P4035 [JSOI2008] Spherical space generator

P4035 [JSOI2008] Spherical space generator title Main topic

The n+1 points on the n-dimensional space are given, and the stores are on a round surface, seeking the coordinates of the circle center.

Defined:

1. Sphere: A point of equal distance to any point on the sphere.

2. Distance formula between two points

\[A (x_1,x_2,x_3,x_4,\cdots x_n) \]

\[B (y_1,y_2,y_3,y_4,\cdots y_n) \]

\[Distance:\sqrt[2]{\sum_{i=1}^{n} (x_i-y_i) ^2} \]

The title should look like a solution to the equation.

We can use the Gauss elimination method .

But here's the problem. This is a two-time multivariate equation set. And our Gauss can only be solved once. And the premise of Gauss is that there are n unknowns, we must have n equations. ( of course, some are not rigorous )

We need to consider moving items and setting an unknown .

One of the original equations: we set a number first, R。 Indicates that the radius a is a point based on the standard equation of the circle, and R is the center

\[a (x_1,x_2,x_3,x_4,\cdots x_n) \]

\[r (y_1,y_2,y_3,y_4,\cdots y_n) \]

\[\sum_{i=1}^{n} (x_i-y_i) ^2=r^2\]

\[\sum_{i=1}^{n}x_i^2-\sum_{i=1}^{n}2x_iy_i+\sum_{i=1}^{n}y_i^2=r^2\]

Please note that the coordinates of a are known quantities. The coordinates of R and R are not

Then we move the item
\[-\sum_{i=1}^{n}2x_iy_i+ (\sum_{i=1}^{n}y_i^2-r^2) =-\sum_{i=1}^{n}x_i^2\]

The most winding step is the substitution of the whole in parentheses (or as an unknown)

So there is an unknown n+1. And after we solve the equation, we only need the first n unknowns. The unknowns behind us were solved. But it's no use. Just one of our helper variables

At the same time, this question also tells us some small tricks.

1. It is impossible for the author to give more conditions. Some of them are for us to set ourselves up.
2. Encountered two-time equations. You can consider splitting parentheses and moving items. And then restore to a power-down purpose

#include <cstdio>#include <algorithm>#include <iostream>#include <cmath>using namespaceStdDoublemap[ the][ the];Doubleans[ the];intNvoidGauss () { for(intI=1; i<=n+1; i++) {intR=i; for(intJ=i+1; j<=n+1; j + +)if(Fabs (Map[r][i]) <fabs (Map[j][i])) r=j;if(r!=i) for(intJ=i;j<=n+2; j + +) Swap (map[i][j],map[r][j]);DoubleDiv=map[i][i]; for(intJ=i;j<=n+2; j + +) Map[i][j]/=div; for(intJ=i+1; j<=n+1; j + +) {Div=map[j][i]; for(intK=i;k<=n+2; k++) map[j][k]-=div*map[i][k]; }} ans[n+1]=map[n+1][n+2]; for(inti=n;i>=1; i--) {Ans[i]=map[i][n+2]; for(intJ=i+1; j<=n+1; j + +) Ans[i]-=map[i][j]*ans[j]; }}intMain () {scanf ("%d", &n); for(intI=1; i<=n+1; i++) {DoubleData for(intj=1; j<=n;j++) {scanf ("%LF", &data); map[i][j]=-2.0*data; Map[i][n+2]-=data*data; } map[i][n+1]=1;    } gauss (); printf"%.3LF", ans[1]); for(intI=2; i<=n;i++) printf (" %.3LF", Ans[i]);}

P4035 [JSOI2008] Spherical space generator