球面坐标的计算

Question

我不得不说，我对基本数学知之甚少感到害怕和惊讶。基本上我所拥有的是原点（0,0,0），我知道圆的半径（10），我知道两个角度（theta和phi）。鉴于这种假设，我想计算球体上的投影点。我通过阅读https://stackoverflow.com/a/969880/1230358，https://stackoverflow.com/a/36369852/1230358，http://tutorial.math.lamar.edu/Classes/CalcII/SphericalCoords.aspx和https://en.wikipedia.org/wiki/Spherical_coordinate_system的答案，提出了底层代码。

我目前的代码：

,     //Match a comma
(?!   //Negative look-ahead. We want to match a comma NOT followed by...
[^(]* //Any number of characters NOT '(', zero or more times
/)    //Followed by the ')' character
)     //Close the lookahead.

代码的输出如下：

#!/usr/bin/env python3
import math


PI = math.pi
PI_2 = PI / 2

def calc_sphere_coordinates(radius, phi, theta):
    # see: https://stackoverflow.com/a/969880/1230358
    # see: https://stackoverflow.com/q/19673067/1230358
    # see: http://mathinsight.org/spherical_coordinates
    # see: https://en.wikipedia.org/wiki/Spherical_coordinate_system
    # see: http://tutorial.math.lamar.edu/Classes/CalcII/SphericalCoords.aspx

    # φ phi is the polar angle, rotated down from the positive z-axis (slope)
    # θ theta is azimuthal angle, the angle of the rotation around the z-axis (aspect)

    # z        
    # |  x
    # | /
    # |/
    # +-------- y

    # both angles need to be in radians, not degrees!
    theta = theta * PI / 180
    phi = phi * PI / 180

    x = radius * math.sin(phi) * math.cos(theta)
    y = radius * math.sin(phi) * math.sin(theta)
    z = radius * math.cos(phi)
    return (x, y, z)

if __name__ == "__main__":

    # calculate point position in hemisphere by rotating down from positive z-axis
    for i in (10, 20, 30, 40, 50, 60, 70 , 80, 90):
        print(calc_sphere_coordinates(10, i, 0))

    print("-"*10)

    # calculate point position in hemisphere by rotating around the z axis
    for i in (10, 20, 30, 40, 50, 60, 70 , 80, 90):
        print(calc_sphere_coordinates(10, 0, i))

    print("-"*10)

    # calculate point position by rotating in both directions
    for i in (10, 20, 30, 40, 50, 60, 70 , 80, 90):
        print(calc_sphere_coordinates(10, i, 90-i))    

(1.7364817766693033, 0.0, 9.84807753012208) (3.420201433256687, 0.0, 9.396926207859085) (4.999999999999999, 0.0, 8.660254037844387) (6.4278760968653925, 0.0, 7.660444431189781) (7.66044443118978, 0.0, 6.427876096865393) (8.660254037844386, 0.0, 5.000000000000001) (9.396926207859083, 0.0, 3.4202014332566884) (9.84807753012208, 0.0, 1.7364817766693041) (10.0, 0.0, 6.123233995736766e-16) ---------- (0.0, 0.0, 10.0) (0.0, 0.0, 10.0) (0.0, 0.0, 10.0) (0.0, 0.0, 10.0) (0.0, 0.0, 10.0) (0.0, 0.0, 10.0) (0.0, 0.0, 10.0) (0.0, 0.0, 10.0) (0.0, 0.0, 10.0) ---------- (0.30153689607045814, 1.7101007166283433, 9.84807753012208) (1.16977778440511, 3.2139380484326963, 9.396926207859085) (2.5, 4.330127018922192, 8.660254037844387) (4.131759111665348, 4.92403876506104, 7.660444431189781) (5.868240888334652, 4.92403876506104, 6.427876096865393) (7.5, 4.330127018922192, 5.000000000000001) (8.83022221559489, 3.2139380484326963, 3.4202014332566884) (9.69846310392954, 1.7101007166283433, 1.7364817766693041) (10.0, 0.0, 6.123233995736766e-16) 的行不应该是(10.0, 0.0, 6.123233995736766e-16)而不是0的z坐标吗？无论使用什么角度6.123233995736766e-16，围绕z轴旋转都会得到相同的结果。

Answer 1

据我所知，你的代码运行得很好。老实说，人们不得不承认6.123233995736766e-16对于所有真实应用来说几乎都是0，对吗？

你的问题归结为

为什么math.cos(math.pi / 2.0)不等于零

原因在于如何生成浮点数并将其存储在计算机中。尝试在python中计算0.1+0.2。 Supprised？如果您想了解更多，只需谷歌浮点错误或任何相关的内容。