A Hyperbola Scriber is a draughting instrument that draws a true hyperbolic curve by mechanically enforcing the focus-directrix or two-foci distance rule. It solves the problem of plotting hyperbolas point-by-point with a flexible curve, which is slow and inaccurate on critical work like optical reflector profiles or hyperboloid stonework. The tool uses a pivoting straightedge, a sliding string anchored at the foci, and a stylus that maintains the constant difference |r₁ − r₂| = 2a as it tracks. The result is a continuous, mathematically exact curve drawn in a single pass.
Hyperbola Scriber Interactive Calculator
Vary the hyperbola size, eccentricity, trace position, and string stretch to see the focus spacing, traced point, distance rule, and stretch error.
Equation Used
The scriber is modeled with the two-foci definition of a hyperbola. The semi-transverse axis is a, eccentricity is e, focus distance is c = ae, and the focus spacing is 2c. For the selected trace position x/a, the calculator computes the branch height and verifies that the distance difference remains 2a.
- Centered hyperbola with foci at +/-c on the transverse axis.
- Calculator traces the right-hand branch using x/a as the point location.
- Eccentricity must be greater than 1.
- String stretch angle error uses the article rule of thumb: 0.3 mm stretch gives about 0.1 deg asymptote opening.
How the Hyperbola Scriber Actually Works
The hyperbola scriber works on the two-foci definition: every point on a hyperbola has the same absolute difference of distances to two fixed foci, and that difference equals 2a, the length of the transverse axis. The classic mechanical version uses a straightedge that pivots at one focus F₁, a string of fixed length tied between the far end of the straightedge and the second focus F₂, and a stylus that holds the string taut against the edge of the rule. As you swing the straightedge, the stylus traces one branch of the hyperbola because the difference between the rule's reach and the string's slack stays constant — that is the focus directrix property in mechanical form.
Why build it this way? Because point-by-point construction with a French curve compounds error. Each plotted point carries its own ±0.2 mm uncertainty, and the curve drawn between them is whatever the flexible spline decides. A hyperbola scriber removes the interpolation step entirely. If the string stretches even 0.3 mm under tension, you'll see the curve drift outward near the vertex and the asymptote angle open up by roughly 0.1°. That is why the string must be braided polyester or fine steel cable, not cotton, and why the focus pins must seat in reamed holes — not punched ones — to keep pivot slop under 0.05 mm.
Common failure modes are predictable. A loose focus pin lets the whole curve walk laterally. A worn string knot at the straightedge end shortens the effective length and pulls the vertex inward. And if the paper lifts under the stylus, you'll get a kinked curve right where the string angle crosses 90° to the rule. The fix is heavy paper, a clean reference plane, and a stylus weight under 50 g.
Key Components
- Focus Pins (F₁, F₂): Two hardened steel pins set at the calculated focal positions, separated by 2c where c² = a² + b². Pin diameter is typically 1.5 mm seated in reamed holes to hold pivot slop under 0.05 mm. Any wobble here translates directly into vertex displacement on the curve.
- Pivoting Straightedge: A rigid rule pivoting at F₁, longer than the maximum r₁ you intend to draw. The working edge must be ground straight to within 0.02 mm over its length, because any bow shows up as a ripple on the asymptote.
- Tensioned String or Wire: Fixed at the far end of the straightedge and at F₂, length chosen so the difference between rule length and string length equals 2a — the transverse axis. Braided polyester or 0.3 mm stainless cable, never cotton, because stretch above 0.3 mm shifts the curve.
- Stylus or Tracing Point: Holds the string taut against the rule and marks the paper. Weight kept under 50 g to avoid paper lift, tip radius 0.2 mm for fine work or 0.5 mm for vellum. The stylus must slide freely along the rule edge — any binding gives a stepped curve.
- Reference Baseplate: A flat board, typically maple or aluminium tooling plate, flat to 0.1 mm over 600 mm. The paper pins down here. Without a true reference, every other tolerance becomes meaningless.
Where the Hyperbola Scriber Is Used
Hyperbola scribers show up wherever a true conic curve has to be laid out at full or working scale. Plotting from a table of coordinates is fine for CAD output, but for hand-drafted templates, stone layout, and optical-element tooling, the scriber draws a curve that no spline can match for fairness. You'll find them in optics, naval architecture, architectural stonework, and historical drafting reproduction work.
- Optical Engineering: Layout of Cassegrain telescope secondary mirror profiles, where the secondary is a convex hyperboloid. Workshops reproducing 19th-century instrument drawings — for example at the Birr Castle Leviathan restoration — use hyperbola scribers to redraft the secondary profile at 1:1.
