Reflections

Time Limit: 2 Seconds
Memory Limit: 65536 KB

Rendering realistic images of imaginary environments or objects is an interesting
topic in computer graphics. One of the most popular methods for this purpose is
ray-tracing.

To render images using ray-tracing, one computes (traces) the path that rays
of light entering a scene will take. We ask you to write a program that computes
such paths in a restricted environment.

For simplicity, we will consider only two-dimensional scenes. All objects in
the scene are totally reflective (mirror) spheres. When a ray of light hits
such a sphere, it is reflected such that the angle of the incoming ray and the
leaving ray against the tangent are the same:

The following figure shows a typical path that a ray of light may take in
such a scene:

Your task is to write a program, that given a scene description and a ray
entering the scene, determines which spheres are hit by the ray.

**Input**

The input consists of a series of scene descriptions. Each description starts
with a line containing the number n (n <= 25) of spheres in the scene. The
following n lines contain three integers xi, yi, ri each, where (xi, yi) is
the center, and ri > 0 is the radius of the i-th sphere. Following this is
a line containing four integers x, y, dx, dy, which describe the ray. The ray
originates from the point (x,y) and initially points in the direction (dx, dy).
At least one of dx and dy will be non-zero.

The spheres will be disjoint and non-touching. The ray will not start within
a sphere, and never touch a sphere tangentially.

A test case starting with n = 0 terminates the input. This case should not
be processed.

**Output**

For each scene first output the number of the scene. Then print the numbers
of the spheres that the ray hits in its first ten deflections (the numbering
of spheres is according to their order in the input).

If the ray hits at most ten spheres (and then heads towards infinity), print
inf after the last sphere it hits. If the ray hits more than 10 spheres, print
three points (...) after the tenth sphere.

Output a blank line after each test case.

**Sample Input **

3

3 3 2

7 7 1

8 1 1

3 8 1 -4

2

0 0 1

5 0 2

2 0 1 0

0

**Sample Output**

Scene 1

1 2 1 3 inf

Scene 2

2 1 2 1 2 1 2 1 2 1 ...

Source:

**Southwestern Europe 1998**
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