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Planimeter
Brief description
A planimeter is a mechanical analog integrating instrument used to measure the surface area of arbitrary, irregular two-dimensional planar figures by tracing their perimeter. Invented by Swiss mathematician Jakob Amsler-Laffon in 1854, the polar planimeter replaced tedious manual geometric approximations and numerical approximations with a single smooth manual trace, becoming an indispensable tool for civil surveyors, cartographers, naval architects, and mechanical engineers calculating engine indicated power.
Use / Function
- Engine Indicator Diagram Analysis: Integrating the pressure-volume $(P\text{-}V)$ indicator cards of steam and internal combustion engines to determine indicated horsepower.
- Topographical Map Surveying: Measuring watershed areas, agricultural plots, forest reserves, and land parcels on cartographic projections.
- Naval Architecture: Calculating cross-sectional waterplane areas, displacement hulls, and centers of buoyancy for vessel design.
- Structural Engineering: Finding the moment of inertia, centroid, and cross-sectional area of irregular architectural arches and structural beams.
Operating principle
- Mechanical Linkage: The instrument consists of two hinged arms: a pole arm anchored at a stationary pivot point, and a tracer arm carrying a fine tracing stylus and a precision measuring wheel.
- Boundary Tracing: The operator uses the tracer stylus to guide the arm clockwise around the entire closed boundary perimeter of the irregular shape.
- Kinematic Wheel Motion: The measuring wheel rests directly on the drawing paper; as the linkage articulates, the wheel simultaneously rolls along its perpendicular axis and slips parallel to its axis.
- Green’s Theorem Integration: Pure rolling displacement registers only movement perpendicular to the tracer arm, mechanically performing line integration $\oint (x,dy - y,dx)$ around the boundary contour in accordance with Green’s theorem.
- Direct Area Readout: A revolution counter drum with a vernier scale attached to the measuring wheel indicates the net integrated area directly in square centimeters, inches, or scaled square meters.
How to create it
- Machine the Linkage Arms: Mill two rigid, non-flexing arms from seasoned brass bars: the pole arm (with a heavy anchored pivot needle) and the tracer arm.
- Fabricate the Elbow Joint: Join the two arms with a hardened steel conical pivot bearing that allows free coplanar articulation without lateral backlash or play.
- Turn the Integrating Wheel: Turn a precision measuring wheel from hardened steel or brass with a knife-edged rim ground to an exact known rolling diameter.
- Mount Precision Bearings: Mount the measuring wheel axle on low-friction adjustable steel screw centers positioned parallel to the tracer arm centerline.
- Engrave the Vernier Drum: Divide the wheel perimeter into 100 equal graduations, and attach a geared revolution disc and stationary vernier plate reading to fractions of a division.
- Balance and Calibrate: Fit a tracing point or magnifying reticle at the outer end of the tracer arm, and calibrate the instrument against an accurately scribed square or circular test template of known area.
Materials needed
- Essential: High-grade Brass for structural arms, weight anchors, and vernier frames; hardened Steel for pivot pivots, needle anchors, tracing pointers, and the knife-edged integrating wheel.
- Tools: Precision dividing engine for rotary scale graduation, metal lathe, jewelers’ screwdrivers, abrasive polishing stones.
- Substitutes: A linear (rolling) planimeter uses wheels mounted on a straight track rather than a fixed polar pivot for very long drawing sheets.
Variants and improvements
- Amsler Polar Planimeter: The classic dual-arm rotating linkage anchored to a single fixed point on the drawing board.
- Linear / Rolling Planimeter: Moves along an external cylindrical roller or guide rail, permitting measurement of long geographic strips without polar limit restrictions.
- Compensating Planimeter: Allows reversal of tracing arms to mathematically cancel slight mechanical eccentricities or wheel alignment errors.
- Radial Planimeter: Specialized for reading circular recording charts from barographs, pressure gauges, and gas meters.
Limits and risks
- Paper Surface Friction: Uneven paper texture, wrinkles, or greasy smudges cause the measuring wheel to slip instead of roll, producing significant integration errors.
- Backlash and Pivot Wear: Any mechanical play or looseness in the hinged elbow bearing alters the effective arm geometry and skews calculations.
- Tracing Drift: Deviations from the drawn boundary line by the human operator directly distort the perimeter integral; skilled, steady tracing is essential.