1. 介绍

这个初学循环的时候，可能会被布置为作业。这里我们介绍的不是找规律，而是这一类问题的通解。

其实这个问题可以被抽象为高中的线性规划，只不过我们这里需要离散化直线和区域内的点。

2. 案例分析

我们以漏斗型星号为例来分析。

如图所示，我们把坐标系给放正，需要($n=4$)输出星号的点就是上面标的那些。显然这些点必须为整点。

实际上需要输出星号的点就是

$$\begin{gathered} N=n-1 \\ \begin{array}{r}\{\begin{array}{l}(r-c)(r+c-2N)\ge 0\\ 0\le r\le 2N\\ 0\le c\le 2N\end{array}\end{array} \end{gathered}$$

$r$是行号，从$0$行开始。
$c$是列号，从$0$行开始。
这里$n$是漏斗中心到底部的距离加一，
即$N=n-1$
$rc$都需要是整数值。

3. 漏斗型

• 形状
  

• 区域表达式
  $$\begin{gathered} N=n-1 \\ \begin{array}{r}\{\begin{array}{l}(r-c)(r+c-2N)\ge 0\\ 0\le r\le 2N\\ 0\le c\le 2N\end{array}\end{array} \end{gathered}$$

• 代码

#include <stdio.h>


void loudou_stars(int n) {


    int r ;
    int c ;

    int N = n - 1;


    for ( r = 0; r <= 2 * N; r++ ) {
        for ( c = 0; c <= 2 * N; c++ ) {
            if ((r - c) * (r + c - 2 * N) >= 0) {
                printf("*");
            }
            else {
                printf(" ");
            }
        }
        printf("\n");
    }
}


int main()
{

    int n = 4;
    
    loudou_stars( n );
    
    return 0;
}

• 结果
  

4. 直角三角形

4.1 左上直角三角形

• 形式
  

• 区域表达式
  $$\begin{gathered} N=n-1 \\ \begin{array}{r}\{\begin{array}{l}r+c-N\le 0\\ 0\le r\le N\\ 0\le c\le N\end{array}\end{array} \end{gathered}$$

• 代码

#include <stdio.h>


void lu_trianle(int n) {

    int r;
    int c;
    int N = n - 1;
    for ( r = 0; r <= N; r++) {
        for ( c = 0; c <= N; c++) {
            if ( r + c - N <= 0  ) {
                printf("*");
            }
            else {
                printf(" ");
            }
        }
        printf("\n");
    }
}
int main()
{

    lu_trianle(5);
    return 0;
}

4.2 右上直角三角形

• 形式
  
• 区域表达式
  $$\begin{gathered} N=n-1 \\ \begin{array}{r}\{\begin{array}{l}r-c\le 0\\ 0\le r\le N\\ 0\le c\le N\end{array}\end{array} \end{gathered}$$
• 代码

#include <stdio.h>


void ru_trianle(int n) {

    int r;
    int c;
    int N = n - 1;

    for ( r = 0; r <= N; r++) {
        for ( c = 0; c <= N; c++) {
            if ( r - c  <= 0  ) {
                printf("*");
            }
            else {
                printf(" ");
            }
        }
        printf("\n");
    }
}
int main()
{

    ru_trianle(5);
    return 0;
}

4.3 左下直角三角形

其实对右上的三角形代码符号变一下就可以了

• 形式
  

• 区域表达式
  $$\begin{gathered} N=n-1 \\ \begin{array}{r}\{\begin{array}{l}r-c\ge 0\\ 0\le r\le N\\ 0\le c\le N\end{array}\end{array} \end{gathered}$$

• 代码

#include <stdio.h>


void lb_trianle(int n) {

    int r;
    int c;
    int N = n - 1;
    for ( r = 0; r <= N; r++) {
        for ( c = 0; c <= N; c++) {
            if ( r - c  >= 0  ) {
                printf("*");
            }
            else {
                printf(" ");
            }
        }
        printf("\n");
    }
}
int main()
{

    lb_trianle(5);
    return 0;
}

4.4 右下直角三角形

将左上三角形换个符号就可以得到

• 形状
  
• 区域表达式
  $$\begin{gathered} N=n-1 \\ \begin{array}{r}\{\begin{array}{l}r+c-N\ge 0\\ 0\le r\le N\\ 0\le c\le N\end{array}\end{array} \end{gathered}$$
• 代码

#include <stdio.h>


void rb_trianle(int n) {

    int r;
    int c;
    int N = n - 1;

    for ( r = 0; r <= N; r++) {
        for ( c = 0; c <= N; c++) {
            if ( r + c - N >= 0  ) {
                printf("*");
            }
            else {
                printf(" ");
            }
        }
        printf("\n");
    }
}
int main()
{

    rb_trianle(5);
    return 0;
}

5. 等腰三角形

5.1 正等腰三角形

• 形状
  
• 区域表达式

$$\begin{gathered} N=n-1 \\ \begin{array}{r}\{\begin{array}{l}r+c-N\ge 0\\ r-c+N\ge 0\\ 0\le r\le N\\ 0\le c\le 2N\end{array}\end{array} \end{gathered}$$

