Key takeaway

Triangle 1 turns three gas percentages into a single point by barycentric (ternary) projection: a constraint plane in 3-space rotated face-on so the three axes sit at 60°. Pentagon 1 turns five gas percentages into a single point by an area-weighted polygon centroid — the Bourke (Shoelace) integral, not the arithmetic mean of the five vertices. Once you can draw the construction on a napkin, the diagrams stop being magic and start being projections of a real measurement.

1. Why this companion exists#

The companion article on Triangle 1 and Pentagon 1 covers what the two diagrams diagnose and how to read the zones. It does not derive the geometry, and several engineers wrote to ask: what is actually going on with the 60° axes? Why does a five-gas reading reduce to one point and not five? And how are the zone vertex tables produced?

Engineers fluent in 2D Cartesian and 3D orthogonal coordinates were never formally taught barycentric (ternary) projection or polygon centroids — this geometry sits in textbooks the average maintenance engineer never opens. The result is that the most-used graphical DGA methods come across as opaque conventions. They are not. The construction is five lines of arithmetic.

This post derives the geometry of both diagrams from first principles, walks the same worked examples used in the companion post through the construction, and shows where the two diagrams' zone-definition conventions are deliberately asymmetric. The arithmetic is short enough that you should be able to reproduce both worked examples on paper before the end of the article.

2. Triangle 1 — the ternary geometry#

Three gases, two degrees of freedom#

Triangle 1 consumes three gas concentrations: methane (CH₄), ethylene (C₂H₄) and acetylene (C₂H₂). The raw triplet lives in 3-space — three independent ppm values, three perpendicular axes.

Any three positive numbers can be viewed as coordinates $(x, y, z)$ in an ordinary Cartesian frame; their sum is the plane $x + y + z = S$. Normalising by $S$ projects every point onto the plane $S = 100$. The three pure-gas extremes $(100, 0, 0)$, $(0, 100, 0)$, $(0, 0, 100)$ are the corners of that plane in the positive octant, and every normalised triplet lies somewhere on the triangle they bound.

The classification step does not care about absolute concentrations; it cares about the ratio between the three gases. So the first move is to normalise (IEC 60599:2022, Annex B, Figure B.3, p. 37):

and analogously for %C₂H₄ and %C₂H₂. By construction these three percentages sum to exactly 100. That single algebraic identity is what makes 2D plotting possible: a point that was free to wander in 3-space is now constrained to the plane %CH₄ + %C₂H₄ + %C₂H₂ = 100 sitting inside the positive octant. A plane is two-dimensional. A faithful 2D display exists.

Why not just plot two gases against each other#

A reader who has only ever seen Cartesian plots might ask: why not just plot CH₄ on the x-axis against C₂H₄ on the y-axis, and drop C₂H₂ as a separate threshold filter?

The geometric answer is that dropping one gas destroys the constraint structure. Consider two transformers reading (CH₄=60, C₂H₄=20, C₂H₂=20) and (CH₄=60, C₂H₄=20, C₂H₂=0). They project to the same (CH₄, C₂H₄) location in a 2-gas Cartesian plot, but their normalisations differ wildly: (60, 20, 20)/100 versus (60, 20, 0)/80 = (75, 25, 0). One has 20% acetylene — in the heart of high-energy discharge territory. The other has no acetylene at all — pure thermal. The constraint plane in 3-space carries this distinction; a 2-axis subspace flattens it.

The diagnostic answer is that the fault zones in Triangle 1 are bounded by lines parallel to the triangle sides — lines along which one gas's percentage is constant (IEC 60599:2022, Figure B.3, "Limits of zones"). On the ternary plot those lines are straight, the zones are convex polygons, and the classification reduces to six inequalities on three percentages. A 2-axis Cartesian projection of the same boundaries would be curved (the dropped gas's percentage is no longer constant along a parallel line), and zone classification would require solving an inequality on a quadratic. Triangle 1 is implementable in a few lines of code because the ternary geometry is what makes the zones convex and the boundaries linear.

