基于R语言实现勾股定理可视化证明的技术求助

问题描述 #

勾股定理的可视化证明图示如下：

需完成以下任务：
a) 编写两个接收正数参数a和b的R函数，分别生成对应两幅示例图（示例参数：a=3,b=4和a=2.5,b=10.6），所有非黑色颜色用hcl()生成。
b) 用layout()函数结合上述两个函数，复现以下组合图形：

用户现有尝试代码：

plot.new()
plot.window(xlim = c(0, 10), ylim = c(0, 10))


half_side_length <- 2.5

 
x_coords_polygon <- c(5, 7.5, 5, 2.5)   
y_coords_polygon <- c(2.5, 7.5, 10, 7.5)   


x_coords_triangles <- list(
  c(5, 6.25, 5),   
  c(6.25, 7.5, 6.25),   
  c(5, 3.75, 5),   
  c(3.75, 2.5, 3.75)  
)

y_coords_triangles <- list(
  c(2.5, 3.75, 5),    
  c(7.5, 8.75, 10),  
  c(7.5, 6.25, 5),    
  c(2.5, 1.25, 0)     
)

 polygon(x_coords_polygon, y_coords_polygon, col = "blue", border = "black")

for (i in 1:4) {
  polygon(x_coords_triangles[[i]], y_coords_triangles[[i]], col = "lightgreen", border = "black")
}

解决方案 #

a) 实现两个可视化函数 #

函数1：中心小正方形+外围直角三角形

此函数绘制包含4个全等直角三角形和中心小正方形的大正方形，直观体现a² + b² = c²的关系：

plot_pythagoras_center <- function(a, b) {
  c <- sqrt(a^2 + b^2)
  max_range <- max(a, b) * 2
  plot.new()
  plot.window(xlim = c(0, max_range), ylim = c(0, max_range))
  
  # 用hcl定义颜色
  tri_color <- hcl(h = 120, c = 40, l = 75)
  square_color <- hcl(h = 240, c = 40, l = 75)
  
  # 绘制4个直角三角形
  polygon(c(max_range - a, max_range, max_range - a), c(0, b, b), col = tri_color, border = "black")
  polygon(c(max_range - c, max_range, max_range - a), c(max_range - b, max_range, max_range), col = tri_color, border = "black")
  polygon(c(0, a, 0), c(max_range - b, max_range, max_range - c), col = tri_color, border = "black")
  polygon(c(0, a, c), c(0, b, 0), col = tri_color, border = "black")
  
  # 绘制中心小正方形
  polygon(c(a, max_range - b, max_range - a, b), c(b, max_range - a, max_range - b, a), col = square_color, border = "black")
  
  axis(1); axis(2)
  title(main = sprintf("a=%.1f, b=%.1f", a, b))
}

# 验证：a=3, b=4
plot_pythagoras_center(3, 4)

函数2：面积拆分对比图

此函数将直角三角形重新排列，展示两个小正方形的面积之和等于大正方形面积：

plot_pythagoras_split <- function(a, b) {
  c <- sqrt(a^2 + b^2)
  max_range <- a + b + c
  plot.new()
  plot.window(xlim = c(0, max_range), ylim = c(0, max_range))
  
  # 用hcl定义颜色
  tri_color <- hcl(h = 30, c = 40, l = 75)
  square_a_color <- hcl(h = 180, c = 40, l = 75)
  square_b_color <- hcl(h = 300, c = 40, l = 75)
  
  # 绘制两个小正方形
  polygon(c(0, a, a, 0), c(0, 0, a, a), col = square_a_color, border = "black")
  polygon(c(a, a + b, a + b, a), c(0, 0, b, b), col = square_b_color, border = "black")
  
  # 绘制重新排列的直角三角形
  polygon(c(a, a + b, a + b - a), c(b, b, b + a), col = tri_color, border = "black")
  polygon(c(a + b, max_range, max_range - a), c(b, b + a, b + a), col = tri_color, border = "black")
  polygon(c(max_range - a, max_range, max_range), c(b + a, b + a, max_range), col = tri_color, border = "black")
  polygon(c(max_range - a, max_range - a + b, max_range), c(max_range, max_range - b, max_range), col = tri_color, border = "black")
  
  axis(1); axis(2)
  title(main = sprintf("a=%.1f, b=%.1f", a, b))
}

# 验证：a=2.5, b=10.6
plot_pythagoras_split(2.5, 10.6)

b) 用layout()组合图形 #

通过布局函数将两个图形并排展示，复现目标组合图：

# 设置1行2列的布局
layout(matrix(c(1, 2), nrow = 1))

# 调用两个函数生成图形（统一用a=3,b=4保持一致性）
plot_pythagoras_center(3, 4)
plot_pythagoras_split(3, 4)

# 恢复默认布局
layout(1)

内容的提问来源于stack exchange，提问作者user