Meta tags:

Headings (most frequently used words):

ternary, plot, plotting, contents, reading, values, on, derivation, from, cartesian, coordinates, example, list, of, notable, diagrams, see, also, references, external, links, the, points,

Text of the page (most frequently used words):
the (108), #ternary (37), plot (35), and (34), pmatrix (32), frac (31), sqrt (21), triangle (18), begin (17), end (17), diagram (16), from (15), this (14), line (14), for (13), edit (13), three (13), lines (13), point (12), sample (12), left (11), right (11), values (11), plotting (10), example (9), are (9), mathbf (9), figure (9), diagrams (8), two (8), wikipedia (7), with (7), retrieved (7), constant (7), coordinates (7), 100 (7), displaystyle (7), grid (7), perpendicular (7), method (7), variables (7), may (6), solution (6), sand (6), clay (6), silt (6), cartesian (6), each (6), along (6), parallel (6), corner (6), page (5), data (5), concentration (5), can (5), points (5), proportion (5), axis (5), that (5), contents (4), search (4), view (4), commons (4), all (4), references (4), list (4), solvent (4), solubility (4), 2020 (4), create (4), software (4), has (4), plots (4), links (4), barycentric (4), soil (4), used (4), find (4), must (4), sum (4), cdot (4), axes (4), distance (4), quad (4), distances (4), showing (4), component (4), graph (4), tools (4), article (4), hide (4), move (4), sidebar (4), toggle (3), table (3), about (3), text (3), additional (3), using (3), wikimedia (3), wikidata (3), articles (3), january (3), information (3), solvation (3), ratio (3), fraction (3), molar (3), related (3), phase (3), university (3), analysis (3), 2010 (3), also (3), step (3), first (3), because (3), shows (3), how (3), equilateral (3), 10px (3), hat (3), any (3), dimensional (3), derivation (3), increments (3), third (3), value (3), other (3), not (3), opposite (3), side (3), determine (3), only (3), main (3), languages (2), contact (2), privacy (2), policy (2), terms (2), use (2), was (2), categories (2), category (2), link (2), needing (2), 2014 (2), short (2), description (2), ion (2), miscibility (2), chart (2), law (2), mass (2), apparent (2), property (2), dilution (2), chemical (2), edu (2), www (2), com (2), department (2), tri (2), serc (2), carleton (2), external (2), pdf (2), morningstar (2), will (2), september (2), original (2), convex (2), optimization (2), game (2), 2009 (2), viviani (2), theorem (2), sciences (2), graphics (2), compositional (2), see (2), flammability (2), finetti (2), notable (2), samples (2), being (2), but (2), its (2), decreases (2), where (2), then (2), textstyle (2), adding (2), linearly (2), gives (2), aligned (2), case (2), vector (2), respectively (2), give (2), restricted (2), cannot (2), unlabeled (2), major (2), plotted (2), intersection (2), altitude (2), preserves (2), motion (2), equal (2), through (2), drawing (2), interest (2), lengths (2), between (2), sides (2), measured (2), ratios (2), edge (2), there (2), reading (2), compositions (2), called (2), simplex (2), which (2), physical (2), gas (2), learn (2), help (2), sources (2), citations (2), appearance (2), upload (2), file (2), changes (2), history (2), read (2), log (2), account (2), donate (2), menu (2), add, topic, mobile, cookie, statement, statistics, developers, code, conduct, legal, safety, contacts, disclaimers, available, under, apply, site, you, agree, registered, trademark, non, profit, organization, foundation, inc, creative, attribution, sharealike, license, rendered, parsoid, last, edited, april, 2026, utc, hidden, matches, triangles, https, org, index, php, title, ternary_plot, oldid, 1347877585, lyate, lyonium, amphiphile, lipophilic, hydrophile, hydrophobe, polarity, partition, coefficient, boiling, freezing, solvents, inorganic, nonaqueous, polar, aprotic, protic, acid, dissociation, henry, raoult, lattice, energy, enthalpy, shell, equilibrium, total, dissolved, solids, mixing, isotopic, abundance, mole, molality, normality, volume, number, quantities, gap, equation, serial, supersaturation, saturation, alloy, eutectic, separation, colloid, suspension, mixture, flory, huggins, buffer, solid, aqueous, ideal, solutions, holland, steven, 2016, georgia, strata, uga, geosciences, developed, language, ternaryplot, generator, quickly, loughborough, geography, resources, gateway, home, lboro, science, education, resource, center, college, excel, template, media, stock, sector, structure, vaughan, archived, december, boyd, vandenberghe, 2004, cambridge, press, karl, tuyls, evolutionary, theoretic, poker, strategies, 1016, entcom, 002, doi, entertainment, computing, weisstein, eric, 