Why an equilateral triangle, not a right triangle#

Three pure-gas extremes — (100, 0, 0), (0, 100, 0) and (0, 0, 100) — bound that plane. They are pairwise equidistant in 3-space, so projecting the plane face-on yields an equilateral triangle with the three vertices at 60° interior angles. A skewed or right triangle would work mathematically — the constraint plane projects to any non-degenerate triangle — but introduces an arbitrary preferred direction (one gas gets a perpendicular axis, the other two share an oblique). The equilateral choice in IEC 60599 Figure B.3 is the only orientation that treats CH₄, C₂H₄ and C₂H₂ as equals. The 60° geometry is not a stylistic flourish; it is a consequence of the symmetric constraint.

Barycentric-to-Cartesian transform#

IEC 60599 Figure B.3 draws Triangle 1 with %CH₄ at the top vertex (apex), %C₂H₂ at the bottom-left, and %C₂H₄ at the bottom-right. The production tool at /en/tools/duval-triangle follows this convention, and so does this article and its accompanying animation.

For a unit equilateral triangle whose apex sits above the baseline, a point with three percentages maps to Cartesian coordinates by

The production tool implements this with the affine scaling $x_{\text{SVG}} = 50 + 5 \cdot \%\text{C}_2\text{H}_4 + 2.5 \cdot \%\text{CH}_4$ and $y_{\text{math}} = 50 + (5\sqrt{3}/2) \cdot \%\text{CH}_4$ in CoordinateTransformer.percentageToScreen, followed by a Y-flip into the SVG viewBox (600 × 533). Three sanity checks: pure CH₄ → SVG (300, 50) apex; pure C₂H₄ → SVG (550, 480) bottom-right; pure C₂H₂ → SVG (50, 480) bottom-left. The standard fixes the percentage convention and the zone diagram graphically; the affine scale and Y-flip are the production pixel convention, not standardised. An implementation choosing a different vertex placement (for example, %C₂H₂ at the apex) would derive different Cartesian coordinates but produce identical fault classifications, because Triangle 1's zones are defined by percentage inequalities, not Cartesian thresholds.

Worked example#

Take the case in IEEE C57.104:2019 Annex D.4 (p. 63, Figure D.2) — CH₄ = 25 ppm, C₂H₄ = 15 ppm, C₂H₂ = 10 ppm — and scale the absolute concentrations 3× to lift the case comfortably above the 10 ppm plotting threshold (IEC 60599:2022, §6.1): CH₄ = 75 ppm, C₂H₄ = 45 ppm, C₂H₂ = 30 ppm. The ratio is identical to the IEEE case, and Triangle 1 zone classification depends only on the ratio, so the normalised point and the zone are unchanged.

1. Sum: 75 + 45 + 30 = 150 ppm.
2. Normalise: %CH₄ = 50, %C₂H₄ = 30, %C₂H₂ = 20. (Check: 50 + 30 + 20 = 100. ✓)
3. Project onto the canonical equilateral triangle. The production tool at /en/tools/duval-triangle places %CH₄ at the apex and uses the affine transform $x = 5 \cdot \\%\text{C}_2\text{H}_4 + 2.5 \cdot \\%\text{CH}_4 + 50$ and $y = (5\sqrt{3}/2) \cdot \\%\text{CH}_4 + 50$, then flips into SVG space. The worked example lands at SVG coordinate $(325, 265)$. The Cartesian numbers depend on the vertex convention; the classification does not.
4. Classify: %C₂H₂ = 20 sits between the D1/D2 boundary at 13 and the D2/DT boundary at 29; %C₂H₄ = 30 sits between the D1/D2 boundary at 23 and the D2/DT boundary at 40. The point lands in zone D2 — high-energy arcing

(IEEE C57.104:2019 Annex D.4, p. 63, verbatim: "in zone D2"). Duval & Buchacz published companion papers in 2022 introducing D1-P/D2-P (paper) and D1-H/D2-H (oil) sub-zones — Part I on Pentagons 1 and 2 (EIM 38(1)), Part II on Triangle 1 (EIM 38(6), Figure 1) — under which this worked example resolves to the D2-H sub-zone.

3. Pentagon 1 — five axes, polygon centroid#

Five gases, the increasing-energy ordering#

Pentagon 1 consumes all five hydrocarbon gases — H₂, CH₄, C₂H₆, C₂H₄ and C₂H₂. As with Triangle 1, normalisation divides each gas by the sum of the five so the relative percentages add to 100. Each gas then occupies one of the pentagon's five vertices.

