Conic Section（圆锥曲线基础）

Conic Section

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Time Limit: 2 Seconds      Memory Limit: 65536 KB

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The conic sections are the nondegenerate curves generated by the intersections of a plane with one or two nappes of a cone. For a plane perpendicular to the axis of the cone, a circle is produced. For a plane that is not perpendicular to the axis and that intersects only a single nappe, the curve produced is either an ellipse or a parabola. The curve produced by a plane intersecting both nappes is a hyperbola.

conic section	equation
circle	x2+y2=a2
ellipse	x2/a2+y2/b2=1
parabola	y2=4ax
hyperbola	x2/a2-y2/b2=1

Input

There are multiple test cases. The first line of input is an integer T ≈ 10000 indicating the number of test cases.

Each test case consists of a line containing 6 real numbers a, b, c, d, e, f. The absolute value of any number never exceeds 10000. It's guaranteed that a²+c²>0, b=0, the conic section exists and it is non-degenerate.

Output

For each test case, output the type of conic section ax²+bxy+cy²+dx+ey+f=0. See sample for more details.

Sample Input

5
1 0 1 0 0 -1
1 0 2 0 0 -1
0 0 1 1 0 0
1 0 -1 0 0 1
2 0 2 4 4 0

Sample Output

circle
ellipse
parabola
hyperbola
circle

References

• Weisstein, Eric W. "Conic Section." From MathWorld--A Wolfram Web Resource. http://mathworld.wolfram.com/ConicSection.html

首先，题目已经提示了b=0，就是不需考虑旋转，大大降低了题目难度。

其次，题目说明了曲线不会退化。

另外需要注意的是，曲线方程不一定如题中所示的标准形式，需要考虑平移的形式。如圆：(x-a)^2+(y-b)^2=c^2。

#include <iostream>
#include <string>
#include <cstdio>
#include <cmath>
#include <cstring>
#include <algorithm>
#include <map>
#include <vector>
#include <set>
#include <queue>
#include <stack>
#define LL long long
#define MAXI 2147483647
#define MAXL 9223372036854775807
#define eps (1e-8)
#define dg(i) cout << "*" << i << endl;
  
using namespace std;
  
int main()
{
    double a, b, c ,d ,e ,f;
    int t;
    cin >> t;
    while(t--)
    {
        cin >> a >> b >> c >> d >> e >> f;
        if(a && a == c && d*d + e*e - 4*a*f > 0) puts("circle");
        else if(a*c < 0 && (d*d/(4*a*a)) + (e*e/(4*c*c)) - f) puts("hyperbola");
        else if((!a && d && c) || (!c && e && a)) puts("parabola");
        else puts("ellipse");
    }
    return 0;
}