PostgreSQL 三角函数的用法举例 - 已知3点求夹角（旋转门续）

背景

前几天写了一篇关于旋转门的数据压缩算法在PostgreSQL中的实现，里面用到了PostGIS里面的ST_Azimuth函数用来计算夹角，其实在PostgreSQL 中，我们还可以使用三角函数，以及三边来求夹角。

https://yq.aliyun.com/articles/59101

文中用到的计算夹角的方法如下

      SELECT 180-ST_Azimuth(
                              ST_MakePoint(o_x, o_val+i_radius),    -- 门上点
                              ST_MakePoint(v_x, v_val)              -- next point
                           )/(2*pi())*360 as degAz,                 -- 上夹角
                 ST_Azimuth(
                              ST_MakePoint(o_x, o_val-i_radius),    -- 门下点
                              ST_MakePoint(v_x, v_val)              -- next point
                           )/(2*pi())*360 As degAzrev               -- 下夹角
      INTO v_angle1, v_angle2; 

余弦定理

cosA=(b²+c²-a²)/(2bc)

定点为A、B、C； 对的边分别为a、b、c；

PostgreSQL 支持的三角函数

https://www.postgresql.org/docs/9.6/static/functions-math.html

例子

已知三个点A(3,2)，B(1,2.5)，C(1,1)。 求夹角B, C。

套用余弦公式

cosB=(a²+c²-b²)/(2ac)   
  
cosC=(b²+a²-c²)/(2ba)     

首先求三条边长

postgres=# select point_distance(point(3,2), point(1,2.5)) as c , point_distance(point(3,2), point(1,1)) as b , point_distance(point(1,1), point(1,2.5)) as a;
        c         |        b         |  a  
------------------+------------------+-----
 2.06155281280883 | 2.23606797749979 | 1.5
(1 row)  

运算如下

cosB=(a²+c²-b²)/(2ac)      
=(1.5^2 + 2.06155281280883^2 - 2.23606797749979^2) / (2*1.5*2.06155281280883)  
=0.24253562503633260164  
  
cosC=(b²+a²-c²)/(2ba)     
=(1.5^2 + 2.23606797749979^2 - 2.06155281280883^2) / (2*2.23606797749979*1.5)  
=0.44721359549995825124  

求夹角 1 度数

postgres=# select acosd(0.24253562503633260164);
      acosd       
------------------
 75.9637565320735
(1 row)

求夹角 2 度数

postgres=# select acosd(0.44721359549995825124);
      acosd      
-----------------
 63.434948822922
(1 row)

比对使用PostGIS计算的结果一致

test=>  SELECT 180-ST_Azimuth(
                              ST_MakePoint(1,2.5),    -- 门上点
                              ST_MakePoint(3,2)              -- next point
                           )/(2*pi())*360 as degAz,                 -- 上夹角
                 ST_Azimuth(
                              ST_MakePoint(1,1),    -- 门下点
                              ST_MakePoint(3,2)              -- next point
                           )/(2*pi())*360 As degAzrev ;
      degaz       |    degazrev     
------------------+-----------------
 75.9637565320735 | 63.434948822922
(1 row)

源码

三角函数属于浮点运算中的函数

src/backend/utils/adt/float.c

/*
 *              acosd_q1                - returns the inverse cosine of x in degrees, for x in
 *                                                the range [0, 1].  The result is an angle in the
 *                                                first quadrant --- [0, 90] degrees.
 *
 *                                                For the 3 special case inputs (0, 0.5 and 1), this
 *                                                function will return exact values (0, 60 and 90
 *                                                degrees respectively).
 */
static double
acosd_q1(double x)
{
        /*
         * Stitch together inverse sine and cosine functions for the ranges [0,
         * 0.5] and (0.5, 1].  Each expression below is guaranteed to return
         * exactly 60 for x=0.5, so the result is a continuous monotonic function
         * over the full range.
         */
        if (x <= 0.5)
        {
                volatile float8 asin_x = asin(x);

                return 90.0 - (asin_x / asin_0_5) * 30.0;
        }
        else
        {
                volatile float8 acos_x = acos(x);

                return (acos_x / acos_0_5) * 60.0;
        }
}


/*
 *              dacosd                  - returns the arccos of arg1 (degrees)
 */
Datum
dacosd(PG_FUNCTION_ARGS)
{
        float8          arg1 = PG_GETARG_FLOAT8(0);
        float8          result;

        /* Per the POSIX spec, return NaN if the input is NaN */
        if (isnan(arg1))
                PG_RETURN_FLOAT8(get_float8_nan());

        INIT_DEGREE_CONSTANTS();

        /*
         * The principal branch of the inverse cosine function maps values in the
         * range [-1, 1] to values in the range [0, 180], so we should reject any
         * inputs outside that range and the result will always be finite.
         */
        if (arg1 < -1.0 || arg1 > 1.0)
                ereport(ERROR,
                                (errcode(ERRCODE_NUMERIC_VALUE_OUT_OF_RANGE),
                                 errmsg("input is out of range")));

        if (arg1 >= 0.0)
                result = acosd_q1(arg1);
        else
                result = 90.0 + asind_q1(-arg1);

        CHECKFLOATVAL(result, false, true);
        PG_RETURN_FLOAT8(result);
}

man asin 
NAME
       asin, asinf, asinl - arc sine function

SYNOPSIS
       #include <math.h>

       double asin(double x);
       float asinf(float x);
       long double asinl(long double x);

///

CONFORMING TO
       C99, POSIX.1-2001.  The variant returning double also conforms to SVr4, 4.3BSD, C89.

SEE ALSO
       acos(3), atan(3), atan2(3), casin(3), cos(3), sin(3), tan(3)

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