The vertex ordering is not arbitrary. Duval and Lamarre arranged the five gases counterclockwise from H₂ at the top in order of increasing energy of formation (D&L 2014, p. 10). H₂ is produced by the lowest-energy bond cleavages — corona discharge, stray gassing. C₂H₂ is produced only at very high temperatures or in the core of an electric arc. Between them sit the C–H scission products in energy order: C₂H₆, CH₄, C₂H₄. A point's angular position around the pentagon therefore encodes the energy spectrum of whatever process produced the gas, which is what makes a five-axis geometric layout diagnostic.

Paulhiac & Duval 2023 §V writes the same angles in signed form (90°, 18°, −54°, −126°, 162°). The two notations differ only by 360°; they describe the same pentagon. The 100% vertex Cartesian coordinates follow by setting radial distance to 100 (D&L 2014, p. 10): H₂ at (0, 100), C₂H₆ at (−95.1, 30.9), CH₄ at (−58.8, −80.9), C₂H₄ at (58.8, −80.9), C₂H₂ at (95.1, 30.9).

Per-gas points — five candidate points, not yet the centroid#

For each gas the plotted point sits on the radial line from the pentagon centre (0, 0) toward that gas's 100% vertex, at radial distance equal to its percentage:

Five gases, five points. This is the part of the construction that matches the intuition many engineers reach for first — project each gas as a vector and read where the tips land. It is correct as far as it goes — those five points exist and matter. But none of them is the diagnostic point. They are five candidate vertices; the diagnostic point is the area-weighted centre of the polygon they form.

The area-weighted polygon centroid — what Duval actually published#

Here is the subtle move, and the one most easily mis-implemented. The single DGA result is not the simple arithmetic mean of the five (x_i, y_i) points. It is the area-weighted centroid of the irregular pentagonal polygon they define, computed via the Bourke (Shoelace) formula. Duval and Lamarre are explicit about the choice (D&L 2014, p. 10):

"It may be noted that the center of an irregular polygon may also be calculated mathematically as its 'center of mass,' by computing the average of its (xi) and (yi) coordinates, but it was preferred here to calculate it as its 'centroid.'"

The arithmetic-mean reading is therefore explicitly rejected. The reason matters: for a regular convex polygon with all vertices equidistant from the centre, the Shoelace centroid and the arithmetic mean coincide. For an irregular polygon — which is exactly what five unequal gas percentages produce — the area-weighted centroid pulls toward the regions where two or more high-percentage vertices cluster. The zone boundaries published in 2014 and corrected in 2023 are defined in the area-weighted centroid's coordinate space; substituting the arithmetic mean would land cases in different zones, as the next paragraph shows. An implementation that averages the five vertex coordinates instead will land at a different point — sometimes, as the worked example below shows, in a different zone entirely.

To see how much the choice can move the diagnostic point, take the worked example below. The simple arithmetic mean of the five (x_i, y_i) per-gas points lands at (5.43, −13.94). The area-weighted Bourke centroid, on the same gas readings, lands at (8.14, −21.99) — about eight Cartesian units further out, in a different direction. Both happen to fall inside zone T3 for this gas vector, so the zone label coincides. But the eight-unit offset is large compared with Pentagon 1 zone widths (the T3/D2 boundary is two units away from the Bourke centroid here), the P&D 2023 zone polygons were calibrated against the area-weighted centroid, and for gas vectors near a zone boundary the same offset will straddle the boundary and the diagnosis flips. Substituting the arithmetic mean is an implementation error wherever it appears, even when it produces a coincidentally correct zone label.

One implementation note before the formula: the five vertices must be traversed counterclockwise in the published gas order — H₂ → C₂H₆ → CH₄ → C₂H₄ → C₂H₂ — or the signed area flips and the centroid flips with it. This is the single most common mis-implementation.

The Bourke (Shoelace) formula with the polygon's signed area $A$, and the centroid $(C_x, C_y)$, is:

with $n = 5$ and indices taken modulo 5 so the polygon closes.

