三视图投影 - 计算空间几何体的三视图
# Three-View Projection\n\n## \u9898\u76ee\u63cf\u8ff0\n\n> \u5168\u6587\u4e0e\u0024x, y, z\u0024\u8f74/\u65b9\u5411\u6709\u5173\u7684\u5185\u5bb9\u5747\u57fa\u4e8e\u4ee5\u4e0b\u7684\u0024x, y, z\u0024\u8f74\u3001\u4f4d\u7f6e\u53ca\u89c2\u5bdf\u65b9\u5411\u3002\u89c2\u5bdf\u8005\u7acb\u5728\u5982\u56fe\u6240\u793a\u7684\u4f4d\u7f6e\uff0c\u6b63\u5728\u5411\u7740\u0024y\u0024\u5468\uff08\u7eff\u7ebf\uff09\u6b63\u65b9\u5411\uff08\u7bad\u52a8\u65b9\u5411\uff09\u89c2\u5bdf\u3002\n> 
\n\n\u73b0\u5728\u5728\u5730\u9762\u4e0a\u6709\u4e00\u4e2a\u957f\u3001\u5e7f\uff08\u5411\u0024x, y\u0024\u8f74\u5ef6\u8fdc\u7684\u957f\u5ea6\uff09\u4e3a\u0024n \text{ cm}, m \text{ cm}\u0024\u7684\u957f\u5706\u5f62\u533a\u57df\u3002\u6211\u4eec\u5c06\u8fd9\u7247\u533a\u57df\u5206\u522b\u5212\u5206\u4e3a\u0024n \times m\u0024\u4e2a\u5c0f\u683c\u5b50\u3002\u6bcf\u4e2a\u5c0f\u683c\u5b50\u4e2d\u53ef\u4ee5\u653e\u7f6e\u4e00\u4e2a\u8fb9\u957f\u4e3a\u00241 \text{ cm}\u0024\u7684\u5c0f\u6b63\u65b9\u4f53\u3002\n\n\u6211\u4eec\u4f7f\u7528\u4e00\u4e2a\u6574\u6570\u4e8c\u5143\u7ec4\uff08\u0024x, y\u0024\uff09\u6765\u8868\u793a\u4e00\u4e2a\u683c\u5b50\u5728\u7a7a\u95f4\u4e2d\u7684\u4f4d\u7f6e\uff0c\u5176\u4ee3\u8868\u4ece\u5de6\u5411\u53f3\u6570\u7b2c\u0024x\u0024\u4e2a\uff0c\u4ece\u524d\u5411\u540e\u6570\u7b2c\u0024y\u0024\u4e2a\u683c\u5b50\u3002\n\n
\n\n\u4f8b\u5982\uff0c\u4e0a\u56fe\u5c55\u793a\u7684\u00243 \times 4\u0024\u533a\u57df\u4e2d\uff0c\u6309\u7167\u89c2\u5bdf\u8005\u7684\u4f4d\u7f6e\uff0c\u53ef\u4ee5\u53d1\u73b0\u7ea2\u84dd\u7ebf\uff08\u0024x, z\u0024\u8f74\uff09\u6240\u5728\u7684\u9762\u4e3a\u524d\u9762\uff0c\u84dd\u7eff\u7ebf\uff08\u0024y, z\u0024\u8f74\uff09\u6240\u5728\u7684\u9762\u4e3a\u5de6\u9762\uff0c\u7ea2\u7eff\u7ebf\uff08\u0024x, y\u0024\u8f74\uff09\u6240\u5728\u7684\u9762\u4e3a\u4e0b\u9762\u3002\u4e94\u4e2a\u5c0f\u6b63\u65b9\u4f53\u7684\u4f4d\u7f6e\u5206\u522b\u4e3a\u0024(1, 1), (1, 3), (3, 1), (3, 3), (3, 4)\u0024\u3002\n\n\u4e09\u89c2\u56fe\u662f\u89c2\u5bdf\u8005\u4ece\u4e0a\u9762\uff08\u4ece\u4e0a\u5411\u4e0b\uff09\u3001\u5de6\u9762\uff08\u4ece\u5de6\u5411\u53f3\uff09\u3001\u524d\u9762\uff08\u4ece\u524d\u5411\u540e\uff09\u4e09\u4e2a\u4e0d\u540c\u89d2\u5ea6\u89c2\u5bdf\u540c\u4e00\u4e2a\u7a7a\u95f4\u51e0\u4f53\u7269\u800c\u753b\u51fa\u7684\u56fe\u5f62\u3002\n\n
\n\n