- Architectural Stonework: Templates for hyperbolic cooling-tower formwork ribs and for hyperboloid masonry features such as the Kobe Port Tower steelwork layout drawings. Stone shops use the scriber on plywood templates before transferring to stone.
- Naval Architecture: Lofting hyperbolic frame curves on hulls with hyperboloid bow sections, particularly on heritage rebuilds where original lines plans specify conic frames rather than B-splines.
- Drafting Education: Engineering graphics courses at institutions like the Indian Institute of Technology Madras still teach conic construction with mechanical scribers as part of the descriptive geometry syllabus.
- Acoustic Engineering: Layout of hyperbolic horn loudspeaker profiles, particularly for restorations of mid-century cinema horns such as Western Electric 15A reproductions, where the throat curve must follow a true hyperbola.
- Cartographic Reproduction: Redrafting LORAN-C navigation chart hyperbolic position-line overlays from archival masters, where each line of position is a hyperbola defined by two transmitter stations.
The Formula Behind the Hyperbola Scriber
The governing relationship is the two-foci definition itself. What matters in practice is how the eccentricity e and the transverse semi-axis a interact across your drawing range — at low eccentricity (e just above 1) the hyperbola opens narrowly and the scriber draws gently, with the stylus moving slowly along the rule. At high eccentricity (e above 3) the curve flares fast toward its asymptotes and small string-length errors get amplified at the extremities. The sweet spot for mechanical scribing sits around e = 1.3 to 2.0, where the curve is well-defined near the vertex and the asymptotes are still reachable on a normal-size board.
Variables
| Symbol | Meaning | Unit (SI) | Unit (Imperial) |
|---|---|---|---|
| r1 | Distance from any point on the curve to focus F₁ | mm | in |
| r2 | Distance from the same point to focus F₂ | mm | in |
| a | Transverse semi-axis (half the vertex-to-vertex distance) | mm | in |
| b | Conjugate semi-axis, controls asymptote slope ±b/a | mm | in |
| c | Focal semi-distance, half the distance between F₁ and F₂ | mm | in |
| e | Eccentricity, e > 1 for any hyperbola | dimensionless | dimensionless |
Worked Example: Hyperbola Scriber in a planetarium projector lens template
A small-instrument workshop in Jena is laying out the full-scale layout drawing for a hyperbolic field-flattener element used in a refurbished Zeiss ZKP-1 planetarium projector. The element profile is specified by 2a = 180 mm and 2c = 240 mm, drawn on a 600 × 400 mm board. The drafter wants to know the vertex-to-asymptote behaviour at three radii along the curve so the scriber's string and rule can be pre-tensioned correctly before tracing.
Given
- 2a = 180 mm
- 2c = 240 mm
- Maximum r1 = 300 mm
Solution
Step 1 — compute the conjugate semi-axis b and the eccentricity e from the given geometry. Half-values are a = 90 mm and c = 120 mm:
Step 2 — at the nominal mid-range plot point, set r1 = 200 mm. The two-foci constraint gives r2 directly:
That is a comfortable working point — the stylus sits well clear of both foci and the rule angle is roughly 35° off the transverse axis. The asymptote slope here is ±b/a = ±0.882, so the curve is climbing at about 41° to the axis at this radius.
Step 3 — at the low end of the typical draw range, r1 = 100 mm (just past the vertex):
Near the vertex the string runs nearly parallel to the rule and the stylus crawls — perfect for fine vertex detail but unforgiving of any pivot slop, which prints straight onto the curve. Step 4 — at the high end, r1 = 300 mm:
Now the curve is well into asymptotic territory, sloping at close to the limit angle of arctan(b/a) ≈ 41.4°. The stylus moves fast along the rule, and any 0.3 mm string stretch shifts the plotted point by roughly 0.5 mm laterally because the rule angle is shallow.
Result
Across the working range the scriber traces a hyperbola with a = 90 mm, b ≈ 79. 4 mm, and eccentricity 1.33 — a moderately open curve well-suited to mechanical scribing. At r1 = 200 mm (nominal) the stylus sits in the cleanest part of the curve, where rule angle and string tension are both well-conditioned and you can draw 100 mm of arc in one smooth pass. Compare that to the vertex region (r1 ≈ 95 mm) where the stylus creeps and any pivot slop is brutally visible, versus the far end (r1 = 300 mm) where the stylus moves fast and string stretch dominates the error budget. If your traced curve sits 0.5–1 mm outside the calculated vertex, suspect (1) a stretched string — swap the cotton for 0.3 mm braided Dyneema, (2) a focus pin seated in an oversized hole letting F1 wander laterally during the swing, or (3) the straightedge edge bowed by more than 0.05 mm over its length, which puts a slow ripple onto the asymptote leg of the curve.
