/* z[0] = (1, 0) z[i + 1] = [[1 + v, -w], [w, 1 + v]] z[i] + u[i] z[1] = r z[0] + u[0] z[2] = r^2 z[0] + r u[0] + u[1] ... minimize sum_i (x[i]^2 + y[i]^2) subject to sum_i (a[i] x[i] + b[i] y[i]) = e sum_i (c[i] x[i] + d[i] y[i]) = f h = sum_i (x[i]^2 + y[i]^2) - p (sum_i (a[i] x[i] + b[i] y[i]) - e) - q (sum_i (c[i] x[i] + d[i] y[i]) - f) 0 = dh/dx[i] = 2 x[i] - a[i] p - c[i] q 0 = dh/dy[i] = 2 y[i] - b[i] p - d[i] q */ import std.conv, std.functional, std.range, std.stdio, std.string; import std.algorithm, std.array, std.bigint, std.complex, std.container, std.math, std.numeric, std.regex, std.typecons; import core.bitop; class EOFException : Throwable { this() { super("EOF"); } } string[] tokens; string readToken() { for (; tokens.empty; ) { if (stdin.eof) { throw new EOFException; } tokens = readln.split; } auto token = tokens.front; tokens.popFront; return token; } int readInt() { return readToken.to!int; } long readLong() { return readToken.to!long; } real readReal() { return readToken.to!real; } bool chmin(T)(ref T t, in T f) { if (t > f) { t = f; return true; } else { return false; } } bool chmax(T)(ref T t, in T f) { if (t < f) { t = f; return true; } else { return false; } } int binarySearch(alias pred, T)(in T[] as) { int lo = -1, hi = cast(int)(as.length); for (; lo + 1 < hi; ) { const mid = (lo + hi) >> 1; (unaryFun!pred(as[mid]) ? hi : lo) = mid; } return hi; } int lowerBound(T)(in T[] as, T val) { return as.binarySearch!(a => (a >= val)); } int upperBound(T)(in T[] as, T val) { return as.binarySearch!(a => (a > val)); } int T; real P, W, V, GX, GY; void main() { try { for (; ; ) { const numCases = readInt(); foreach (caseId; 0 .. numCases) { T = readInt(); P = readReal(); W = readReal(); V = readReal(); GX = readReal(); GY = readReal(); const real[][] r = [[1.0 + V, -W], [W, 1.0 + V]]; auto rr = new real[][][](T + 1, 2, 2); rr[0] = [[1.0, 0.0], [0.0, 1.0]]; foreach (i; 0 .. T) { foreach (j; 0 .. 2) foreach (k; 0 .. 2) { rr[i + 1][j][k] = 0.0; } foreach (j; 0 .. 2) foreach (l; 0 .. 2) foreach (k; 0 .. 2) { rr[i + 1][j][k] += rr[i][j][l] * r[l][k]; } } /* x[i] = (1/2) (rr[T - 1 - i][0][0] p + rr[T - 1 - i][1][0] q) y[i] = (1/2) (rr[T - 1 - i][0][1] p + rr[T - 1 - i][1][1] q) */ real[][] a = [[0.0, 0.0], [0.0, 0.0]]; foreach (i; 0 .. T) { foreach (j; 0 .. 2) foreach (k; 0 .. 2) foreach (l; 0 .. 2) { a[j][k] += (rr[T - 1 - i][j][l] * rr[T - 1 - i][k][l]) / 2.0; } } real[] b = [GX, GY]; foreach (j; 0 .. 2) { b[j] -= (rr[T][j][0] * 1.0 + rr[T][j][1] * 0.0); } const det = a[0][0] * a[1][1] - a[0][1] * a[1][0]; const p = (b[0] * a[1][1] - a[0][1] * b[1]) / det; const q = (a[0][0] * b[1] - b[0] * a[1][0]) / det; debug { writeln(a, " ", [p, q], " ", b, "; ", a[0][0] * p + a[0][1] * q, " ", a[1][0] * p + a[1][1] * q); } auto ansX = new real[T]; auto ansY = new real[T]; foreach (i; 0 .. T) { ansX[i] = (rr[T - 1 - i][0][0] * p + rr[T - 1 - i][1][0] * q) / 2.0; ansY[i] = (rr[T - 1 - i][0][1] * p + rr[T - 1 - i][1][1] * q) / 2.0; } debug { real sum = 0.0; foreach (i; 0 .. T) { sum += ansX[i]^^2 + ansY[i]^^2; } writefln("P = %s; %s", P, sum); real x = 1.0, y = 0.0; foreach (i; 0 .. T) { const xx = x - y * W + x * V + ansX[i]; const yy = y + x * W + y * V + ansY[i]; x = xx; y = yy; } writefln("(GX, GY) = (%s, %s); (%s, %s)", GX, GY, x, y); } foreach (i; 0 .. T) { writefln("%.20f %.20f", ansX[i], ansY[i]); } } } } catch (EOFException e) { } }