Now that we have explored Polar Coordinates and Cylindrical Coordinates, let's look at another common three-dimensional coordinate system - Spherical Coordinates.
To locate a point in the Spherical coordinate system, let's imagine we are standing
at the origin of a three-dimensional coordinate system. With the positive z-axis
pointing "upwards", we face towards the positive x-axis and then rotate θ
degrees counter-clockwise. We then point our finger straight up so that it is
parallel with the z-axis, then lower it until it makes an angle of φ degrees.
The point we are looking for is now ρ units away from the origin in the direction
we are pointing.
Note that while the image uses r to denote distance, we will use ρ in this problem.
So θ is the direction we are facing on the x-y plane (called
the azimuth), while φ is the angle from the positive z-axis (called the
inclination), and a single point is defined in the format (ρ, θ, φ).
Note that while this naming convention is typical for math applications, it is
common in other applications (such as physics) to use φ to represent the azimuth
and θ to represent the inclination. For consistency, this site will use θ to represent the azimuth and φ to represent the inclination.
Problem Statement
Input Data
First line will be Q, the quantity of testcases.
Q lines will then follow with a single testcase each, either in the format
C x y z or S ρ θ φ, where C or S indicate if the values are in Cartesian
or Spherical coordinates, and the following three values are the coordinates of the
point.
Answer
Should consist of 3 * Q space-separated values, corresponding to the given
coordinates after being converted into the other coordinate system.
Error should be no more than 1e-6
Example
input data:
5
C 0 3 0
C 4.5 -6.7 8.9
S 3 456 0
S 4.5 -6.7 8.9
S -9.8 7.6 -5.4
answer:
3 90 90 12.014574 -56.113041 42.203209 0 0 3 0.691442 -0.081226 4.445819 0.914160 0.121975 -9.756507