Worked example#

A mineral-oil DGA dominated by ethylene with secondary methane and acetylene: CH₄ = 30 ppm, C₂H₄ = 60 ppm, C₂H₂ = 10 ppm, H₂ = 0, C₂H₆ = 0. Σ_G = 100 ppm so each percentage equals its ppm value.

1. Per-gas Cartesian points (radius = %gas):

   • H₂ at 90°: (0, 0) — at origin because %H₂ = 0
   • C₂H₆ at 162°: (0, 0) — at origin because %C₂H₆ = 0
   • CH₄ at 234°: (−17.63, −24.27)
   • C₂H₄ at 306°: (35.27, −48.54)
   • C₂H₂ at 18°: (9.51, 3.09)

2. Apply Bourke. Traversing H₂ → C₂H₆ → CH₄ → C₂H₄ → C₂H₂ → H₂, three of the five cross-terms $c_i = x_i y_{i+1} - x_{i+1} y_i$ vanish because their vertices sit at the origin. The two non-trivial terms are:

   • $c_3$ (CH₄ → C₂H₄): $(-17.63)\cdot(-48.54) - 35.27\cdot(-24.27) \approx 1711.90$
   • $c_4$ (C₂H₄ → C₂H₂): $35.27\cdot 3.09 - 9.51\cdot(-48.54) \approx 570.63$

   The cross-terms sum to $\Sigma c_i = 2282.54$, so $A = 1141.27$. The $C_x$ and $C_y$ sums then give $(C_x, C_y) = (8.14, -21.99)$.

3. The simple arithmetic mean of the five points is $(\bar{x}, \bar{y}) = (5.43, -13.94)$ — measurably different from the Bourke centroid (about eight units further out, in a different direction).

4. Classify against the 2023 unified Pentagon 1 zones (Paulhiac & Duval 2023 Table 5): the Bourke centroid (8.14, −21.99) sits inside the T3 polygon — thermal fault > 700 °C. The arithmetic-mean point (5.43, −13.94) also falls inside T3 for this case, so the zone label coincides; but the two centroids differ by about eight Cartesian units, the Bourke value is what P&D 2023 calibrated the zones against, and the same eight-unit offset straddles a zone boundary for gas vectors that sit close to one — at which point the diagnosis flips.

An independent end-to-end re-derivation — cross-checked against the Grok 4.3 reasoning model — confirms $(C_x, C_y) = (8.14, -21.99)$ to within IEEE 754 rounding. tribotech.dk's Duval Pentagon tool accepts the same input and reproduces the result.

Maximum centroid radius — 100/3, not 40#

Even if one gas is 100% and the other four are 0%, the centroid does not reach the 100% vertex. Paulhiac & Duval 2023 derive the limit by setting H₂ = 100, the other four gases to ε, and taking ε → 0 in the Bourke formula (P&D 2023, §V). The result is exactly 100 / 3 ≈ 33.33%. The 2014 paper's stated practical limit of 40% was a small over-approximation, corrected in 2023.

The geometric intuition is that the centroid of a polygon with one vertex at (0, 100) and the other four collapsed to the origin is one-third of the way along the median from the origin to the (0, 100) vertex. Every DGA centroid in Pentagon 1 therefore lives inside a smaller pentagon of radius 33.33%; the fault zones occupy this inner region, and the outer ring between 33.33% and 100% radius is geometrically inaccessible to any real DGA result.

These two constructions therefore publish their zone edges in incompatible coordinate languages — one vertex-driven, one inequality-driven — which is the deliberate asymmetry §4 examines.

4. Zone vertex coordinates — the deliberate asymmetry#

Triangle 1 zones and Pentagon 1 zones are published in fundamentally different forms, and the asymmetry is itself a useful pedagogical point.

Triangle 1 — percentage thresholds, no Cartesian table#

IEC 60599:2022 Annex B Figure B.3 (p. 37) publishes the seven Triangle 1 zones as a "Limits of zones" tabulation in the form of percentage thresholds on the three gases:

(IEC 60599:2022 Figure B.3 (p. 37) labels the same mixed zone "D+T"; CIGRE TB771:2019 App Table H.1 (p. 69) labels it "DT" with vertex coordinates %C₂H₄ at 40, 50 and %C₂H₂ at 4, 13, 15, 29. The zone is the same; only the label differs by a "+".)