2021, mathworld, wolfram, trilemma, project, sigmaplot, dedicated, package, maintained, comprehensive, archive, network, cran, origin, igor, pro, earth, mathematics, finance, investment, portfolio, depicted, ray, uses, defensive, sensitive, cyclical, sectors, alternative, pie, asset, allocation, usda, particle, sizes, texture, uigs, classification, ultramafic, rock, hydrochemistry, piper, jensen, cation, dalitz, chromaticity, types, programmed, mathematica, dependent, intersect, same, proportions, swapped, reflected, vertical, rises, correspondingly, together, make, components, notes, works, hypothetical, set, colorized, united, states, agriculture, textural, useful, consider, placed, triple, analogue, slope, scale, cross, denotes, respective, proportional, similar, calculation, substituting, formula, bigl, bigr, form, unit, denoted, rotated, viewed, face, appears, isometric, positive, plane, containing, negative, bounded, space, oblique, projection, minor, indication, work, empty, second, without, displacement, while, increases, amount, half, decrease, increase, does, require, straight, drawn, these, well, segments, corresponding, individually, possess, easiest, way, shortest, height, consisting, edges, locus, situated, vertex, opposed, decreasing, increasing, regular, intervals, zero, fine, divisions, established, easy, estimation, equivalent, methods, correspond, trilinear, advantage, depicting, conveniently, outlining, composition, regions, different, phases, exist, some, usually, represented, substances, graphed, one, variable, independent, others, known, instance, numerical, vary, independently, possible, combinations, dimensions, degrees, freedom, genotype, frequencies, often, theory, population, genetics, graphically, depicts, positions, show, systems, composed, species, analyzing, metallurgy, mineralogy, petrology, chemistry, gibbs, mixtures, inert, burn, nitrogen, oxygen, methane, when, remove, message, please, unsourced, material, challenged, jstor, scholar, books, newspapers, news, removed, reliable, improve, needs, more, redirected, free, encyclopedia, item, projects, printable, version, download, print, export, switch, legacy, parser, get, shortened, url, cite, permanent, what, here, general, actions, english, talk, українська, română, português, polski, nederlands, 한국어, 日本語, italiano, français, فارسی, eesti, español, ελληνικά, deutsch, العربية, subsection, top, personal, special, pages, recent, community, portal, contribute, random, current, events, navigation, jump, content,

Text of the page (random words):
ternary plot wikipedia jump to content main menu main menu move to sidebar hide navigation main page contents current events random article about wikipedia contact us contribute help learn to edit community portal recent changes upload file special pages search search appearance donate create account log in personal tools donate create account log in contents move to sidebar hide top 1 reading values on a ternary plot 2 derivation from cartesian coordinates 3 plotting a ternary plot 4 example toggle example subsection 4 1 plotting the points 5 list of notable ternary diagrams 6 see also 7 references 8 external links toggle the table of contents ternary plot 15 languages العربية deutsch ελληνικά español eesti فارسی français italiano 日本語 한국어 nederlands polski português română українська edit links article talk english read edit view history tools tools move to sidebar hide actions read edit view history general what links here related changes upload file permanent link page information cite this page get shortened url switch to legacy parser print export download as pdf printable version in other projects wikimedia commons wikidata item appearance move to sidebar hide from wikipedia the free encyclopedia redirected from triangle plot barycentric plot on three variables this article needs more citations please help improve this article by adding citations to reliable sources unsourced material may be challenged and removed find sources ternary plot news newspapers books scholar jstor january 2014 learn how and when to remove this message a ternary flammability diagram showing which mixtures of methane oxygen gas and inert nitrogen gas will burn a ternary plot ternary graph triangle plot simplex plot or gibbs triangle is a barycentric plot on three variables which sum to a constant 1 it graphically depicts the ratios of the three variables as positions in an equilateral triangle it is used in physical chemistry petrology mineralogy metallurgy and other physical sciences to show the compositions of systems composed of three species ternary plots are tools for analyzing compositional data in the three dimensional case in population genetics a triangle plot of genotype frequencies is called a de finetti diagram in game theory 2 and convex optimization 3 it is often called a simplex plot in a ternary plot the values of the three variables a b and c must sum to some constant k usually this constant is represented as 1 0 or 100 because a b c k for all substances being graphed any one variable is not independent of the others so only two variables must be known to find a sample s point on the graph for instance c must be equal to k a b because the three numerical values cannot vary independently there are only two degrees of freedom it is possible to graph the combinations of all three