Hyperbola Scriber vs Alternatives
A hyperbola scriber is one of three practical ways to draft a hyperbola at scale. The choice depends on accuracy demands, drawing size, and whether the curve will be transferred to stone, metal or paper.
| Property | Hyperbola Scriber (string method) | Point-by-Point with French Curve | CAD Plot + Cutter |
|---|---|---|---|
| Curve accuracy (typical) | ±0.2 mm over 500 mm | ±0.5 mm (interpolation error) | ±0.05 mm (plotter limited) |
| Setup time | 20–40 min (pin and tension) | 5–10 min (just the table) | 2–5 min (file open) |
| Tooling cost | £40–200 shop-built | £15 (curves + table) | £800+ for A1 plotter |
| Maximum practical span | ~1.5 m board | Limited by curve length | Limited by plotter bed |
| Suits hand transfer to stone/wood | Excellent — direct trace | Good | Poor — paper print fragile |
| Eccentricity range | 1.1 to ~3.0 practical | Any (table-driven) | Any |
| Skill required | Moderate — string handling | Low | CAD literacy needed |
Frequently Asked Questions About Hyperbola Scriber
Almost always string elongation under increasing tension. As r1 grows, the angle between string and rule shallows, so the tension component pulling the stylus into the rule edge climbs sharply. A cotton or braided nylon string can stretch 0.3–0.8 mm under that load, and because the stylus is far from the foci, that stretch projects outward as a millimetre-scale curve error.
Switch to 0.3 mm braided Dyneema or a fine stainless cable. Pre-tension it overnight before you draw. If the bow persists, check that the string anchor at the far end of the rule has not migrated — a slipped knot adds length that mimics stretch.
No — and trying to is the single most common beginner mistake. The string-and-rule arrangement enforces r1 − r2 = +2a, which gives only the branch nearer F2. To draw the other branch you must swap the roles: pivot the rule on F2 and run the string back to F1, which enforces r2 − r1 = +2a.
For symmetric layouts this means a complete reset including re-tensioning. Mark the transverse axis on the baseplate first so both branches share the same a-line — any axis mismatch between setups shows as a stagger at the vertices.
Three factors push you toward the scriber: large size (over A1), transfer to a non-paper substrate like stone or plywood, and curves you will scribe directly into a workpiece. A plotted print is fine for reference but tears, stretches with humidity, and won't survive a router base running over it.
Print-based work is the right call when you need 0.05 mm accuracy, when the curve is one of many on a busy drawing, or when you'll digitise the result downstream. For a one-off optical or stonework template at full scale, the scriber wins on both fairness of curve and durability of the line.
Practical range is roughly e = 1.1 to 3.0. Below 1.1 the foci sit so close to the vertices that the rule barely articulates and pin slop dominates everything — you'd be better off treating the curve as a slightly modified parabola. Above 3.0 the curve flares so steeply toward its asymptotes that the rule angle becomes very shallow, and any string stretch projects laterally with huge gain.
If you must draw e > 3, switch to a coordinate-table method or build a rigid linkage scriber that replaces the string with a slotted bar. The slotted-bar version trades setup time for elimination of string-stretch error.
The asymptote angle is set by the ratio b/a, and the only way it tilts is if either a or c is wrong on the baseplate. The most likely culprit is focal distance: if 2c was set with a steel rule rather than a vernier, you can easily be 0.5–1 mm off, and that propagates into b through b = √(c2 − a2) and then into the asymptote angle.
Re-measure 2c with a calliper directly between the focus pin shanks, not between visible pin tops. A bent pin shank also offsets the effective focus by 0.2–0.4 mm. Re-pin the foci into freshly reamed holes and the asymptote should snap back to spec.
Always on-axis with the transverse axis aligned to the optical axis of the system. The Cassegrain secondary's vertex sits on the optical axis, and one focus of the hyperboloid coincides with the prime focus of the primary mirror — that's the entire point of the geometry.
When laying out the template, mark the optical axis first on the baseplate, then place F1 at the prime-focus position and F2 at the eyepiece-focus position scaled to drawing units. Any tilt of the transverse axis relative to the optical axis becomes coma in the finished telescope, so the alignment of the axis line on the baseplate is the most critical setup step — flat to 0.1 mm over 600 mm minimum.
References & Further Reading
- Wikipedia contributors. Hyperbola. Wikipedia
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