area: Triangle Surface Area Calculation

Vector-Based Surface Measurement

Triangle area calculation employs the cross product relationship between two edge vectors to determine surface magnitude. This fundamental geometric operation provides precise measurements regardless of triangle orientation in three-dimensional space.

The calculation utilizes the mathematical property that cross product magnitude equals the parallelogram area formed by two vectors. Since triangles represent half of such parallelograms, the final result requires multiplication by 0.5.

Mathematical Definition

The area formula derives from vector cross product properties:

$A = \frac{1}{2} |(\vec{b} - \vec{a}) \times (\vec{c} - \vec{a})|$

Where:

• $\vec{a}$, $\vec{b}$, $\vec{c}$ represent triangle vertex positions
• The cross product generates a vector perpendicular to both edge vectors
• Vector magnitude provides the parallelogram area
• Division by 2 yields triangle area

Cross Product Geometry

Component	Vector Calculation	Geometric Meaning
Edge AB	$\vec{b} - \vec{a}$	First triangle edge from vertex A
Edge AC	$\vec{c} - \vec{a}$	Second triangle edge from vertex A
Cross Product	$\vec{AB} \times \vec{AC}$	Perpendicular vector with parallelogram magnitude
Area Result	$\frac{1}{2}\|\vec{AB} \times \vec{AC}\|$	Triangle surface measurement

The cross product direction follows the right-hand rule, indicating surface normal orientation. However, for area calculation, only the magnitude matters.

ライブエディター

const fragment = () => Scope(() => {
      const scale = iTime.mul(0.5).sin()
      const tri = Triangle({
              a: vec3(-0.4, -0.4, 0),
              b: vec3(0.4, -0.4, 0),
              c: vec3(0, scale.mul(0.4), 0)
      })
      const triArea = area(tri)
      const pos = uv.mul(2).sub(1)
      const isInside = triangleContain(tri, vec3(pos, 0))
      const areaColor = vec3(triArea.mul(2), triArea.mul(1.5), 0.3)
      return vec4(vec3(0.1, 0.1, 0.15).select(areaColor, isInside), 1)
})