Summary
| Description |
English: Derivation of a ternary plot from Cartesian coordinates:
Figure (1) shows an oblique projection of point P(a,b,c) in a 3-dimensional Cartesian space with axes a, b and c, respectively. If a + b + c = K (a positive constant), P is restricted to a plane containing A(K,0,0), B(0,K,0) and C(0,0,K). If a, b and c each cannot be negative, P is restricted to the triangle bounded by A, B and C, as in (2). In (3), the axes are rotated to give an isometric view. The triangle, viewed face-on, appears equilateral. In (4), the distances of P from lines BC, AC and AB are denoted by a’ , b’ and c’ , respectively. For any line l = s + t n̂ in vector form (n̂ is a unit vector) and a point p, the distance from a point to a line from p to l is . In this case, point P is at . Line BC has and . Using the perpendicular distance formula,
Substituting K = a + b + c, . Similar calculation on lines AC and AB gives and . This shows that the distance of the point from the respective lines is linearly proportional to the original values a, b and c.[1] |
| Source | Own work |
| Author | Cmglee |
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