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One free scan finds every outdated or missing driver and matches the right update for your exact hardware.Free scan · exact hardware matchUse a lattice-coordinate transform, not a visual redraw. A regular hexagonal PCB pattern is a triangular center lattice: two equal-length basis vectors meet at 60 degrees. Choose the lattice origin, the nearest-neighbor center pitch d, and the orientation; represent every location with integer coordinates; then transform those coordinates to the PCB editor’s Cartesian x/y plane. The editor’s rectangular grid supplies coordinates and snapping, but it does not turn the hexagonal point pattern into an orthogonal grid.
This guide gives forward and inverse equations for pointy-top and flat-top conventions, handles staggered row/column indexes, shows how to round measured points safely, and outlines a repeatable PCB-editor workflow.
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What “hexagonal grid” means on a PCB
In layout work, “hexagonal grid” usually means the centers of pads, vias, holes, or other repeated features lie on a regular triangular lattice. Each center has six nearest neighbors. The six directions are separated by 60 degrees, even though the PCB editor normally displays positions as Cartesian coordinates.
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A conversion therefore needs four explicit choices:
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- Origin: Cartesian coordinates (x0, y0) assigned to lattice index (0, 0).
- Pitch: d, the center-to-center distance between neighboring lattice points.
- Orientation: the selected pair of basis vectors (called pointy-top or flat-top below).
- Indexing: axial integers (q, r), or a defined staggered row/column convention that can be converted to axial coordinates.
Keep the pitch definition straight. If a drawing gives the hexagon’s side length s rather than center pitch, use d = √3 s for a regular hexagon. Substituting s directly into the lattice equations makes every neighbor too close.
Choose the basis and keep units consistent
The equations below are linear combinations of two basis vectors. Millimeters, inches, and mils all work, but d, x0, y0, and the resulting coordinates must use the same unit. Do not mix a metric pitch with an origin entered in mils.
Pointy-top convention
Use the basis vectors
a = (d, 0) and b = (d/2, √3 d/2).
Axial coordinate (q, r) means q steps along a and r steps along b. The third neighbor direction is b − a; all three have length d. This basis is the definition used here for the pointy-top label, so do not substitute formulas from a library that uses a different axis orientation without checking its convention.
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Flat-top convention
Use the rotated basis vectors
a = (√3 d/2, d/2) and b = (0, d).
The same integer pair now produces a lattice rotated relative to the pointy-top basis. A pattern can be geometrically identical after rotation, but its numeric x/y values are different.
Forward conversion: axial hex coordinates to Cartesian x/y
Pointy-top equations
| Quantity | Equation | Meaning |
|---|---|---|
| Cartesian x | x = x0 + d(q + r/2) | Origin plus the horizontal components of both basis steps |
| Cartesian y | y = y0 + (√3 d/2)r | Only the second basis contributes vertical displacement |
For example, with d = 2 mm, origin (10 mm, 5 mm), and (q, r) = (2, 3):
- x = 10 + 2(2 + 3/2) = 17 mm.
- y = 5 + (√3 × 2/2) × 3 ≈ 10.196 mm.
The six nearest centers are obtained by adding the axial steps (1, 0), (0, 1), (−1, 1), (−1, 0), (0, −1), and (1, −1); each is exactly 2 mm away before any editor snapping or rounding.
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Flat-top equations
| Quantity | Equation | Meaning |
|---|---|---|
| Cartesian x | x = x0 + (√3 d/2)q | Horizontal displacement comes from the first basis |
| Cartesian y | y = y0 + (d/2)q + dr | Both basis steps contribute vertically |
With the same origin, pitch, and index (2, 3), the flat-top result is x ≈ 13.464 mm and y = 13 mm. That difference is expected: the orientation changed, not the lattice spacing.
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Many drawings label locations as row and column rather than axial (q, r). A staggered scheme shifts every other row or column by half a pitch. There is no universal meaning for “odd row” or “even column”; the parity rule, whether the first row is numbered zero or one, and the axis direction must be recorded with the drawing.
Convert the offset indexes to axial integers first, then apply the pointy-top or flat-top equations. The following are common zero-based mappings; they are examples of conventions, not a license to guess the convention used by a board file.
