0x4073,0x4121,? ??0x41CE,0x427A,0x4325,0x43D0,0x447A,0x4524,0x45CD,0x4675,? ??0x471C,0x47C3,0x4869,0x490F,0x49B4,0x4A58,0x4AFB,0x4B9D,? ??0x4C3F,0x4CE0,0x4D81,0x4E20,0x4EBF,0x4F5D,0x4FFB,0x5097,? ??0x5133,0x51CE,0x5268,0x5302,0x539B,0x5432,0x54C9,0x5560,? ??0x55F5,0x568A,0x571D,0x57B0,0x5842,0x58D3,0x5964,0x59F3,? ??0x5A82,0x5B0F,0x5B9C,0x5C28,0x5CB3,0x5D3E,0x5DC7,0x5E4F,? ??0x5ED7,0x5F5D,0x5FE3,0x6068,0x60EB,0x616E,0x61F0,0x6271,? ??0x62F1,0x6370,0x63EE,0x646C,0x64E8,0x6563,0x65DD,0x6656,? ??0x66CF,0x6746,0x67BC,0x6832,0x68A6,0x6919,0x698B,0x69FD,? ??0x6A6D,0x6ADC,0x6B4A,0x6BB7,0x6C23,0x6C8E,0x6CF8,0x6D61,? ??0x6DC9,0x6E30,0x6E96,0x6EFB,0x6F5E,0x6FC1,0x7022,0x7083,? ??0x70E2,0x7140,0x719D,0x71F9,0x7254,0x72AE,0x7307,0x735E,? ??0x73B5,0x740A,0x745F,0x74B2,0x7504,0x7555,0x75A5,0x75F3,? ??0x7641,0x768D,0x76D8,0x7722,0x776B,0x77B3,0x77FA,0x783F,? ??0x7884,0x78C7,0x7909,0x794A,0x7989,0x79C8,0x7A05,0x7A41,? ??0x7A7C,0x7AB6,0x7AEE,0x7B26,0x7B5C,0x7B91,0x7BC5,0x7BF8,? ??0x7C29,0x7C59,0x7C88,0x7CB6,0x7CE3,0x7D0E,0x7D39,0x7D62,? ??0x7D89,0x7DB0,0x7DD5,0x7DFA,0x7E1D,0x7E3E,0x7E5F,0x7E7E,? ??0x7E9C,0x7EB9,0x7ED5,0x7EEF,0x7F09,0x7F21,0x7F37,0x7F4D,? ??0x7F61,0x7F74,0x7F86,0x7F97,0x7FA6,0x7FB4,0x7FC1,0x7FCD,? ??0x7FD8,0x7FE1,0x7FE9,0x7FF0,0x7FF5,0x7FF9,0x7FFD,0x7FFE};
// 正余弦值结果结构体typedef?struct?{? ??int16_t?hSin;?// 正弦值，Q15格式? ??int16_t?hCos;?// 余弦值，Q15格式} Sin_Cos_Value;
/**?* @brief 通过查表法计算任意角度的正弦和余弦值。?* @param angle 输入角度，Q15格式，范围[-32768, 32767] 对应 [-π, π] 弧度或 [-180°, 180°]。?* @return Sin_Cos_Value 包含正弦值(hSin)和余弦值(hCos)的结构体，均为Q15格式。?*/Sin_Cos_Value?CalSinCos(int16_t?angle){? ? Sin_Cos_Value result = {0};? ??uint16_t?hindex;
? ??// 1. 角度标准化：将输入角度从[-32768, 32767]转换到[0, 65535]无符号范围。? ??// 这使得角度值可以方便地进行整数运算和象限判断。? ? hindex = (uint16_t)((int32_t)angle +?32768U);
? ??// 2. 计算10位索引并确定象限：将角度范围从65536缩小到1024。? ??// 右移6位（相当于除以64）得到0-1023的索引值，其高2位代表象限。? ? hindex /=?64U;?// 或者使用效率更高的位操作: hindex >>= 6;
? ??// 3. 根据角度所在象限，利用三角函数的对称性进行查表和符号处理。? ??switch?(hindex & SIN_MASK)? ? {? ? ? ??case?U0_90:?// 第一象限 (0° to 90°): sin正, cos正? ? ? ? ? ? result.hSin = hSin_Cos_Table[hindex];? ? ? ? ? ? result.hCos = hSin_Cos_Table[255?- hindex];?// cos(θ) = sin(90°-θ)? ? ? ? ? ??break;
? ? ? ??case?U90_180:?// 第二象限 (90° to 180°): sin正, cos负? ? ? ? ? ? result.hSin = hSin_Cos_Table[255?- hindex];? ? ? ? ? ? result.hCos = -hSin_Cos_Table[hindex];? ? ? ? ? ??break;
? ? ? ??case?U180_270:?// 第三象限 (180° to 270°): sin负, cos负? ? ? ? ? ? result.hSin = -hSin_Cos_Table[hindex];? ? ? ? ? ? result.hCos = -hSin_Cos_Table[255?- hindex];? ? ? ? ? ??break;
? ? ? ??case?U270_360:?// 第四象限 (270° to 360°): sin负, cos正? ? ? ? ? ? result.hSin = -hSin_Cos_Table[255?- hindex];? ? ? ? ? ? result.hCos = hSin_Cos_Table[hindex];? ? ? ? ? ??break;
