A Parabola Scriber is a draughting instrument that traces a true parabolic curve onto paper, plywood, or sheet stock by mechanically enforcing the focus-directrix property — every point on the curve sits an equal distance from a fixed focus point and a fixed straight directrix line. It solves the problem of drawing parabolas without point-by-point plotting from a table of coordinates. A pencil rides on a sliding T-square while a taut string anchored at the focus keeps the geometry exact. Workshops use it for solar concentrator templates, antenna dish layouts, and headlamp reflector tooling where a 0.5 mm profile error shifts the focus measurably.
Parabola Scriber Interactive Calculator
Vary focal length, aperture, and pencil station to see the scribed parabola depth and focus-directrix equality update live.
Equation Used
The calculator uses the parabola equation from the focus-directrix geometry. For a focal length p, the pencil depth at station x is y = x^2/(4p). The mechanism is correct when the focus-to-pencil distance d1 equals the pencil-to-directrix distance d2.
- Vertex is at x=0, y=0.
- Focus is at x=0, y=p and directrix is y=-p.
- Aperture is the full width of the scribed profile.
- Station x is measured from the parabola centerline.
Inside the Parabola Scriber
The mechanism rests on one piece of conic-section geometry: a parabola is the locus of points equidistant from a fixed point (the focus) and a fixed line (the directrix). The classic string-and-square parabola method physically enforces that rule. You clamp a T-square so its blade slides along the directrix line. You fix one end of a non-stretch string to the focus pin and the other end to the top corner of the T-square's stock. The string length equals the distance from that corner down to the blade edge. A pencil traps the string against the blade and slides along it. As the T-square slides left or right along the directrix, the pencil traces a curve where the focus-to-pencil distance always equals the blade-to-pencil distance — that is the parabola.
Why build the tool this way instead of plotting y = x² / 4p from a coordinate table? Two reasons. First, a plotted curve is only as smooth as your point density and your French curve fairing — for a 1200 mm solar dish template, the eye picks up the kinks. Second, the focus-directrix scriber gives you the focal point as a physical pin location on the drawing, which is exactly the dimension a fabricator cares about. You are dimensioning the curve from its functional reference, not from an arbitrary vertex coordinate.
Tolerances matter. The string must be braided polyester or aramid, not cotton — cotton stretches under pencil pressure and the curve grows by 1-2 mm at the wings of a 1 m draw. The T-square blade must run dead straight against the directrix; a 0.3 mm bow over 600 mm rotates the local tangent enough to throw the focal length off by a couple of millimetres on a shallow dish. If your scribed curve fails to close symmetrically when you flip the T-square to the other side of the focus, your string length is wrong or your focus pin has walked under tension.
Key Components
- Focus pin: A hardened steel pin, typically 1.5-2.0 mm diameter, driven into the board at the parabola's focus. The string anchors here under tension. Pin walk of even 0.2 mm shifts the entire curve by the same amount, so we recommend a pre-drilled pilot hole and a brass collar in plywood layout boards.
- Directrix-aligned T-square: A standard draughting T-square whose head slides along a straightedge clamped to the directrix line. The blade extends perpendicular to the directrix. Blade straightness must be within 0.1 mm over 500 mm — anything sloppier shows up as a visible wobble in the scribed curve.
- Inextensible string: Braided polyester or Kevlar fishing line, 0.3-0.5 mm diameter, with stretch under 0.1% at typical pencil pressures. Length is set equal to the distance from the T-square's top anchor down to the blade edge at the directrix. Cotton string is unusable — it elongates 1-2% under load and ruins the geometry.
- Pencil or scriber tip: A 0.5 mm clutch pencil or hardened scriber tip that traps the string against the T-square blade. The tip must sit exactly on the blade edge — offset of 1 mm shifts the curve inward by the same amount across the full draw.
- Directrix straightedge: A clamped aluminium or steel rule defining the directrix. The T-square head rides along this. It must be parallel to the eventual axis of symmetry of the parabola within 0.1°, otherwise the curve tilts and the focal length is no longer where the focus pin sits.
Real-World Applications of the Parabola Scriber
Parabola Scribers earn their keep wherever a full-scale parabolic profile must transfer onto physical stock with the focal point pinned as a real reference. Coordinate-table plotting works for a CAD draughtsman with a plotter, but in a workshop laying out a 1:1 plywood template, the scriber is faster, more accurate at the wings, and gives you the focus location for free. You see them in solar concentrator workshops, antenna fabrication, optical lab tooling, and theatrical lighting reflector restoration. The common thread is that the focal point matters more than the curve shape itself — a parabola whose focus is 5 mm off-axis is functionally a worse reflector than a slightly imperfect curve whose focus is dead-on.
