用 C 打印毕达哥拉斯三元组的表示

Question

我正在尝试创建一个程序来打印在 C 中找到的毕达哥拉斯三元组的映射数据。到目前为止，我已经编写了能够找到三元组的程序。

#include <stdio.h>
#include <math.h>

int main (int argc, char * argv[]) {
    int a, b, c;
    int a2, b2, c2;
    int limit = 60;

    for (a = 1; a <= limit; a++) {
        a2 = a * a;
        for (b = 1; b <= limit; b++) {
            b2 = b * b;
            for (c = 0; c <= ((int)sqrt(a2+b2)); c++) {
                c2 = c * c;
                if (a < b && (c2 == (a2 + b2))) {
                    printf("triple: %d %d %d\n", a, b, c);
                }
            }
        }
    }
}

预期输出格式为：

123456789012345678901234567890123456789012345678901234567890
1\
2 \
3  \
4  *\
5    \
6     \
7      \
8     * \
9        \
0         \
1          \
2    *   *  \

我正在尝试编写一个执行此操作的循环，但想不出如何以这种方式打印。有什么建议吗？

更新：我设法打印了 x 轴和 y 轴（x = a 和 y = b）。值是正确的，现在剩下映射部分。

    for (int x = 0; x < a; x++) { // x-axis = a
        printf("%d ", x);
    }

    printf("\n");

    for (int y = 0; y < b; y++) { // y-axis = b

        printf("%d\n", y);
    }

更新：修改了代码，输出正在打印，但打印空格有问题。尝试手动向“\”和“*”添加空格，但这只会扭曲整个图像。

#include <stdio.h>
#include <math.h>
#include <stdbool.h>

bool is_perfect_square(int num);

int main (int argc, char * argv[]) {

    int a, b, c;
    int a2, b2, c2;
    int limit = 60;
    bool flag = false;

    for (a = 1; a <= limit; a++) {
        a2 = a * a;
        for (b = 1; b <= limit; b++) {
            b2 = b * b;
            for (c = 0; c <= ((int)sqrt(a2+b2)); c++) {
                c2 = c * c;
                if (a < b && (c2 == (a2 + b2))) {
                    // printf("triple: %d %d %d\n", a, b, c);
                }
            }
        }
    }

    for (int x = 0; x < a; x++) {
        for (int y = 0; y < b; y++) {
            if (x == 0) {
                printf("%d ", ((y+1)% 10));
            } else if (y == 0) {
                printf("%d ", (x % 10));
            } else if (x == y) {
                printf("\");
            } else if (is_perfect_square((x*x) + (y*y))) {
                printf("*");
            }
        }
        printf("\n");
    }
}

bool is_perfect_square (int num) {
    int root = (int)(sqrt(num));

    if (num == root * root) {
        return true;
    } else {
        return false;
    }
}

仍在研究可能的解决方案。

Answer 1

提示：

有一个索引为 i,j 的嵌套循环；

for i 0.. limit {
  for j 0 .. limit {
   /*Implement the below algorithm here!!*/
  }
    printf("\n")
}

循环内使用的算法：

if i == 0 通过打印 (j+1)% 10 来打印 x 轴值 [见末尾的注释]
else if j == 0 通过打印 i % 10
else if i == j 打印 '\'
else if is_perfect_square((i*i) + (j*j)) returns 1, 打印'*'
否则打印 space.

is_perfect_square 函数的说明： returns 如果输入是正方形则为 1，否则为 0 的函数。例如：

is_perfect_square(25) 应该 return 1
is_perfect_square(7) 应该 return 0
is_perfect_square(49) 应该 return 1

注意：i == 0 案例应打印 j%10 以 0 开始输出以表示原点。但是所提供的输出以 1 开头。因此使用 (j+1)%10

您可能需要处理一些极端情况，一旦在代码中实现了该算法，这些情况就应该很简单了。

Answer 2

简单的方法

如果您设法以正确的顺序打印所有内容，您可以轻松避免不必要的 if 语句并优化数学计算作为副作用。

为了找到毕达哥拉斯三元组，我修改了你的第一个方法，这样我就可以避免在每个位置调用 sqrt。

#include <stdio.h>
#include <math.h>

// all the loops will test numbers from 1 to 60 included
#define LIMIT 61

int main ( int argc, char * argv[] ) {
    int i, j, k, k2, sum;

    // print the first line of reference numbers
    // start with a space to allign the axes
    putchar(' ');
    // print every digit... 
    for ( i = 1; i < LIMIT; ++i )         
        putchar('0' + i%10);
    // then newline
    putchar('\n');

    // now print every row
    for ( i = 1; i < LIMIT; ++i ) {
        // first print the number
        putchar('0' + i%10);
        // then test if the indeces (i,j) form a triple: i^2 + j^2 = k^2
        // I don't want to call sqrt() every time, so I'll use another approach
        k = i;
        k2 = k * k;
        for ( j = 1; j < i; ++j ) {
            // this  ^^^^^ condition let me find only unique triples and print
            // only the bottom left part of the picture