variables in only two dimensions the advantage of using a ternary plot for depicting chemical compositions is that three variables can be conveniently plotted in a two dimensional graph ternary plots can also be used to create phase diagrams by outlining the composition regions on the plot where different phases exist the values of a point on a ternary plot correspond up to a constant to its trilinear coordinates or barycentric coordinates reading values on a ternary plot edit there are three equivalent methods that can be used to determine the values of a point on the plot parallel line or grid method the first method is to use a diagram grid consisting of lines parallel to the triangle edges a parallel to a side of the triangle is the locus of points constant in the component situated in the vertex opposed to the side each component is 100 in a corner of the triangle and 0 at the edge opposite it decreasing linearly with increasing distance perpendicular to the opposite edge from this corner by drawing parallel lines at regular intervals between the zero line and the corner fine divisions can be established for easy estimation perpendicular line or altitude method for diagrams that do not possess grid lines the easiest way to determine the values is to determine the shortest i e perpendicular distances from the point of interest to each of the three sides by viviani s theorem the distances or the ratios of the distances to the triangle height give the value of each component corner line or intersection method the third method does not require the drawing of perpendicular or parallel lines straight lines are drawn from each corner through the point of interest to the opposite side of the triangle the lengths of these lines as well as the lengths of the segments between the point and the corresponding sides are measured individually the ratio of the measured lines then gives the component value as a fraction of 100 a displacement along a parallel line grid line preserves the sum of two values while motion along a perpendicular line increases or decreases the two values an equal amount each half of the decrease increase of the third value motion along a line through a corner preserves the ratio of the other two values figure 1 altitude method figure 2 intersection method figure 3 an example ternary diagram without any points plotted figure 4 an example ternary diagram showing increments along the first axis figure 5 an example ternary diagram showing increments along the second axis figure 6 an example ternary diagram showing increments along the third axis figure 7 empty ternary plot figure 8 indication of how the three axes work unlabeled triangle plot with major grid lines unlabeled triangle plot with major and minor grid lines derivation from cartesian coordinates edit 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 n ̂ in vector form n ̂ is a unit vector and a point p the perpendicular distance from p to l is s p s p n n displaystyle left mathbf s mathbf p bigl mathbf s mathbf p cdot mathbf hat n bigr mathbf hat n right in this case point p is at p a b c displaystyle mathbf p begin pmatrix a b c end pmatrix line bc has s 0 k 0 and n 0 k 0 0 0 k 0 k 0 0 0 k 0 k k 0 2 k 2 k 2 0 1 2 1 2 displaystyle mathbf s begin pmatrix 0 k 0 end pmatrix quad text and quad mathbf hat n frac begin pmatrix 0 k 0 end pmatrix begin pmatrix 0 0 k end pmatrix left begin pmatrix 0 k 0 end pmatrix begin pmatrix 0 0 k end pmatrix right frac begin pmatrix 0 k k end pmatrix sqrt 0 2 k 2 k 2 begin pmatrix 0 frac 1 sqrt 2 frac 1 sqrt 2 end pmatrix using the perpendicular distance formula a a k b c a k b c 0 1 2 1 2 0 1 2 1 2 a k b c 0 k b 2 c 2 0 1 2 1 2 a k b k b c 2 c k b c 2 a k b c 2 k b c 2 a 2 k b c 2 2 k b c 2 2 a 2 k b c 2 2 displaystyle begin aligned a left begin pmatrix a k b c end pmatrix left begin pmatrix a k b c end pmatrix cdot begin pmatrix 0 frac 1 sqrt 2 frac 1 sqrt 2 end pmatrix right begin pmatrix 0 frac 1 sqrt 2 frac 1 sqrt 2 end pmatrix right 10px left begin pmatrix a k b c end pmatrix left 0 frac k b sqrt 2 frac c sqrt 2 right begin pmatrix 0 frac 1 sqrt 2 frac 1 sqrt 2 end pmatrix right 10px left begin pmatrix a k b frac k b c 2 c frac k b c 2 end pmatrix right left begin pmatrix a frac k b c 2 frac k b c 2 end pmatrix right 10px sqrt a 2 left frac k b c 2 right 2 left frac k b c 2 right 2 sqrt a 2 frac k b c 2 2 end aligned substituting k a b c a a 2 a b c b c 2 2 a 2 a 2 2 a 3 2 displaystyle a sqrt a 2 frac a b c b c 2 2 sqrt a 2 frac a 2 2 a sqrt frac 3 2 similar calculation on lines ac and ab gives b b 3 2 and c c 3 2 displaystyle b b sqrt frac 3 2 quad text and quad c c sqrt frac 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 4 plotting a ternary plot edit analogue on a cartesian grid by adding lines of slope 1 the scale of the c axis is 1 2 textstyle frac 1 sqrt 2 that of the a and b axes the cross denotes the point a b c cartesian coordinates are useful for plotting points in the triangle consider an equilateral ternary plot