- Renewable energy fabrication: A community solar workshop in Ladakh laying out a 1.4 m parabolic dish concentrator template on 18 mm plywood, with the focal point marked for absorber-tube placement at 350 mm focal length
- Amateur radio and antenna fabrication: A radio club in Ohio scribing the rib profile for an offset-fed 2.4 GHz dish based on a 600 mm aperture, 240 mm focal length design
- Optical instrument restoration: A telescope restoration shop in Oxford redrawing the meridian-section profile of a damaged 250 mm searchlight reflector for a wartime carbon-arc lamp rebuild
- Theatrical lighting: A scenic-lighting workshop laying out a replacement parabolic reflector profile for a vintage Strand Pattern 23 ellipsoidal fixture conversion
- Architectural acoustics: An acoustics consultancy laying out a full-scale plywood form for a parabolic whisper-dish installation in a science museum, where the focal point seating must be precise to the visitor's ear height
- Automotive headlamp tooling: A classic-car restoration shop in Coventry scribing a master profile for a Lucas P700 tripod headlamp reflector replacement
The Formula Behind the Parabola Scriber
The scriber draws the curve mechanically, but you still need to compute the focal length p from the aperture and depth you want, or vice versa. The standard parabola equation in vertex form ties depth, half-aperture, and focal length together. At a shallow focal ratio (f/D around 0.25, which is the deep-dish end of the range you typically see in solar concentrators) the dish wraps around the focus and the focal point sits inside the rim — useful for absorber tubes but awkward for feed horns. At a long focal ratio (f/D around 1.0, common in radio astronomy prime-focus dishes) the dish is shallow and the focus sits far above the rim, which simplifies feed support but increases spillover sensitivity. The sweet spot for general workshop concentrators sits around f/D = 0.4 to 0.5.
Variables
| Symbol | Meaning | Unit (SI) | Unit (Imperial) |
|---|---|---|---|
| y | Depth of the parabola at horizontal distance x from the vertex (sagitta) | m | in |
| x | Horizontal distance from the axis of symmetry (half-aperture at the rim) | m | in |
| p | Focal length — distance from vertex to focus, equals one-quarter of the latus rectum | m | in |
| D | Full aperture diameter (D = 2x at the rim) | m | in |
Worked Example: Parabola Scriber in a 1.4 m solar dish concentrator template
A small renewable-energy cooperative in northern Ladakh is laying out a full-scale plywood template for a 1.4 m aperture parabolic solar dish concentrator. The receiver is a thermal absorber head positioned at the focus, and the design calls for a focal length of 350 mm, giving an f/D of 0.25. They need to know the dish depth at the rim, set the string length on the scriber, and understand what shifts if the design moves to f/D = 0.40 or f/D = 0.50.
Given
- D = 1.4 m
- pnom = 0.350 m
- xrim = 0.700 m
Solution
Step 1 — at the nominal design point, p = 0.350 m (f/D = 0.25), compute the dish depth at the rim:
So the dish is 350 mm deep — the focal point sits exactly at the rim plane. That is a deep dish, and the absorber tube will be inside the rim aperture, well shielded from wind but harder to access for cleaning.
Step 2 — set the string length on the scriber. The string length L equals the distance from the directrix to the focus, which for this parabola is 2p:
Cut the polyester string to 700 mm ± 0.5 mm. Anything looser than ±1 mm and the rim depth comes out wrong by a visible amount.
Step 3 — at the long-focal-length end of the range, p = 0.700 m (f/D = 0.50), the same 1.4 m aperture gives:
That's a shallow dish only 175 mm deep with the focus floating 525 mm above the rim — easier to mount a feed but the absorber needs a tripod and is fully exposed to crosswind heat loss. At a mid-range p = 0.560 m (f/D = 0.40), depth comes out to y = 0.219 m and the focus sits 341 mm above the rim — this is the practical sweet spot most field workshops settle on for cooker-grade concentrators.
Result
The nominal scribed curve has a rim depth of 350 mm with the focal point exactly at the rim plane, and the scriber string is set to 700 mm. Across the operating range of f/D = 0.25 to 0.50, the same aperture goes from a deep 350 mm dish wrapping around its focus, to a shallow 175 mm dish with the focus floating well above the rim — for a Ladakh-style cooker workshop, the f/D = 0.40 mid-range gives the best compromise between absorber accessibility and wind shielding. If your scribed curve at the rim measures 345 mm instead of the predicted 350 mm, the most common causes are: (1) string stretch from using cotton or nylon instead of braided polyester, which elongates 1-2% under pencil pressure and pulls the rim depth shallow; (2) focus pin walking under tension because the pilot hole was oversized — drill 1.5 mm for a 1.6 mm pin and tap it home; (3) the directrix straightedge sitting at a slight angle to the intended axis of symmetry, which tilts the whole curve and shifts the apparent rim depth on one side relative to the other.
