Problem E
The Story of Two Coins
Input: Standard Input
Output: Standard Output
In this problem we have two coins (perfect
circles) in a two dimensional world. The radius of first coin is R and
the radius of the second coin is r. The first coin is static and the
location of its center Q is at the origin (0,0). The second coin
(with radius r) is rotating in counter clock-wise direction like a wheel
on the perimeter of the first circle, as shown in the picture below. As it is
rotating counter clock wise so it is also moving along the surface of the first
circle in counter clock-wise direction. After the second coin has rotated
exactly once around the first coin, it stops.
At the beginning, the center of the second
coin (Denoted by O in the figure above) is on the x-axis and on
the right hand side of the first coin. So initially the coordinate of O
is (R+r, 0). At this moment P is the topmost point of the second
coin. So obviously the coordinate of P is (R+r, r). Now as the
second coin starts rotating the location of P will change due to both
circular translation and its own rotation. For most positions of the second
coin, P will not be its topmost point. T is any arbitrary point
on the perimeter of the moving second coin. If the Cartesian coordinate of T,
(x,y) is given, then it is possible to determine the position of the second
coin. Of course there can be two possible positions as shown in the figure
above. Now your job is a little bit difficult than this. Given the coordinate
of T, you will have to find the coordinate of P for two possible
positions of the second coin.
Input
The input file contains at most 1000 lines of inputs. Each line contains two positive integers and two floating-point numbers. The two positive integers are R, r (0<|r|<=|R|<1000) and the two floating point numbers are x and y (0<|x|,|y|<4000.0). The meanings of these symbols are given in the problem statement above. All the floating-point numbers in the input will have fifteen digits after the decimal point.
Input is terminated by a line where the values of R and r are zero. Your program should not process this line.
Output
For
each line of input produce three or two lines of output. The first line
contains the serial of output .The next two lines contain the possible
coordinates of P for two possible positions of the second coin. The
coordinates of P should be printed in chronological order (If a position
is reached by the second coin before the other position then the corresponding
coordinates of P should be printed first). If it is not possible to find
the coordinate of P for the given data, print “IMPOSSIBLE”
instead. There will be no such input that will cause the second coin to have
exactly one possible position. Also note that there will be no such input,
which will allow precision error to cause difference in the output. Assume that
pi=2cos-1(0), and use at least double precision (64-bit)
floating-point numbers for your calculations. All floating-point numbers should
be rounded to three digits after the decimal point. Look at the output for
sample input for details.
Sample Input Output for Sample Input
5 5 10.000000000000000 3.0000000000000005 5 20.000000000000000 10.0000000000000000 0 0.000000000000000 0.000000000000000 |
Case 1:2.082 7.07611.755 2.606Case 2:
IMPOSSIBLE |
Problem setter: Shahriar Manzoor
Special Thanks: Derek Kisman