\n\n\u4f8b\u5982\uff0c\u4ee5\u4e0a\u662f\u793a\u4f8b\u533a\u57df\u7684\u4e09\u89c2\u56fe\u3002\n\n\u73b0\u5728\uff0c\u7ed9\u5b9a\u4e00\u4e2a\u533a\u57df\u53ca\u5176\u4e2d\u7684\u6b63\u65b9\u4f53\u6392\u5e03\u60c5\u51b5\uff0c\u8bf7\u4f60\u5e2e\u52a9\u8ba1\u7b97\u51fa\u8fd9\u4e2a\u533a\u57df\u7684\u4e09\u89c2\u56fe\u3002\n\n## \u8f93\u5165\u683c\u5f0f\n\n\u8f93\u5165\u5171\u0024n + 1\u0024\u884c\u3002\n\n\u7b2c\u4e00\u884c\u4e3a\u4e24\u4e2a\u6574\u6570\u0024n, m\u0024\uff0c\u4ee3\u8868\u957f\u5706\u5f62\u533a\u57df\u7684\u957f\u3001\u5e7f\u3002\n\u63a5\u7740\u0024n\u0024\u884c\uff0c\u6bcf\u884c\u0024m\u0024\u4e2a\u6574\u6570\u3002\u5176\u4e2d\u7b2c\u0024x\u0024\u884c\u7b2c\u0024y\u0024\u5217\u7684\u6574\u6570\u0024a _ {x, y}\u0024\u4ee3\u8868\uff08\u0024x, y\u0024\uff09\u5904\u683c\u5b50\u7684\u60c5\u51b5\uff0c\u5982\u679c\u0024a _ {x, y} = 1\u0024\uff0c\u4ee3\u8868\u8fd9\u4e2a\u683c\u5b50\u6709\u5c0f\u6b63\u65b9\u4f53\uff1b\u5426\u5219\uff0c\u4ee3\u8868\u8fd9\u4e2a\u683c\u5b50\u6ca1\u6709\u5c0f\u6b63\u65b9\u4f53\u3002\n\n## \u8f93\u51fa\u683c\u5f0f\n\n\u8f93\u51fa\u5171\u00242 + m\u0024\u884c\u3002\n\n\u7b2c\u4e00\u884c\u4e3a\u0024n\u0024\u4e2a\u6574\u6570\uff0c\u4ee3\u8868**\u81ea\u5de6\u81f3\u53f3**\u6b63\u89c2\u56fe\u7684\u60c5\u51b5\u3002\u5982\u679c\u6b63\u89c2\u56fe\u4e2d\u67d0\u4e2a\u4f4d\u7f6e\u6709\u586b\u5145\uff0c\u5219\u8f93\u51fa\u00241\u0024\uff0c\u5426\u5219\u8f93\u51fa\u00240\u0024\u3002\u540c\u4e00\u884c\u6574\u6570\u4e4b\u95f4\u4e24\u4e24\u4ee5\u4e00\u4e2a\u7a7a\u683c\u9694\u5f00\u3002\n\u7b2c\u4e8c\u884c\u4e3a\u0024m\u0024\u4e2a\u6574\u6570\uff0c\u4ee3\u8868**\u81ea\u540e\u81f3\u524d**\u5de6\u89c2\u56fe\u7684\u60c5\u51b5\u3002\u5982\u679c\u5de6\u89c2\u56fe\u4e2d\u67d0\u4e2a\u4f4d\u7f6e\u6709\u586b\u5145\uff0c\u5219\u8f93\u51fa\u00241\u0024\uff0c\u5426\u5219\u8f93\u51fa\u00240\u0024\u3002\u540c\u4e00\u884c\u6574\u6570\u4e4b\u95f4\u4e24\u4e24\u4ee5\u4e00\u4e2a\u7a7a\u683c\u9694\u5f00\u3002\n\u63a5\u7740\u0024m\u0024\u884c\uff0c\u6bcf\u884c\u0024n\u0024\u4e2a\u6574\u6570\uff0c\u4ee3\u8868**\u81ea\u540e\u81f3\u524d\u81ea\u5de6\u81f3\u53f3**\u5012\u89c2\u56fe\u7684\u60c5\u51b5\u3002\u5982\u679c\u5012\u89c2\u56fe\u4e2d\u67d0\u4e2a\u4f4d\u7f6e\u6709\u586b\u5145\uff0c\u5219\u8f93\u51fa\u00241\u0024\uff0c\u5426\u5219\u8f93\u51fa\u00240\u0024\u3002\u540c\u4e00\u884c\u6574\u6570\u4e4b\u95f4\u4e24\u4e24\u4ee5\u4e00\u4e2a\u7a7a\u683c\u9694\u5f00\u3002\n\n**\u4ee5\u4e0a\u65b9\u5411\u5747\u57fa\u4e8e\u89c2\u5bdf\u8005\u521d\u59cb\u7acb\u7684\u4f4d\u7f6e\u53ca\u65b9\u5411\u3002\u5982\u679c\u5bf9\u8f93\u51fa\u7684\u987a\u5e8f\u6709\u7591\u95ee\uff0c\u8bf7\u53c2\u7167\u6837\u4f8b\u8f85\u52a9\u7406\u89e3\u3002**\n\n## \u6837\u4f8b\u00231\n\n### \u6837\u4f8b\u8f93\u5165\u00231\n\n3 