            // compilers should be (and they are) smart enough to optimize this
            sum = i * i  +  j * j;

            // find the next big enough k^2
            if ( sum > k2 ) {
                ++k;
                k2 = k * k;
            }

            // test if the triple i,j,k matches the Pythagorean equation
            if ( sum == k2 )
                // it's a match, so print a '*'
                putchar('*');
            else
                // otherwise print a space
                putchar(' ');
        }
        // the line is finished, print the diagonal (with a '\') and newline
        printf("\\n");
        // An alternative could be to put the reference numbers here instead:
        //      printf("%c\n",'0' + i%10);
    }

    return 0;
}

这个程序的输出是：

 123456789012345678901234567890123456789012345678901234567890
1\
2 \
3  \
4  *\
5    \
6     \
7      \
8     * \
9        \
0         \
1          \
2    *   *  \
3            \
4             \
5       *      \
6           *   \
7                \
8                 \
9                  \
0              *    \
1                   *\
2                     \
3                      \
4      *  *       *     \
5                        \
6                         \
7                          \
8                    *      \
9                            \
0               *             \
1                              \
2                       *       \
3                                \
4                                 \
5           *                      \
6              *           *        \
7                                    \
8                                     \
9                                      \
0        *                    *         \
1                                        \
2                                       * \
3                                          \
4                                *          \
5                       *   *                \
6                                             \
7                                              \
8             *     *               *           \
9                                                \
0                                                 \
1                                                  \
2                                      *            \
3                                                    \
4                                                     \
5                                               *      \
6                                *        *             \
7                                                        \
8                                                         \
9                                                          \
0          *             *      *            *              \

更简单的方法

我将向您展示另一种打印出您想要的内容的方法。

考虑使用字符串数组作为绘图 space，存储在结构中。这看起来很复杂，但您可以简化甚至概括输出过程：

#include <stdio.h>
#include <stdlib.h>
#include <string.h>

typedef struct {
    char **img;
    int rows;
    int cols;
} Image;

Image *init_image( int r, int c, char s ) {
    int i;
    char *tmp = NULL;

    Image *pi = malloc(sizeof(Image));
    if ( !pi ) {
        perror("Error");
        exit(-1);
    }
    pi->rows = r;  
    pi->cols = c;
    pi->img = malloc(r * sizeof(char*));
    if ( !pi->img ) {
        perror("Error");
        exit(-1);
    }
    for ( i = 0; i < r; ++i ) {
        tmp = malloc(c + 1);
        if ( !tmp ) {
            perror("Error");
            exit(-1);
        }
        // fill with initial value (spaces) and add the terminating NULL
        memset(tmp,s,c);
        tmp[c] = '[=12=]';
        pi->img[i] = tmp;
    }
    return pi;
}

void free_image( Image *pi ) {
    int i;
    if ( !pi || !pi->img ) return;
    for ( i = 0; i < pi->rows; ++i ) {
        free(pi->img[i]);
    }
    free(pi->img);
    free(pi);
}

void draw_axes( Image *pi ) {
    int i;
    if ( !pi ) return;
    // I use to loops because cols can be != rows, but if it's always a square...
    for ( i = 1; i < pi->cols; ++i ) {
        pi->img[0][i] = '0' + i%10;
    }
    for ( i = 1; i < pi->rows; ++i ) {
        pi->img[i][0] = '0' + i%10;
    }
}

void draw_diagonal ( Image *pi, char ch ) {
    int i, m;

    if ( !pi ) return;
    m = pi->rows < pi->cols ? pi->rows : pi->cols;
    for ( i = 1; i < m; ++i ) {
        pi->img[i][i] = ch;
    }
}

void print_image( Image *pi ) {
    int i;

    if ( !pi ) return;
    for ( i = 0; i < pi->rows; ++i ) {
        printf("%s\n",pi->img[i]);
    }
}

void draw_triples( Image *pi, char ch ) {
    int i, j, k, k2, sum;

    for ( i = 1; i < pi->rows; ++i ) {
        k = i;
        k2 = k * k;
        // print only the left bottom part
        for ( j = 1; j < i && j < pi->cols; ++j ) {
            sum = i * i  +  j * j;
            if ( sum > k2 ) {
                ++k;
                k2 = k * k;
            }
            if ( sum == k2 ) {
                pi->img[i][j] = ch;
                // printf("%d %d %d\n",i,j,k);
            }
        }
    }
}

int main(int argc, char *argv[]) {
    // Initialize the image buffer to contain 61 strings of 61 spaces
    Image *img = init_image(61,61,' ');

    // draw the reference numbers at row 0 and column 0
    draw_axes(img);
    // draw a diagonal with character '\'
    draw_diagonal(img,'\');
    // put a '*' if a couple of coordinates forms a Pythagorean triple
    draw_triples(img,'*');
    // print out the image to stdout
    print_image(img);

    free_image(img);
    return 0;
}

此代码的输出与前面的代码片段相同，但是不管你信不信，由于打印到 stdout.

