AtCoder Beginner Contest 259 - B - Counterclockwise Rotation - 题解

Posted 2022-07-13 Tisfy

Time Limit: 2 sec / Memory Limit: 1024 MB

Score : $200$ points

Problem Statement

In an $xy$-coordinate plane whose $x$-axis is oriented to the right and whose $y$-axis is oriented upwards, rotate a point $(a,b)$ around the origin $d$ degrees counterclockwise and find the new coordinates of the point.

Constraints

• $-1000\le a,b\le 1000$
• $1\le d\le 360$
• All values in input are integers.

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Input

Input is given from Standard Input in the following format:

$a$ $b$ $d$

Output

Let the new coordinates of the point be (a′,b′)(a', b')(a′,b′). Print a′a'a′ and b′b'b′ in this order, with a space in between.
Your output will be considered correct when, for each value printed, the absolute or relative error from the answer is at most 10−610^-610−6.

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Sample Input 1

2 2 180

Sample Output 1

-2 -2

When $(2,2)$ is rotated around the origin $180$ degrees counterclockwise, it becomes the symmetric point of $(2,2)$ with respect to the origin, which is $(-2,-2)$.

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Sample Input 2

5 0 120

Sample Output 2

-2.49999999999999911182 4.33012701892219364908

When $(5,0)$ is rotated around the origin $120$ degrees counterclockwise, it becomes (−52,532)(-\\frac 52 , \\frac 5\\sqrt32)(−25​,253 ​​).
This sample output does not precisely match these values, but the errors are small enough to be considered correct.

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Sample Input 3

0 0 11

Sample Output 3

0.00000000000000000000 0.00000000000000000000

Since $(a,b)$ is the origin (the center of rotation), a rotation does not change its coordinates.

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Sample Input 4

15 5 360

Sample Output 4

15.00000000000000177636 4.99999999999999555911

A $360$-degree rotation does not change the coordinates of a point.

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Sample Input 5

-505 191 278

Sample Output 5

118.85878514480690171240 526.66743699786547949770

题目大意

$x$轴正向朝右，$y$轴正向朝上的二维坐标平面上有一点的坐标为$(a,b)$

求 将这个点绕坐标原点逆时针旋转$d$°后 的坐标

解题思路

C++内置了$\sin 、\cos 、\arctan$等函数，但这些函数都是以弧度制为基础的。

先计将直角坐标转换为极坐标，然后把角度加上$d$，再转换为直角坐标即可。

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AC代码

赛时：较为复杂

后面会有简化及技巧

#include <bits/stdc++.h>
using namespace std;
#define mem(a) memset(a, 0, sizeof(a))
#define dbg(x) cout << #x << " = " << x << endl
#define fi(i, l, r) for (int i = l; i < r; i++)
#define cd(a) scanf("%d", &a)
typedef long long ll;

const double PI = acos(-1);

int main() 
    double a, b;
    cin >> a >> b;
    double l = sqrt(a * a + b * b);
    if (l < 1e-8) 
        puts("0 0");
        return 0;
    
    double alpha;
    if (a == 0) 
        if (b > 0)
            alpha = PI / 2;
        else
            alpha = PI / 2 * 3;
    
    alpha = atan(b / a);
    if (a < 0)
        alpha += PI;
    double d;
    cin >> d;
    double belta = alpha + d / 180 * PI;
    belta += 4 * PI;
    while (belta > 2 * PI)
        belta -= 2 * PI;
    double x = l * cos(belta);
    double y = l * sin(belta);
    printf("%.9lf %.9lf\\n", x, y);
    return 0;

简化及技巧

用 hypot(a, b) 代替 sqrt(a * a + b * b)

double l = sqrt(a * a + b * b);

👇

double l = hypot(a, b);

直接用atan2求出四个象限下的角度

double alpha;
if (a == 0) 
    if (b > 0)
        alpha = PI / 2;
    else
        alpha = PI / 2 * 3;

alpha = atan(b / a);
if (a < 0)
    alpha += PI;

👇

double alpha = atan2(b, a);

取消掉映射到[0, 2π)

belta += 4 * PI;
while (belta > 2 * PI)
    belta -= 2 * PI;

👇

最终代码

#include <bits/stdc++.h>
using namespace std;
#define mem(a) memset(a, 0, sizeof(a))
#define dbg(x) cout << #x << " = " << x << endl
#define fi(i, l, r) for (int i = l; i < r; i++)
#define cd(a) scanf("%d", &a)
typedef long long ll;

const double PI = acos(-1);

int main() 
    double a, b;
    cin >> a >> b;
    double l = hypot(a, b);  // 代替sqrt(a * a + b * b);
    double alpha = atan2(b, a);  // 非常方便地求角
    double d;
    cin >> d;
    double belta = alpha + d / 180 * PI;
    double x = l * cos(belta);
    double y = l * sin(belta);
    printf("%.9lf %.9lf\\n", x, y);  // 也可以：cout<<fixed<<setprecision(9)<<x<<' '<<y<<endl;
    return 0;

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Tisfy：https://letmefly.blog.csdn.net/article/details/125699603