When to Use a Parabola Scriber and When Not To
The Parabola Scriber is one of three practical ways to get a parabolic curve onto stock. The choice between them comes down to aperture size, accuracy demand, and whether you need the focus pinned as a physical reference. Here's how the scriber stacks up against coordinate-table plotting and CNC routing from a CAD curve.
| Property | Parabola Scriber (focus-directrix) | Coordinate-table plotting | CNC router from CAD curve |
|---|---|---|---|
| Profile accuracy at 1 m draw | ±0.5 mm with polyester string and 0.1 mm-straight T-square | ±1-2 mm depending on point density and French-curve fairing | ±0.05 mm limited by router and stock flatness |
| Setup time | 10-15 minutes — pin focus, clamp directrix, cut string | 30-60 minutes for plotting and fairing | 1-3 hours for CAD draw, CAM toolpath, machine setup |
| Cost (tooling) | Under $30 — T-square, string, pins | Under $20 — French curves, ruler, calculator | $5,000+ for entry CNC router plus software |
| Focal-point reference | Pinned physically on the drawing — automatic | Must be added by separate dimensioning step | Marked in CAD, transferred via reference hole |
| Practical aperture range | 100 mm to about 2 m before string sag dominates | Any size, but smoothness degrades at large draws | Limited by router bed — typically up to 1.2 × 2.4 m |
| Skill required | Basic draughting — anyone can run it after one demo | Patience and a good eye for fairing curves | CAD/CAM literacy plus machine operation |
Frequently Asked Questions About Parabola Scriber
Asymmetry between halves almost always means the focus pin shifted between the two passes, or the string was re-tied with a slightly different effective length when you swapped sides. The string knot at the T-square anchor changes length by 1-2 mm depending on how you tie it, and that shows up as a visible mismatch at the rim.
Fix it by scribing both halves in one continuous setup — pin the focus once, tie the string once, and slide the T-square through the full range without re-anchoring. If you must re-tie, mark the anchor point with a sharpie before untying so you can return to the same effective length.
The decision is driven by the absorber, not the curve. A short focal length (f/D ≈ 0.25) puts the absorber inside the rim aperture, which shelters it from wind and reduces convective heat loss — good for thermal cookers and steam generators. The cost is a deep dish that's harder to fabricate from flat stock and a focus that's awkward to access for maintenance.
A long focal length (f/D ≈ 0.50) gives a shallow dish that lays up easily on plywood ribs and an exposed focus that's easy to service, but the absorber loses 10-20% more heat to crosswind. For a Ladakh-style outdoor cooker, f/D = 0.40 is the practical compromise most field builders converge on after a season of use.
Around 2 m of total draw is where string sag and stretch start to dominate. A 0.4 mm polyester line under 2 N pencil tension sags about 1.5 mm at the midspan of a 1 m horizontal run — manageable. Stretch it to 2 m and sag grows to roughly 6 mm, which is a visible profile error at the wings.
Above 2 m aperture, switch to a segmented approach: scribe the curve in 1 m sections with the focus relocated mathematically for each section, or move to CNC routing. Some workshops use Kevlar bowstring material above 1.5 m draws because its modulus is roughly 5× polyester and sag/stretch drop accordingly.
Yes, and this is one of the scriber's quiet strengths. Scribe the full symmetric parabola at the design focal length, then mark out the off-axis section as a chord between two x-coordinates on one side of the axis. The focal point you pinned is still the true focus of that off-axis section — that's the geometric truth of an offset-fed dish.
The trick is keeping enough of the parent parabola on your layout board to reach the off-axis chord. For a 600 mm offset dish with feed clearance, you typically need a parent parabola layout 1.2-1.4 m wide, even though the final stock is only 600 mm.
This is the classic directrix-not-parallel-to-axis error. If the directrix straightedge sits at even 0.5° off the intended axis of symmetry, the scribed curve is still a true parabola — geometrically perfect — but its axis is rotated relative to the dish frame you're going to mount it in. The focus is in the geometrically correct place for the curve, but in the wrong place for your hardware.
Diagnose it by measuring the rim height at both ends — if one rim is higher than the other by more than 1 mm on a 1 m aperture, your directrix is tilted. Re-clamp it parallel to the dish's intended optical axis using two reference points 1 m apart, not just one.
Counterintuitive but real. A CNC router holds the curve coordinates to ±0.05 mm, but the focal point is a derived dimension — it's calculated from the curve shape after the fact, and any cumulative error in the toolpath, stock flatness, or zero-reference walks the focus by 2-5 mm in a 1 m dish.
The scriber, by contrast, treats the focus as the primary reference and derives the curve from it. So the focal point sits exactly where the pin sits, every time. For applications where the focus matters more than the curve geometry — concentrators, antennas, headlamp reflectors — the scriber method is functionally more accurate even though its raw geometric tolerance is coarser.
References & Further Reading
- Wikipedia contributors. Parabola. Wikipedia
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