The standard does not publish (X, Y) Cartesian vertex coordinates for the zones. It does not need to. Every boundary in Figure B.3 is a straight line parallel to one of the triangle's three sides — that is, a locus of constant percentage on a single gas — so the classification is six inequalities on three percentages and the implementation reads if (%CH₄ ≥ 98) return "PD"; else if (%C₂H₂ ≤ 4 && %C₂H₄ ≤ 20) return "T1"; .... There is no need to test point-in-polygon against a Cartesian vertex list.

Pentagon 1 — Cartesian vertex coordinates, no percentage form#

Pentagon 1 zones do require an explicit (X, Y) vertex table. The Bourke centroid is a single (X, Y) coordinate produced from five gas percentages, and the relationship between any single gas's percentage and the centroid's location is non-linear because the centroid depends on the polygon's shape, not on any one vertex's distance from the centre. So there is no equivalent "%H₂ ≤ N" form that could replace the Cartesian polygons.

The corrected Pentagon 1 zone vertices, from P&D 2023 Table 5 (p. 4):

Classification is a point-in-polygon test against this list — ray-casting or winding-number. Small transcription errors in the original 2014 vertex tables propagated into TB771 and IEEE C57.104 before being corrected; implementations should use the 2023 coordinates.

Why the asymmetry matters#

The form in which a zone boundary is published shapes how robust the diagnosis is to convention. Triangle 1's zones, defined intrinsically by percentage inequalities, are immune to vertex-placement choices: an implementation that puts %CH₄ at the top instead of %C₂H₂ at the top will derive different Cartesian coordinates for the same point, but the classification is unchanged because it tests percentages, not Cartesian coordinates. Pentagon 1's zones depend on the canonical vertex ordering and the canonical centroid formula. Change the angle assignments or substitute the arithmetic mean for the Bourke centroid, and the same gas vector classifies into a different zone.

This is the practical reason the Pentagon 1 implementation in the production tool is more constrained than the Triangle 1 implementation. Triangle 1 can be reverse-engineered from a screenshot. Pentagon 1 cannot.

5. Why this matters in practice#

Once you can derive the geometry, three things change in how you read a diagnostic tool's output.

First, near-boundary cases stop being mysterious. A Triangle 1 point that sits 1.5% inside the D2/DT boundary is one rate-of-change cycle away from changing classification. A Pentagon 1 centroid that sits a unit of Cartesian distance from the D2/T3 line is similarly fragile. The geometric distance to the nearest boundary is itself diagnostic information that the zone label alone discards. Tools that surface that distance (the production Duval Triangle and Duval Pentagon, ours included) give the engineer something to act on when the point sits at a boundary.

Second, the Triangle/Pentagon disagreement signal makes sense. The two diagrams use different geometric projections of the same five-gas reading — Triangle 1 throws away H₂ and C₂H₆, Pentagon 1 weights all five through the polygon centroid. They are not redundant; they are two views of a five-dimensional fault space. When they agree, the diagnosis is robust against single-gas measurement error. When they disagree, the disagreement is a real diagnostic signal — a multi-fault case or a stray-gassing artefact — and the geometry of the disagreement (which way each method pulled, how far) carries information beyond a zone label.

Third, the algorithm files in a competitor's implementation become inspectable. If an implementation classifies the D&L 2014 reference case as anything other than T1 at (−17.3, −9.1), it is either using the wrong centroid formula or the wrong vertex angles. If it classifies the IEEE Annex D.4 case as anything other than D2, it has its Triangle 1 zone boundaries wrong. The construction also provides cheap end-to-end self-consistency tests against the source standards. Take CIGRE TB771:2019 Table 3.5 (p. 20), "Example 1 of a Fault O" (6 kV / 25 MVA transformer, overheated iron sheets of the lower frame): the published gas vector reads H₂ = 29, CH₄ = 204, C₂H₂ = 0, C₂H₄ = 17, C₂H₆ = 264 ppm — column order in Table 3.5 is C₂H₂, C₂H₄, C₂H₆ left-to-right, and the case is ethane-dominant, not ethylene-dominant. The three Triangle 1 gases (CH₄, C₂H₄, C₂H₂) sum to 221 ppm and normalise to %CH₄ = 92.3 %, %C₂H₄ = 7.7 %, %C₂H₂ = 0 % — well inside zone T1 per TB771 App Table H.1 (p. 69). The published Tr1 = T1 is correct. Tr4 = O and Pent2 = O are likewise consistent: C₂H₆ at 264 ppm against C₂H₄ at 17 ppm is exactly the ethane-dominant signature TB771 Table 3.4 (p. 19) gives for Fault O (overheating < 250 °C, ethane prefailure 4 460 ppm). Reading the gas-column order from the source table is the first end-to-end test; reproducing the diagnosis from the published gas vector is the only way to be sure a tool implements the same zone boundaries as the standard.