? ? ? ??default:? ? ? ? ? ??// 理论上不会执行到此，因为hindex的高2位只有00,01,10,11四种情况。? ? ? ? ? ??break;? ? }
? ??return?result;}

最后，再提供一个浮点查表的代码，这里就等于把三角函数的结果计算成浮点，查表方式相同，只是返回值变成了浮点数。

const?float IQSin_Cos_Table[256]={0.0,?0.006135884649154475,?0.012271538285719925,?0.01840672990580482,?0.024541228522912288,?0.030674803176636626,?0.03680722294135883,?0.04293825693494082,0.049067674327418015,	0.055195244349689934,?0.06132073630220858,?0.06744391956366405,?0.07356456359966743,?0.07968243797143013,?0.0857973123444399,?0.09190895649713272,0.0980171403295606,?0.10412163387205459,?0.11022220729388306,?0.11631863091190475,?0.1224106751992162,?0.12849811079379317,?0.13458070850712617,?0.1406582393328492,0.14673047445536175,?0.15279718525844344,?0.15885814333386145,?0.1649131204899699,?0.17096188876030122,?0.17700422041214875,?0.18303988795514095,?0.1890686641498062,0.19509032201612825,?0.2011046348420919,?0.20711137619221856,?0.21311031991609136,?0.2191012401568698,?0.22508391135979283,?0.2310581082806711,?0.2370236059943672,0.24298017990326387,?0.24892760574572015,?0.25486565960451457,?0.2607941179152755,?0.26671275747489837,?0.272621355449949,?0.27851968938505306,?0.2844075372112719,0.29028467725446233,?0.2961508882436238,?0.3020059493192281,?0.30784964004153487,?0.3136817403988915,?0.3195020308160157,?0.3253102921622629,?0.33110630575987643, 0.33688985339222005,?0.3426607173119944,?0.34841868024943456,?0.35416352542049034,?0.3598950365349881,?0.36561299780477385,?0.37131719395183754,?0.37700741021641826, 0.3826834323650898,?0.38834504669882625,?0.3939920400610481,?0.3996241998456468,?0.40524131400498986,?0.4108431710579039,?0.41642956009763715,?0.4220002707997997, 0.4275550934302821,?0.43309381885315196,?0.43861623853852766,?0.4441221445704292,?0.44961132965460654,?0.45508358712634384,?0.46053871095824,?0.4659764957679662, 0.47139673682599764,?0.4767992300633221,?0.4821837720791227,?0.487550160148436,?0.49289819222978404,?0.49822766697278187,?0.5035383837257176,?0.508830142543107,0.5141027441932217,?0.5193559901655896,?0.524589682678469,?0.5298036246862946,?0.5349976198870972,?0.5401714727298929,?0.5453249884220465,?0.5504579729366048, 0.5555702330196022,?0.560661576197336,?0.5657318107836131,?0.5707807458869673,?0.5758081914178453,?0.5808139580957645,?0.5857978574564389,?0.5907597018588742, 0.5956993044924334,?0.600616479383869,?0.6055110414043255,?0.6103828062763095,?0.6152315905806268,?0.6200572117632891,?0.6248594881423863,?0.629638238914927, 0.6343932841636455,?0.6391244448637757,?0.6438315428897914,?0.6485144010221124,?0.6531728429537768,?0.6578066932970786,?0.6624157775901718,?0.6669999223036375, 