41 0 1 00 0 0 01 0 1 1### \u6837\u4f8b\u8f93\u51fa\u00231\n\n1 0 11 1 0 10 0 11 0 10 0 01 0 1## \u6837\u4f8b\u00232\n\n### \u6837\u4f8b\u8f93\u5165\u00232\n\n10 70 1 0 0 0 0 00 0 0 0 1 0 00 0 0 0 0 0 00 0 0 0 0 0 01 0 0 0 0 0 00 0 0 0 0 1 00 0 1 0 1 0 00 1 1 0 0 0 01 0 0 0 0 0 00 0 0 0 0 0 0### \u6837\u4f8b\u8f93\u51fa\u00232\n\n1 1 0 0 1 1 1 1 1 00 1 1 0 1 1 10 0 0 0 0 0 0 0 0 00 0 0 0 0 1 0 0 0 00 1 0 0 0 0 1 0 0 00 0 0 0 0 0 0 0 0 00 0 0 0 0 0 1 1 0 01 0 0 0 0 0 0 1 0 00 0 0 0 1 0 0 0 1 0## \u63d0\u793a\n\n### \u6837\u4f8b 1 \u89e3\u91ca\n\n\u6837\u4f8b 1 \u5c31\u662f\u9898\u76ee\u4e2d\u7684\u793a\u4f8b\u3002\n\n
\n\n### \u6570\u636e\u89c4\u6a21\u53ca\u7ea6\u5b9a\n\n\u5bf9\u4e8e\u0024100\u0025\u0024\u7684\u6570\u636e\uff0c\u4fdd\u8bc1\u00241 \leq n, m \leq 200\u0024\uff0c\u00240 \leq a _ {x, y} \leq 1\u0024\u3002\n\n| \u6d4b\u8bd5\u70b9\u7f16\u53f7 | \u0024n\u0024 | \u0024m\u0024 | \u0024a _ {x, y}\u0024 |\n| :-----------: | :-----------: | :-----------: | :-----------: |\n| \u00241\u0024 | \u003d 1 | \u003d 1 | \u65e0\u7279\u6b8a\u9650\u5236 |\n| \u00242 \sim 4\u0024 | \u0026lt;= 10 | \u0026lt;= 10 | \u65e0\u7279\u6b8a\u9650\u5236 |\n| \u00245 \sim 6\u0024 | \u0026lt;= 200 | \u003d 1 | \u65e0\u7279\u6b8a\u9650\u5236 |\n| \u00247\u0024 | \u0026lt;= 200 | \u0026lt;= 200 | \u003d 1 |\n| \u00248 \sim 10\u0024 | \u0026lt;= 200 | \u0026lt;= 200 | \u65e0\u7279\u6b8a\u9650\u5236 |cpp code\u5185\u5bb9\uff1a```cpp#include
int main() { int n, m; cin >> n >> m; vector<vector
// 正视图 vector<int> front_view(n); for (int i = 0; i < n; i++) { for (int j = 0; j < m; j++) { if (grid[i][j] == 1) { front_view[i] = 1; break; } } }
// 左视图 vector<int> left_view(m); for (int j = 0; j < m; j++) { for (int i = 0; i < n; i++) { if (grid[i][j] == 1) { left_view[j] = 1; break; } } }
// 俯视图 vector<vector<int>> top_view(m, vector<int>(n)); for (int j = 0; j < m; j++) { for (int i = 0; i < n; i++) { top_view[j][i] = grid[i][j]; } }
// 输出结果 for (int i = 0; i < n; i++) { cout << front_view[i] << ' '; } cout << endl; for (int j = 0; j < m; j++) { cout << left_view[j] << ' '; } cout << endl; for (int j = 0; j < m; j++) { for (int i = 0; i < n; i++) { cout << top_view[j][i] << ' '; } cout << endl; }
return 0
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