free_triangle

This module computes the area of the triangle or trapezium formed between a triangle with a fixed vertex on a reference line and a line found at a given distance distance from this reference line.

The various cases are described by the images given in the following documentation, where the area which we want to compute is that of the filled shape.

Solution

Let $\mathrm{xOy}$ be an orthogonal coordinate system and let $\triangle ABC$ be an arbitrary triangle so that $A \in \mathrm{Ox}$ and

where $a, b, c \in \mathbb{R}_+^*$ are fixed numbers.

Consider $\alpha = \angle CAX$ and $d \parallel \mathrm{Ox}$ so that the distance between the line d and $\mathrm{Ox}$ equals $h \in \mathbb{R}_+$. Also let $d \cap AB = \{B_1\}$ and $d \cap AC = \{C_1\}$.

We will first find the lengths of the projections of B and C on $\mathrm{Ox}$ because we can separate different cases based on these values. Let $d(B,\mathrm{Ox}) = h_b$ and $d(C,\mathrm{Ox}) = h_c$. Since these values are independent of the relations set between them and h we can consider, without loss of generality, the following case: $h_b > h$ and $h_c > h$.

Let S be the area which we must determine. Based on the values of $h_b$, $h_c$ and $h$, we find the following solutions:

1) $h_b \leq h$, $h_c \leq h \Rightarrow S = \mathcal{A}_{\triangle ABC}$

2) $h_b \geq h$, $h_c \geq h \Rightarrow S = \mathcal{A}_{\triangle AB_1C_1}$

3) $h_b < h$, $h_c > h\f \Rightarrow S = \mathcal{A}_{[ABB_1C_1]}$

4) $h_b > h$, $h_c < h\f \Rightarrow S = \mathcal{A}_{[ABB_1C_1]}$

#include <codecogs/maths/geometry/area/free_triangle.h>
#include <stdio.h>

int main()
{
  // the lengths of the sides
  double a = 2.613, b = 1.878, c = 1.209;

  // the value of the alpha angle in radians
  double alpha = 0.61086;

  // display the lenghts of the sides and the alpha angle
  printf("a = %.3lf\nb = %.3lf\nc = %.3lf\nalpha = %.5lf \n\n", a, b, c, alpha);

  // display the area for different values of h
  for (double h = 0.0; h < 1.6; h += 0.3)
  printf("h = %.1lf   Area = %.2lf\n", h,
  Maths::Geometry::Area::free_triangle(a, b, c, h, alpha));

  return 0;
}

a = 2.613
b = 1.878
c = 1.209
alpha = 0.61086

h = 0.0   Area = 0.00
h = 0.3   Area = 0.14
h = 0.6   Area = 0.56
h = 0.9   Area = 0.97
h = 1.2   Area = 1.04
h = 1.5   Area = 1.04

Parameters

Returns

The vertices must be denoted such that the abscissa of $B$ is less than that of $C$, i.e. as in the given examples.