1. Practice Contest II Problem H: Two Rings
Problem Created by: Kitamura
Solved by: Kitamura, Hirano
Statement by: Kitamura, Sakuraba

2. The Problem
The Problem: Find the coordinates of two rings on a unit sphere
This problem requires hand calculation rather than tough coding
There are some ways to solve the problem; the solution shown here is just one example

3. Basic Idea of the Solution
Find the plane on which a ring is
Let two planes be P1 and P2
Find the intersection line of P1 and P2
Let the line be L
Find the intersection point of L and a unit sphere

4. Coordinate Conversion
θ [NS] φ [EW] → (x, y, z)
z = sz * sinθ
sz = +1 if [NS] = N, -1 if [NS] = S
x = cosθcosφ
y = sy * cosθsinφ
sy = +1 if [EW] = E, -1 if [EW] = W

5. Find the Planes P1 and P2
Equation of a plane: a(x - x0) + b(y - y0) + c(z - z0) = 0 where:
Normal vector N = (a, b, c)
Point on the plane P = (x0, y0, z0)
We can calculate the equations of P1 and P2 using this formula

6. Find the Intersection Line of P1 and P2
Let the equations of P1 and P2 be:
P1: a1x + b1y + c1z + d1 = 0
P2: a2x + b2y + c2z + d2 = 0
The intersection line L can be expressed as: p + tv = 0 (t: parameter)
v is the cross product of two normal vectors:
v = (a1, b1, c1)×(a2, b2, c2)
Let v be (vx, vy, vz)

7. Find the Intersection Line of P1 and P2 (Cont'd)
If v = 0, there are no intersection line (two planes are parallel)
If vx≠0, L passes through a point p1 = (0, y1, z1)
y1 = (c1d2 - c2d1) / vx
z1 = (b1d2 - b2d1) / (-vx)
If vy≠0, L passes through a point p2 = (x2, 0, z2)
If vz≠0, L passes through a point p3 = (x3, y3, 0)
Choose p1, p2 or p3 as p

8. Find the Intersection Points of L and the unit sphere
We have to find the value of t
The distance between an intersection point and the origin is 1;
|| p + tv || = 1
⇔ || p + tv ||2 = 1
This is the quadratic equation of t

9. The Final Step (x, y, z) → θ [NS] φ [EW]
Reverse function of the previous conversion
Be careful in using asin and acos
Value range
Precision

10. Judge Status
1 solved / 1 submission
First accept: Wx (162 min.)