TÄNA AJALOOS, 28. juuli ⟩ Hiinas toimus 20. sajandi suurim maavärin

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DescriptionTernary plot visualisation.svg	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 $\\|(\mathbf {s} -\mathbf {p} )-((\mathbf {s} -\mathbf {p} )\cdot \mathbf {\hat {n}} )\mathbf {\hat {n}} \\|$ . In this case, point P is at $\mathbf {p} ={\begin{pmatrix}a\\b\\c\end{pmatrix}}$ . Line BC has $\mathbf {s} ={\begin{pmatrix}0\\K\\0\end{pmatrix}}$ and $\mathbf {\hat {n}} ={\frac {{\Big (}{\begin{smallmatrix}0\\K\\0\end{smallmatrix}}{\Big )}-{\Big (}{\begin{smallmatrix}0\\0\\K\end{smallmatrix}}{\Big )}}{{\Big \|}{\Big \|}{\Big (}{\begin{smallmatrix}0\\K\\0\end{smallmatrix}}{\Big )}-{\Big (}{\begin{smallmatrix}0\\0\\K\end{smallmatrix}}{\Big )}{\Big \|}{\Big \|}}}={\frac {{\Big (}{\begin{smallmatrix}0\\K\\-K\end{smallmatrix}}{\Big )}}{\sqrt {0^{2}+K^{2}+(-K)^{2}}}}={\begin{pmatrix}0\\\;\;1/{\sqrt {2}}\\-1/{\sqrt {2}}\end{pmatrix}}$ . Using the perpendicular distance formula, ${\begin{aligned}a'&={\bigg \|}{\bigg \|}{\Big (}{\begin{smallmatrix}-a\\K-b\\-c\end{smallmatrix}}{\Big )}-{\bigg (}{\Big (}{\begin{smallmatrix}-a\\K-b\\-c\end{smallmatrix}}{\Big )}\cdot {\Big (}{\begin{smallmatrix}0\\\;\;1/{\sqrt {2}}\\-1/{\sqrt {2}}\end{smallmatrix}}{\Big )}{\bigg )}{\Big (}{\begin{smallmatrix}0\\\;\;1/{\sqrt {2}}\\-1/{\sqrt {2}}\end{smallmatrix}}{\Big )}{\bigg \|}{\bigg \|}\\&={\bigg \|}{\bigg \|}{\Big (}{\begin{smallmatrix}-a\\K-b\\-c\end{smallmatrix}}{\Big )}-{\Big (}0+{\tfrac {K-b}{\sqrt {2}}}+{\tfrac {c}{\sqrt {2}}}{\Big )}{\Big (}{\begin{smallmatrix}0\\\;\;1/{\sqrt {2}}\\-1/{\sqrt {2}}\end{smallmatrix}}{\Big )}{\bigg \|}{\bigg \|}\\&={\bigg \|}{\bigg \|}{\bigg (}{\begin{smallmatrix}-a\\K-b-{\tfrac {K-b+c}{2}}\\-c+{\tfrac {K-b+c}{2}}\end{smallmatrix}}{\bigg )}{\bigg \|}{\bigg \|}={\bigg \|}{\bigg \|}{\bigg (}{\begin{smallmatrix}-a\\{\tfrac {K-b-c}{2}}\\{\tfrac {K-b-c}{2}}\end{smallmatrix}}{\bigg )}{\bigg \|}{\bigg \|}\\&={\sqrt {(-a)^{2}+{\big (}{\tfrac {K-b-c}{2}}{\big )}^{2}+{\big (}{\tfrac {K-b-c}{2}}{\big )}^{2}}}={\sqrt {a^{2}+{\tfrac {(K-b-c)^{2}}{2}}}}\\\end{aligned}}$ Substituting K = a + b + c, $a'={\sqrt {a^{2}+{\tfrac {(a+b+c-b-c)^{2}}{2}}}}={\sqrt {a^{2}+{\tfrac {a^{2}}{2}}}}=a{\sqrt {\tfrac {3}{2}}}$ . Similar calculation on lines AC and AB gives $b'=b{\sqrt {\tfrac {3}{2}}}$ and $c'=c{\sqrt {\tfrac {3}{2}}}$ . This shows that the distance of the point from the respective lines is linearly proportional to the original values a, b and c.^[1] ↑ Ternary plots
Source	Own work
Author	Cmglee

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	Date/Time	Dimensions	User	Comment
current	00:03, 24 June 2013	512 × 128 (5 KB)	Cmglee	Fix svg viewBox.
	00:01, 24 June 2013	512 × 512 (5 KB)	Cmglee	Finish graphic.
	19:58, 21 June 2013	512 × 154 (6 KB)	Cmglee	{{Information \|Description ={{en\|1=Derivation of ternary plots from Cartesian coordinates: For a line '''l''' = '''s''' + ''t'' '''n''' in vector form, where '''n''' is a unit vector, and a point '''p''', the [[distance from a point to a line\|perp...

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Short title	Derivation of ternary plots from Cartesian coordinates
Image title	Derivation of ternary plots from Cartesian coordinates, drawn by CMG Lee. (1) shows a point P(a,b,c) in a 3-dimensional Cartesian plane. In (2), if a + b + c = K, and a, b and c are non-negative, P is restricted to the triangle bounded by A, B and C. In (3), the axes are rotated to give an isometric view (the triangle appears equilateral). In (4), the distances of P from lines BC, AC and AB are denoted by a', b' and c'.
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