Geometry, measurement and traceable exports

Draw the chord.
Measure the rise.

Arcspan reconstructs an ideal minor circular arc from two measurements: the straight chord between its endpoints, and the perpendicular rise at the chord's midpoint. The result supports craft templates, drawings and geometric layout. It does not assess a structure or the behaviour of a material.

What the two inputs actually measure

The chord is the direct straight-line distance from one endpoint to the other. A measurement taken along a curved edge is the arc length, which is longer for a non-flat curve. The rise, also called the sagitta, runs perpendicular to the chord from its midpoint to the arc. These inputs assume a symmetric circular curve.

A measured shape may be elliptical, uneven, offset or simply uncertain. Two numbers do not establish that a real object is circular. Arcspan shows the circle implied by the inputs, rather than fitting or inspecting an object. If you are copying a real curve, compare several intermediate points with the proposed layout before using the template.

Recovering the radius without a distant centre mark

Let c be chord length and h be positive midpoint rise. The radius R is c² ÷ (8h) + h ÷ 2. This follows by taking half the chord as one leg of a right triangle, R − h as another, and R as the hypotenuse: (c/2)² + (R − h)² = R². Cancelling R² and rearranging gives the radius expression.

Wolfram's authored Sagitta reference defines this midpoint distance and gives the equivalent radius relation. Arcspan uses the relation for its stated minor-arc scope, with a separate straight-line branch when h is zero.

The half-angle relation gives the angle formula used here. Wolfram's Circular Arc reference describes the radius-times-radian-angle arc-length relation and the distinction between an arc and its chord. Degrees are provided for display; the length calculation uses radians.

Worked example: 600 mm chord, 75 mm rise

With ten equal chord intervals, stations are spaced 60 mm apart horizontally. There are eleven stations because both endpoints are included. Station zero is x = 0, y = 0. The midpoint station is x = 300 mm, y = 75 mm. The final station is x = 600 mm, y = 0. These are points on the geometric centreline, not proposed fastener positions or cut allowances.

Semicircle and straight-line boundaries

The tool accepts rise from zero through half the chord. If the chord is 200 mm and the rise is 100 mm, the radius is exactly 100 mm and the circle centre lies on the chord midpoint. The central angle is 180 degrees, or π radians, so arc length is 100π = 314.1592653589… mm. This is the semicircle boundary of the model.

A larger rise would describe a different major-arc branch that is deliberately outside this calculator's scope. Such a value is rejected rather than silently choosing another interpretation. A negative rise is also rejected; reflect the layout yourself if you need the identical curve on the other side of a reference line.

At exactly zero rise, the output is a straight line whose length equals the chord. There is no finite circle radius or defined circle centre in that limiting case. The station heights are all zero and the SVG draws a line. The tool does not place an enormous arbitrary radius into the result to conceal division by zero.

Station spacing follows the chord

The interval setting divides the straight chord into equal horizontal steps. Choosing N intervals gives N + 1 stations, including the two endpoints. These are not equal-length intervals along the curved arc. The distinction matters because the horizontal distance represented by a given arc length varies along the curve.

The station origin is the left chord endpoint: x increases to the right, and y is the perpendicular height above the chord. Station coordinates are displayed in the chosen input unit. CSV contains both that chosen-unit view and canonical millimetre coordinates. Each view rounds to six decimal places, and the export labels this display convention.

The tool accepts 2–100 chord intervals. An even number places a station at the midpoint; an odd number does not, although the rise still defines the actual midpoint and the SVG curve includes that point. Increasing the station count does not change the radius or arc length. It provides more sampled reference coordinates for the same ideal curve.

Shallow curves need careful arithmetic

A very shallow arc can have a radius far larger than the chord. Subtracting two nearly equal huge circle heights can discard useful numerical detail. Arcspan uses stable equivalent expressions for station heights and sampled SVG coordinates rather than relying only on that direct subtraction.

For example, a 10,000 mm chord with a 0.000001 mm rise implies a radius of approximately 12,500,000,000,000 mm. That rise is a mathematical input far below ordinary workshop measuring resolution. The model keeps it positive and its midpoint height remains 0.000001 mm, but it does not imply that a physical template or printer can reproduce such a distinction.

Input dimensions are plain nonnegative decimals with up to six decimal places. Chord length must be equivalent to 0.001–10,000 mm. The rise may be zero and must not exceed half the chord. These are computational boundaries, not material-size or construction limits. Geometry uses ordinary finite floating-point calculations; displayed decimal detail is not a measurement accuracy guarantee.

Unit changes preserve the measured dimensions

Millimetres, centimetres and inches are views of the same physical inputs. The tool retains the canonical millimetre dimensions when you change units, while the visible fields round to six decimal places. A status note explains that the physical values remain in use until an individual field is edited.

This avoids repeated conversions gradually changing a chord or shallow rise. Editing one dimension establishes a new value for that dimension in the currently selected unit, while preserving the other canonical dimension until it too is edited. Invalid chord/rise values must be corrected before a unit change, so an invalid entry is not silently reinterpreted.

What the SVG actually contains

The downloaded SVG uses physical millimetre dimensions. It contains the ideal centreline approximated by short straight segments, a dashed chord and midpoint-rise reference, and a 50 mm calibration line. It does not contain an offset cut path, material thickness, glue tab, kerf allowance or machine instructions. Stroke widths are drawing marks, not allowances.

The sampling adapts to the radius and angle so the maximum ideal radial deviation between a curve segment and its corresponding circular arc is no more than 0.02 mm. SVG coordinates are then rounded to six decimal places in millimetres, which adds coordinate-rounding error. That mathematical approximation bound does not cover measurement error, tracing, paper movement, printer scaling, cutter behaviour or material deformation.

The on-page preview is scaled to the screen and is not an actual-size ruler. Open the saved SVG in suitable software, print at actual size rather than fit-to-page, then measure its 50 mm line. Large layouts can exceed a sheet. Arcspan does not automatically tile the drawing across pages, choose a printer, or verify software support. Check a small test before relying on a printed template.

Save coordinates with their assumptions

Calculate a valid result before saving SVG, saving station CSV or copying the layout. The copied plan records dimensions, radius or flat-line state, circle-centre convention, angle, length, station coordinates, units and approximation limits. Changing an input clears the old preview and disables exports until recalculation, so an earlier valid plan is not silently reused after an invalid edit.

Restore example resets to 600 mm chord, 75 mm rise and ten intervals. All calculations and exports run locally, without an account or uploading measurements. Return to Arcspan, or use the shared contact inbox for a reproducible geometry correction.