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Triangle

  • Triangle is made up of three points v1, v2 and v3.
  • The triangle have three edges e1, e2, e3. 
  • The length of each edge is l1, l2 and l3.
  • The triangle lies in a plane.
  • The perimeter of the triangle the sum of the length of it's sides. p = l1 + l2+ l3.
  • The semi perimeter s is half of the of the perimeter. s = (l1 + l2+ l3) / 2.
  • Area:
    • The area of a triangle can be calculated with the length of the sides with herons formula. A = sqrt( s(s-l1) * (s-l2) * (s-l3) ),
    • In 3D we can use the cross product. A = (||e1 x e2 ||) / 2.
  • Center of gravity or centroid or geometric center
    • g = v1+ v2 + v3 / 3.
    • Barycentric coo = 0.33,0.33,0.33
  • Circumcenter
    • The circumcenter is the center of a circle passing through the three vertices of the triangle.
  • Incenter
    • s = l1+l2+l3
    • i = l1v1 + l2v2 + l3v3 / s
    • Barycentric coo: l1/ s, l2/s, l3/s
    • Radius of inscribed circle: r = A/s
Barycentric Space
  • This is a coordinate space that moves around on the plane of the triangle.
  • Any point in the plane of the triangle can be given as a weighted average of the vertices in the triangle.
  • p = b1v1 + b2v2 + b3v3.
  • (b1,b2,b3) are the barycentric coordinates. There sum should always add up to one.
  • v1 = (1,0,0), v2 = (0,1,0) and v3 = (0,0,1)
  • All points inside the triangle will have the coordinates in range of 0...1. A point outside will have at least one coordinated negative.
  • Barycentric space splits the plane into triangles of the same size as the orginal triangle.

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