Drawing mechanisms · Geometric inversion

Peaucellier Inversor

Swing a circular crank and watch a hinged rhombus guide the pen along a straight line. Change the circle to bend the path, inspect the stacked pivots, and test the geometric invariant behind the drawing.

Pen horizontal position $0$

Pen vertical position $0$

Pen speed $0$

Pen acceleration magnitude $0$

Inversion distance product $0$

Straight-reference deviation $0$

Local path curvature $0$

Largest link closure error $0$

Move & trace

Make the geometry draw.

Crank angle

\theta

35.00\,{}^\circ

Prescribed angular rate

\omega

40.00\,{}^\circ/\mathrm s

Rhombus side length

r

1.25\,\mathrm m

Distance between fixed pivots

D

1.00\,\mathrm m

Crank radius / pivot distance

R/D

1.00\,

The drawing plane is horizontal–vertical: right and up are positive, depth points toward you. Angles use degrees in the controls; internal calculations and CSV use radians, metres and seconds.

Two blue links have fixed length 1.6 metres. Four gold links share the selected rhombus side length. The coral crank is attached to the other fixed pivot; the pen has no rail. Aligned pin axes join the visibly stacked link layers.

A radius ratio of unity makes the input circle pass through the fixed inversion pivot, producing an exact straight path. Other ratios produce circular arcs. The deviation plot compares all paths with that same straight reference, not with a fitted line. The dashed input circle is a full construction, not a reachable full crank turn.

Playback traces one displayed phase interval and stops. The pale path is a complete geometric reference; coral ink shows phases up to the current cursor. Moving backward retraces this phase view rather than erasing a physical recording. CSV has 721 kinematic samples, not measured data.

Input motion is imposed. These models solve geometry and derivatives, not contact forces or a motor drive. Parts, pen supports and clearances are schematic; material penetration, inertia, friction, backlash, tolerances and structural safety are outside scope.

Ideal planar rigid links with prescribed crank angle. Only a nonsingular working arc is shown. Thicknesses and pin stacking are schematic; no friction, inertia, collision, contact load or tolerance model.

Start with the essentials

Focus question: What must be true of the input circle for the output to be straight?
One-sentence intuition: The rhombus keeps the input and pen on the same radial line, with a constant distance product.

Core mathematical model

Geometric inversion

K=\ell^2-r^2,\quad\mathbf Q=\frac{K\mathbf P}{|\mathbf P|^2}

The fixed pivot is the origin; two long links and four equal rhombus sides impose this relation.

The input circle

\mathbf P=(D+R\cos\theta,R\sin\theta)

Its center is the other fixed pivot. Equal radius and pivot distance put the origin on the circle.

The exact straight path

R=D\quad\Longrightarrow\quad Q_x=\frac K{2D},\quad Q_y=\frac K{2D}\tan\frac\theta2

Only the safe crank arc is shown. Extending the crank through the inversion center is singular.

The more general output circle

(D^2-R^2)|\mathbf Q|^2-2KDQ_x+K^2=0

The same constraint becomes a line when the quadratic coefficient vanishes.

Common difficulties

The reference is not a measurement

Typical misconceptionThe pale curve is an independent experimental check.

Better mental modelIt is the complete ideal model path. Link-closure residuals and independent geometric tests check consistency; real data are not acquired.

Motion is prescribed

Typical misconceptionA smooth trace proves that a real mechanism can be driven safely at this rate.

Better mental modelMass, friction, contact forces, clearances and drive dynamics require a separate analysis. The three-dimensional part thicknesses only explain assembly.

Run the experiment

01
Trace the default path
Use the drawing-plane view, then play or drag the coral handle.
What to observe: Every bar keeps its length while the unguided pen follows a vertical line.
02
Break the straight path
Change the crank radius ratio, keeping the two fixed pivots in place.
What to observe: The pen follows a circular arc even though the inversion product remains constant.
03
Separate geometry from speed
Increase the angular rate and export the phase sweep.
What to observe: The same path is traversed faster. Velocities scale with angular rate and accelerations with its square.

Geometry & apparatus: Il Giardino di Archimede · Peaucellier · Caltech · Elliptical Trammel