附录

这离题太远了，但我很高兴调整以前的代码来实际输出一个图像文件（作为灰度 512x512 PGM 二进制格式的文件），代表边长最大为 8192 的所有三元组。

每个像素对应一个 16x16 的方块，如果没有匹配项则颜色为黑色，或者根据算法在块中找到的三元组的数量而变亮。

#include <stdio.h>
#include <stdlib.h>
#include <string.h>

typedef struct {
    char *img;      // store data in a 1D array this time
    int dim;
    int samples;
} Image;

Image *init_image( int d, int z );
void free_image( Image *pi );
void add_sample( Image *pi, int r, int c );
void draw_triples_sampled( Image *pi );
void save_image( char *file_name, Image *pi );

int main(int argc, char *argv[]) {
    // store the results in a 512x512 image, with each pixel corresponding
    // to a 16x16 square, so the test area is 8192x8192 wide
    Image *img = init_image(512,16);

    draw_triples_sampled(img);
    save_image("triples.pgm",img);
    free_image(img);

    return 0;
}

Image *init_image( int d, int z ) {

    Image *pi = malloc(sizeof(Image));
    if ( !pi ) {
        perror("Error");
        exit(-1);
    }
    pi->dim = d;
    pi->samples = z;
    pi->img = calloc(d*d,1);
    if ( !pi->img ) {
        perror("Error");
        exit(-1);
    }
    return pi;
}

void free_image( Image *pi ) {
    if ( !pi ) free(pi->img);
    free(pi);
}

void add_sample( Image *pi, int r, int c ) {
    // each pixel represent a square block of samples
    int i = r / pi->samples;                
    int j = c / pi->samples;
    // convert 2D indeces to 1D array index
    char *pix = pi->img + (i * pi->dim + j);
    ++(*pix);
}

void draw_triples_sampled( Image *pi ) {
    int i, j, k, k2, sum;
    int dim;
    char *val;

    if ( !pi ) return;
    dim = pi->dim * pi->samples + 1;
    for ( i = 1; i < dim; ++i ) {
        k = i;
        k2 = k * k;
        // test only bottom left side for simmetry...
        for ( j = 1; j < i; ++j ) {
            sum = i * i + j * j;
            if ( sum > k2 ) {
                ++k;
                k2 = k * k;
            }
            if ( sum == k2 ) {              
                add_sample(pi,i-1,j-1);
                // draw both points, thanks to simmetry
                add_sample(pi,j-1,i-1);
            }
        }
    }
}

void save_image( char *file_name, Image *pi ) {
    FILE *pf = NULL;
    char v;
    char *i = NULL, *end = NULL;

    if ( !pi ) {
        printf("Error saving image, no image data\n");
        return;
    }
    if ( !file_name ) {
        printf("Error saving image, no file name specified\n");
        return;
    }
    pf = fopen(file_name,"wb");
    if ( !pf ) {
        printf("Error saving image in file %s\n",file_name);
        perror("");
        return;
    }
    // save the image as a grayscale PGM format file
    // black background, pixels from gray to white   
    fprintf(pf,"P5 %d %d %d ",pi->dim,pi->dim,255);
    end = pi->img + pi->dim * pi->dim;
    for ( i = pi->img; i != end; ++i ) {
        if ( *i == 0 )
            v = 0;
        else if ( *i < 10 )
            v = 105 + *i * 15;
        else
            v = 255; 
        fprintf(pf,"%c",v);
    }
    close(pf);
}

输出的图片是（在这里转成PNG到post）这个。注意新出现的模式：

Answer 3

#include <stdio.h>
#include <math.h>
#include <stdbool.h>

bool is_perfect_square(int num);

int main (int argc, char * argv[]) {

    int a, b, c;
    int a2, b2, c2;
    int limit = 60;
    bool flag = false;

    for (a = 1; a <= limit; a++) {
        a2 = a * a;
        for (b = 1; b <= limit; b++) {
            b2 = b * b;
            for (c = 0; c <= ((int)sqrt(a2+b2)); c++) {
                c2 = c * c;
                if (a < b && (c2 == (a2 + b2))) {
                    // printf("triple: %d %d %d\n", a, b, c);
                }
            }
        }
    }

    printf(" ");
    for (int x = 0; x < a; x++) {
        for (int y = 0; y < b; y++) {
            if (x == 0) {
                printf("%d", ((y+1) % 10));
            } else if (y == 0) {
                printf("%d", (x % 10));
            } else if (x == y) {
                printf("\");
            } else if (is_perfect_square((x*x) + (y*y))) {
                printf("*");
            } else {
                printf(" ");
            }
        }
        printf("\n");
    }
}

bool is_perfect_square (int num) {
    int root = (int)(sqrt(num));

    if (num == root * root) {
        return true;
    } else {
        return false;
    }
}

最终更新：终于想出了这个答案。非常感谢 Ravichandra 先生。这是为以后可能需要这样的东西的任何人准备的。代码并不完美，可以改进。输出令人满意并打印出与所需输出模式几乎相似的模式。

用 C 打印毕达哥拉斯三元组的表示

Print a representation of the Pythagorean Triple in C

c

plot

graph

pythagorean