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| Offset convention | Axial conversion |
|---|---|
| Odd-r (row is the offset axis) | q = column − (row − (row mod 2))/2; r = row |
| Even-r | q = column − (row + (row mod 2))/2; r = row |
| Odd-q (column is the offset axis) | q = column; r = row − (column − (column mod 2))/2 |
| Even-q | q = column; r = row − (column + (column mod 2))/2 |
Before converting a complete array, verify two adjacent rows or columns against the source drawing. A one-row parity error leaves most points apparently regular while shifting every other band onto the wrong lattice.
Inverse conversion: Cartesian x/y back to hex coordinates
Pointy-top inverse
Given measured or editor coordinates, first subtract the same origin:
r = 2(y − y0)/(√3 d)
q = (x − x0)/d − r/2
Flat-top inverse
For the rotated basis:
q = 2(x − x0)/(√3 d)
r = (y − y0)/d − q/2
An exact lattice point should produce integers, apart from floating-point noise. If a measured point is intended to be the nearest center, do not round x and y independently. Independent Cartesian rounding can create a point that belongs to no valid hex location.
Nearest-center rounding
- Apply the appropriate inverse equations to obtain real-valued q and r.
- Convert to cube coordinates: cx = q, cz = r, and cy = −cx − cz.
- Round all three cube values to the nearest integers.
- Compute the three rounding errors. Replace the coordinate with the largest error so that cx + cy + cz = 0.
- Use the resulting integer (q, r) in the forward equation and place that regenerated Cartesian point.
This round-trip makes the selected center explicit and prevents accumulated drift when many points are measured from an image or a board.
Independent reader supportYour contribution helps us test, update, and keep practical guides available for everyone.A repeatable PCB-editor workflow
- Document the source convention. Write down pointy-top or flat-top, axial or offset indexing, zero-based or one-based labels, units, nearest-neighbor pitch, and the physical location represented by (q, r) = (0, 0).
- Fix the origin. Choose a stable pad, hole, board datum, or drawing intersection for (x0, y0). Use the same datum for every conversion and for later inspection.
- Generate Cartesian coordinates. Convert every integer index with full numeric precision. Keep extra digits internally; round only when the editor’s coordinate resolution requires it.
- Set the editor’s coordinate and grid units. Use millimeters, inches, or mils consistently. If snapping is enabled, choose a grid fine enough that snapping cannot move a generated point by a meaningful fraction of the pitch.
- Enter or import the positions. Place pads or footprints at the generated x/y values, or use the editor’s scripting/import facility if the array is large. Do not rely on dragging until the pattern has been validated numerically.
- Round-trip a sample. Pick the origin, several interior points, and edge points. Apply the inverse transform and confirm that the recovered indexes are the intended integers within a defined tolerance.
- Check geometry and manufacturing rules. Measure several neighbor distances, inspect the alternating stagger direction, and run the PCB editor’s design-rule checks. A mathematically correct lattice can still violate clearance, annular-ring, drill, or board-edge rules.
Using KiCad as a concrete example
The KiCad 10 PCB Editor manual documents a movable grid origin, Cartesian coordinate display, coordinate entry in inches, mils, and millimeters, and grid snapping. Its footprint-editor guidance explains setting the grid origin at the first pad and setting grid size to the pad pitch in the footprint editor documentation. The reviewed KiCad GRID API describes separate x and y grid dimensions.
Those facilities support entering and repeating generated Cartesian coordinates. They do not establish a native hexagonal snap grid. Treat the hex positions as calculated coordinates unless the exact KiCad release you are using explicitly documents a hex-snap feature. Menu names and scripting interfaces can change between releases, so confirm the labels in the installed version before automating a production board.