• 代码

#include <stdio.h>

void iso_trangle(int n) {


    int N = n - 1;
    int r;
    int c;
    for (r = 0; r <= N; r++) {
        for (c = 0; c <= 2 * N; c++) {
            if ((r + c  - N >= 0 ) && (r - c + N >= 0)) {
                printf("*");
            }
            else {
                printf(" ");
            }
        }
        printf("\n");
    }

}

int main() {

    iso_trangle(5);
    return 0;
}

5.2 倒等腰三角形

• 形状
  

• 区域表达式
  $$\begin{gathered} N=n-1 \\ \begin{array}{r}\{\begin{array}{l}r-c\le 0\\ r+c-2N\le 0\\ 0\le r\le N\\ 0\le c\le 2N\end{array}\end{array} \end{gathered}$$

• 代码

#include <stdio.h>

void iso_trangle_rev(int n) {


    int N = n - 1;
    int r;
    int c;
    for (r = 0; r <= N; r++) {
        for (c = 0; c <= 2 * N; c++) {
            if ((r - c <= 0 ) && (r + c - 2 * N <= 0)) {
                printf("*");
            }
            else {
                printf(" ");
            }
        }
        printf("\n");
    }

}

int main() {

    iso_trangle_rev(5);
    return 0;
}

6. 平行四边形

6.1 纵向左下平行四边形

• 形状
  
• 区域表达式
  $$\begin{gathered} N=n-1 \\ \begin{array}{r}\{\begin{array}{l}r+c-N\ge 0\\ r+c-2N\le 0\\ 0\le r\le 2N\\ 0\le c\le N\end{array}\end{array} \end{gathered}$$
• 代码

#include <stdio.h>


void lb_parall(int n) {

    int N = n - 1;
    int r;
    int c;


    for (r = 0; r <= 2 * N; r++) {
        for (c = 0; c <= N; c++) {
            if ((r + c - N >= 0) && (r + c - 2 * N <= 0)) {
                printf("*");
            }
            else {
                printf(" ");
            }
        }
        printf("\n");
    }

}

int main()
{

    lb_parall(5);


    return 0;
}

6.2 纵向左上平行四边形

• 形状
  
• 区域表达式
  $$\begin{gathered} N=n-1 \\ \begin{array}{r}\{\begin{array}{l}r-c\ge 0\\ r-c-N\le 0\\ 0\le r\le 2N\\ 0\le c\le N\end{array}\end{array} \end{gathered}$$
• 代码

#include <stdio.h>


void lu_parall(int n) {

    int N = n - 1;
    int r;
    int c;


    for (r = 0; r <= 2 * N; r++) {
        for (c = 0; c <= N; c++) {
            if ((r - c >= 0) && (r - c - N <= 0)) {
                printf("*");
            }
            else {
                printf(" ");
            }
        }
        printf("\n");
    }

}

int main()
{

    lu_parall(5);


    return 0;
}

6.3 横向左上平行四边形

• 形状
  

• 区域表达式
  $$\begin{gathered} N=n-1 \\ \begin{array}{r}\{\begin{array}{l}r-c\le 0\\ r-c+N\ge 0\\ 0\le r\le N\\ 0\le c\le 2N\end{array}\end{array} \end{gathered}$$

• 代码

#include <stdio.h>


void lu_parall_h(int n) {

    int N = n - 1;
    int r;
    int c;


    for (r = 0; r <= N; r++) {
        for (c = 0; c <= 2 * N; c++) {
            if ((r - c  <= 0) && (r - c + N >= 0)) {
                printf("*");
            }
            else {
                printf(" ");
            }
        }
        printf("\n");
    }

}

int main()
{

    lu_parall_h(5);


    return 0;
}

6.4 横向左下平行四边形

• 形状
  

• 区域表达式
  $$\begin{gathered} N=n-1 \\ \begin{array}{r}\{\begin{array}{l}r+c-N\ge 0\\ r+c-2N\le 0\\ 0\le r\le N\\ 0\le c\le 2N\end{array}\end{array} \end{gathered}$$

• 代码

#include <stdio.h>


void lb_parall_h(int n) {

    int N = n - 1;
    int r;
    int c;


    for (r = 0; r <= N; r++) {
        for (c = 0; c <= 2 * N; c++) {
            if ((r + c - N >= 0) && (r + c - 2 * N <= 0)) {
                printf("*");
            }
            else {
                printf(" ");
            }
        }
        printf("\n");
    }

}

int main()
{

    lb_parall_h(5);


    return 0;
}

7. 菱形

• 形状
  

• 区域表达式
  $$\begin{gathered} N=n-1 \\ \begin{array}{r}\{\begin{array}{l}r+c-N\ge 0\\ r+c-3N\le 0\\ r-c+N\ge 0\\ r-c-N\le 0\\ 0\le r\le 2N\\ 0\le c\le 2N\end{array}\end{array} \end{gathered}$$

• 代码

#include <stdio.h>

void rhom_stars(int n)
{

    int N = n - 1;
    int r;
    int c;


    for ( r = 0;r <= 2 * N; ++r) {
        for ( c = 0; c <= 2 * N; ++c) {
            if ( (r + c - N >= 0 && r + c - 3 * N <= 0) && (r - c + N >= 0 && r - c - N <= 0)) {
                printf("*");
            }
            else {
                printf(" ");
            }
        }
        printf("\n");
    }

}


int main()
{

    rhom_stars(3);

    return 0;
}