The fault-classification work in the companion T1/P1 article rests on this geometry. Engineers who internalise the construction stop reading the diagrams as conventions and start reading them as projections of a real measurement.

Bibliography#

Full expansion of abbreviated citations used above.

IEC 60599:2022. International Electrotechnical Commission. Mineral oil-filled electrical equipment in service — Guidance on the interpretation of dissolved and free gases analysis. IEC, 2022. Annex B (Figures B.1–B.4, pp. 35–38); Figure B.3 (Triangle 1 with normalisation formulae and "Limits of zones" tabulation, p. 37).

Duval & Lamarre 2014 ("D&L 2014"). Duval, M., and Lamarre, L. "The Duval Pentagon — A New Complementary Tool for the Interpretation of Dissolved Gas Analysis in Transformers." IEEE Electrical Insulation Magazine, vol. 30, no. 6, pp. 9–12, November/December 2014. DOI: 10.1109/MEI.2014.6943428. Pentagon vertex coordinates (p. 10), area-weighted centroid choice (p. 10, verbatim quote in §3 above), worked example landing at centroid (−17.3, −9.1) (Figure 1, p. 9).

Paulhiac & Duval 2023 ("P&D 2023"). Paulhiac, L., and Duval, M. "Unified Pentagon for DGA in Mineral Insulating Oil." CIGRE International Conference on Transformers (ICTRAM), Paper ICTRAM06, 2023. §V (per-gas formulae, ε → 0 derivation of the 100/3 ≈ 33.33% maximum centroid radius); Table 5 (corrected zone vertex coordinates for Pentagon 1, p. 4).

IEEE C57.104:2019. Institute of Electrical and Electronics Engineers. IEEE Guide for the Interpretation of Gases Generated in Mineral Oil-Immersed Transformers. IEEE Std C57.104-2019. Annex D.1 (Key Gas error-rate figures, p. 61), Annex D.4 (Triangle 1 worked example, p. 63, Figure D.2 — CH₄=25, C₂H₄=15, C₂H₂=10 → zone D2).

Duval & Buchacz 2022 Part I. Duval, M., and Buchacz, J. "Identification of Arcing Faults in Paper and Oil in Transformers — Part I: Using the Duval Pentagons." IEEE Electrical Insulation Magazine, vol. 38, no. 1, pp. 19–21, January/February 2022. Introduces D1-P/D2-P (paper) and D1-H/D2-H (oil) arcing sub-zones on Pentagons 1 and 2 with explicit (X, Y) summit coordinates.

Duval & Buchacz 2022 Part II. Duval, M., and Buchacz, J. "Gas Formation from Arcing Faults in Transformers — Part II." IEEE Electrical Insulation Magazine, vol. 38, no. 6, pp. 12–14, November/December 2022. Figure 1 extends the D1-H/D1-P/D2-H/D2-P sub-zone scheme to Triangle 1.

CIGRE TB771:2019. CIGRE Joint Working Group D1/A2.47 (M. Duval, convenor). Technical Brochure 771: Advances in DGA interpretation. CIGRE, Paris, June 2019. App Table H.1 (Triangle 1 zone boundary numerics including the DT zone, p. 69).

Bourke. Bourke, P. "Calculating the Area and Centroid of a Polygon." paulbourke.net/geometry/polyarea. Geometric reference for the Shoelace centroid formula, cited by D&L 2014 [reference 5] and P&D 2023 §V Steps 2–3.