0.6715589548470183,?0.6760927035753159,?0.680600997795453,?0.6850836677727004,?0.6895405447370668,?0.693971460889654,?0.6983762494089729,?0.7027547444572253, 0.7071067811865475,?0.7114321957452164,?0.7157308252838186,?0.7200025079613817,?0.7242470829514669,?0.7284643904482252,?0.7326542716724128,?0.7368165688773698, 0.7409511253549591,?0.745057785441466,?0.7491363945234593,?0.7531867990436124,?0.7572088465064845,?0.7612023854842618,?0.765167265622459,?0.7691033376455796, 0.773010453362737,?0.7768884656732324,?0.7807372285720944,?0.7845565971555752,?0.7883464276266062,?0.7921065773002124,?0.7958369046088835,?0.799537269107905, 0.8032075314806448,?0.8068475535437992,?0.8104571982525948,?0.8140363297059483,?0.8175848131515837,?0.8211025149911046,?0.8245893027850253,?0.8280450452577558, 0.8314696123025452,?0.83486287498638,?0.838224705554838,?0.8415549774368983,?0.844853565249707,?0.8481203448032971,?0.8513551931052652,?0.8545579883654005, 0.8577286100002721,?0.8608669386377673,?0.8639728561215867,?0.8670462455156926,?0.8700869911087113,?0.8730949784182901,?0.8760700941954066,?0.8790122264286334, 0.8819212643483549,?0.8847970984309378,?0.8876396204028539,?0.8904487232447579,?0.8932243011955153,?0.8959662497561851,?0.8986744656939538,?0.901348847046022, 0.9039892931234433,?0.9065957045149153,?0.9091679830905223,?0.9117060320054299,?0.9142097557035307,?0.9166790599210427,?0.9191138516900578,?0.9215140393420419, 0.9238795325112867,?0.9262102421383113,?0.9285060804732155,?0.9307669610789837,?0.9329927988347388,?0.9351835099389475,?0.937339011912575,?0.9394592236021899, 0.9415440651830208,?0.9435934581619604,?0.9456073253805213,?0.9475855910177411,?0.9495281805930367,?0.9514350209690083,?0.9533060403541938,?0.9551411683057707, 0.9569403357322089,?0.9587034748958716,?0.9604305194155658,?0.9621214042690416,?0.9637760657954398,?0.9653944416976894,?0.9669764710448521,?0.9685220942744173, 0.970031253194544,?0.9715038909862518,?0.9729399522055601,?0.9743393827855759,?0.9757021300385286,?0.9770281426577544,?0.9783173707196277,?0.9795697656854405, 0.9807852804032304,?0.9819638691095552,?0.9831054874312163,?0.984210092386929,?0.9852776423889412,?0.9863080972445987,?0.9873014181578584,?0.9882575677307495, 0.989176509964781,?0.9900582102622971,?0.99090263542778,?0.9917097536690995,?0.99247953459871,?0.9932119492347945,?0.9939069700023561,?0.9945645707342554, 0.9951847266721968,?0.9957674144676598,?0.996312612182778,?0.9968202992911657,?0.9972904566786902,?0.9977230666441916,?0.9981181129001492,?0.9984755805732948, 0.9987954562051724,?0.9990777277526454,?0.9993223845883495,?0.9995294175010931,?0.9996988186962042,?0.9998305817958234,?0.9999247018391445,?0.9999811752826011};