A traditional parallel ruler joins two straight edges with equal pivoted arms. Moving one edge changes its position while preserving its direction. Explore the ideal linkage and calculate longitudinal shift, perpendicular separation and the enclosed parallelogram area.
Parallel-Ruler Geometry Calculator
Enter ruler pivot spacing, equal cross-arm length and an inspection angle. The worksheet reports exact shift, separation and enclosed parallelogram area.
Equation Used
- The two ruler edges are rigid and their corresponding pivot spacing is C.
- Both connecting arms have equal length L and remain mutually parallel.
- All four joints are perfect coplanar revolute pins.
- Scale markings, tolerance, friction, clearance, compliance, wear and drawing accuracy are excluded.
The dashed upper edge marks the entered inspection angle. Numerical results describe that selected position while the moving linkage illustrates parallel translation over a range of angles.
Equal pivot arms preserve the ruler direction
Museum terminology describes a parallel ruler as a pair of rulers connected by two pivoting equal-length crosspieces. European Patent EP0168948A1 likewise describes parallel straight edges linked by a pair of mutually parallel equal-length arms pivoted at their ends.
In the declared ideal construction, each point on the moving ruler is translated by the same vector (L cos theta, L sin theta). Its orientation therefore does not change.
Transferring parallel lines
The Smithsonian documents parallel rules among historical drawing instruments, while the Louvre records a pair of parallel rulers connected by two openwork pieces. Sir John Soane's Museum records a 380 mm mahogany example with a brass parallel lower rail.
The mechanism transfers direction geometrically; this page does not assign an accuracy tolerance to a real instrument.
Exact ideal translation
The first two expressions are the components of either equal cross-arm. The area is base C times perpendicular separation.
Declared geometry example
For C=380 mm, L=100 mm and theta=45 degrees, shift and separation are each 70.71 mm, enclosed area is about 26,870.06 mm squared and ideal edge-angle error is zero.
What ideal geometry omits
A physical ruler has pin clearance, arm and rail stiffness, wear, surface contact and marking error. Those properties require measured dimensions and loads.
The ideal zero angle error follows from equal link lengths and perfect pivots. Estimating drift in a physical ruler requires its actual link dimensions and joint clearances.
Parallel-ruler geometry questions
Why does the moving ruler stay parallel?
The two equal cross-arms and equal pivot spacing form a parallelogram.
Does zero ideal error mean a real ruler is perfect?
No; it is a geometric result for perfect pins and rigid members.
What does the arm angle control?
It sets both longitudinal shift and perpendicular separation.
Does the model include measurement scale error?
No.
Parallel-ruler references
- Musee du Louvre, Pair of parallel rulers: two parallel rulers connected by two openwork pieces.
- Sir John Soane's Museum, Parallel ruler X283: documented mahogany and brass example, 380 mm long.
- Fitzwilliam Museum terminology, parallel ruler: two rulers joined by pivoting equal-length crosspieces so they remain parallel.
- European Patent EP0168948A1: parallel straight edges linked by mutually parallel equal-length arms.
- Smithsonian, Parallel Rules: instrument history, forms and uses.
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