Automating the conversion
A small script eliminates transcription errors and makes the convention auditable. The following Python functions implement the equations above; they return coordinates in the same unit supplied for d and the origin.
from math import sqrt
def pointy_xy(q, r, d, x0=0.0, y0=0.0):
return (x0 + d * (q + r / 2.0),
y0 + sqrt(3.0) * d * r / 2.0)
def flat_xy(q, r, d, x0=0.0, y0=0.0):
return (x0 + sqrt(3.0) * d * q / 2.0,
y0 + d * (q / 2.0 + r))
def pointy_qr(x, y, d, x0=0.0, y0=0.0):
r = 2.0 * (y - y0) / (sqrt(3.0) * d)
q = (x - x0) / d - r / 2.0
return q, r
def flat_qr(x, y, d, x0=0.0, y0=0.0):
q = 2.0 * (x - x0) / (sqrt(3.0) * d)
r = (y - y0) / d - q / 2.0
return q, rFor production use, add the cube-coordinate rounding step rather than calling round(x) and round(y). Export the resulting list with an explicit header such as index_q,index_r,x,y,unit, and retain the source convention alongside the file so another designer can reproduce it.
Precision, tolerances, and snapping
The irrational factor √3 means many valid coordinates cannot be represented as short decimal fractions. Retain sufficient precision through generation and inverse checks. A practical tolerance should be smaller than the placement error your fabrication process and editor resolution can detect, not an arbitrary number copied between projects.
Quick Recap
- Do not round early: early rounding changes later inverse results and can create a visible drift across a large array.
- Snap deliberately: snapping to a coarse rectangular grid can move every calculated point. Temporarily disable snapping, use a finer grid, or enter exact coordinates when the editor permits it.
- Check pitch statistically: measure several pairs of known neighbors in different lattice directions; all should be approximately d, within the declared tolerance.
- Keep the origin reproducible: changing the origin by even a small amount shifts the entire array, so store its physical reference and numeric coordinates with the design.
Common failure modes and fixes
| Symptom | Likely cause | Fix |
|---|---|---|
| Every other row is displaced in the wrong direction | Odd/even stagger convention or row numbering was assumed | Verify parity on two adjacent rows, convert to axial coordinates, then regenerate the array. |
| Neighbor distance is √3 times too small or too large | Hex side length was used as center pitch, or vice versa | For a regular hexagon, convert side length with d = √3 s. |
| Pattern is rotated or mirrored | Pointy/flat basis or axis direction does not match the source | Identify two known neighbor vectors and choose the basis whose signs and angle match them. |
| Inverse coordinates are nearly, but not exactly, integers | Floating-point representation, measurement noise, or a wrong origin | Use a stated tolerance; if errors grow with distance, recheck pitch and origin before rounding. |
| Points look correct in the script but move in the PCB editor | Rectangular-grid snapping or unit conversion changed the entered values | Match units, use a sufficiently fine grid, and compare the editor’s reported coordinates with the export file. |
| Only edge features fail design-rule checks | The lattice transform is correct but clearances or board boundaries are not | Run design-rule checks after placement and adjust the board outline, feature size, or pitch as a design decision. |
Which method should you use?
| Decision axis | Best choice when | Trade-off |
|---|---|---|
| Pointy-top basis | The source’s primary neighbor direction is horizontal and the accompanying direction rises at 60 degrees | Uses the pointy equations; incompatible axis labels must be remapped. |
| Flat-top basis | The source is rotated so one basis is vertical and the other slopes at 30 degrees | Uses different inverse equations; copied pointy formulas give wrong coordinates. |
| Axial indexes | You control the data format or are generating a new array | Requires two integer coordinates but avoids parity ambiguity. |
| Staggered row/column indexes | You must reproduce a drawing, table, or legacy layout | Readable to humans, but parity and zero/one-based numbering must be recorded. |
| Manual coordinate entry | The pattern is small and the editor has reliable numeric placement | Slow and vulnerable to transcription mistakes. |
| Generated coordinates | The pattern is large, repeated, or must be revised | Requires a checked script, explicit units, and an import or placement procedure. |
Validation checklist before fabrication
- Pitch d is center-to-center and uses the same unit as every coordinate.
- Origin and orientation are written in the board’s documentation.
- Offset indexes, if any, have a named odd/even convention and a stated starting index.
- At least one forward/inverse round trip returns the original integer pair.
- Neighbor distances and stagger direction match the source drawing.
- Cartesian snapping has not altered generated positions.
- Design-rule checks pass for clearances, drills, copper, and board edges.
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