where a 100 is placed at x y 0 0 and b 100 at 1 0 then c 100 is 1 2 3 2 textstyle frac 1 2 frac sqrt 3 2 and the triple a b c is 1 2 2 b c a b c 3 2 c a b c displaystyle left frac 1 2 cdot frac 2b c a b c frac sqrt 3 2 cdot frac c a b c right example edit a colorized soil textural triangle from the united states department of agriculture this example shows how this works for a hypothetical set of three soil samples sample clay silt sand notes sample 1 50 20 30 because clay and silt together make up 70 of this sample the proportion of sand must be 30 for the components to sum to 100 sample 2 10 60 30 the proportion of sand is 30 as in sample 1 but as the proportion of silt rises by 40 the proportion of clay decreases correspondingly sample 3 10 30 60 this sample has the same proportion of clay as sample 2 but the proportions of silt and sand are swapped the plot is reflected about its vertical axis plotting the points edit plotting sample 1 step 1 find the 50 clay line plotting sample 1 step 2 find the 20 silt line plotting sample 1 step 3 being dependent on the first two the intersect is on the 30 sand line plotting all the samples ternary triangle plot of soil types sand clay and silt programmed with mathematica list of notable ternary diagrams edit chromaticity diagram de finetti diagram dalitz plot flammability diagram jensen cation plot piper diagram used in hydrochemistry uigs classification diagram for ultramafic rock usda soil texture diagram by particle sizes in finance the asset allocation of an investment portfolio can be depicted as a ternary plot for example the morningstar x ray 5 uses the defensive sensitive and cyclical sectors as an alternative to a pie chart see also edit apparent molar property barycentric coordinates mathematics compositional data list of information graphics software earth sciences graphics software igor pro origin data analysis software r has a dedicated package ternary maintained on the comprehensive r archive network cran sigmaplot project triangle trilemma viviani s theorem references edit weisstein eric w ternary diagram mathworld wolfram com retrieved 2021 06 05 karl tuyls an evolutionary game theoretic analysis of poker strategies entertainment computing january 2009 doi 10 1016 j entcom 2009 09 002 p 9 boyd s and vandenberghe l 2004 convex optimization cambridge university press vaughan will september 5 2010 ternary plots archived from the original on december 20 2010 retrieved september 7 2010 morningstar stock sector structure pdf external links edit wikimedia commons has media related to ternary plots excel template for ternary diagrams serc carleton edu science education resource center serc carleton college retrieved 14 may 2020 tri plot ternary diagram plotting software www lboro ac uk loughborough university department of geography resources gateway home tri plot retrieved 14 may 2020 ternary plot generator quickly create ternary diagrams on line www ternaryplot com retrieved 14 may 2020 holland steven 2016 data analysis in the geosciences ternary diagrams developed in the r language strata uga edu university of georgia retrieved 14 may 2020 v t e chemical solutions solution ideal solution aqueous solution solid solution buffer solution flory huggins mixture suspension colloid phase diagram phase separation eutectic point alloy saturation supersaturation serial dilution dilution equation apparent molar property miscibility gap concentration and related quantities molar concentration mass concentration number concentration volume concentration normality molality mole fraction mass fraction isotopic abundance mixing ratio ternary plot total dissolved solids solubility solubility equilibrium solvation solvation shell enthalpy of solution lattice energy raoult s law henry s law solubility table data solubility chart miscibility solvent category acid dissociation constant protic solvent polar aprotic solvent inorganic nonaqueous solvent solvation list of boiling and freezing information of solvents partition coefficient polarity hydrophobe hydrophile lipophilic amphiphile lyonium ion lyate ion retrieved from https en wikipedia org w index php title ternary_plot oldid 1347877585 categories diagrams triangles hidden categories articles with short description short description matches wikidata articles needing additional references from january 2014 all articles needing additional references commons category link is on wikidata this page was last edited on 9 april 2026 at 10 53 utc page was rendered with parsoid text is available under the creative commons attribution sharealike 4 0 license additional terms may apply by using this site you agree to the terms of use and privacy policy wikipedia is a registered trademark of the wikimedia foundation inc a non profit organization privacy policy about wikipedia disclaimers contact wikipedia legal safety contacts code of conduct developers statistics cookie statement mobile view search search toggle the table of contents ternary plot 15 languages add topic