Drawing mechanisms · Geometric inversion

Peaucellier Inversor

How can hinged bars draw a straight line without a guide? By geometric inversion. The rhombus and the two long bars keep the pen and the input point on one line through the fixed pivot, and the product of their distances from that pivot stays fixed. Inversion turns a circle through the pivot into a straight line. So when the crank's circle passes through that pivot, the pen draws an exact straight line; any other crank length gives a circular arc.

Swing the crank around its circle and watch a hinged rhombus steer the pen along a straight line. Change the input circle to bend the path, look closely at the stacked pivots, and check 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 constraint residual $0$

Move & trace

Let the linkage draw the curve

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 holds the horizontal and vertical directions, with right and up positive and depth pointing toward you. The controls give angles in degrees; the calculations and the CSV use radians, metres and seconds.

The two blue links are fixed at 1.6 metres. The four gold links all have the chosen rhombus side length. The coral crank turns about the other fixed pivot, and the pen runs on no rail. The links are drawn in stacked layers, joined by pins on shared axes.

When the radius ratio is one, the input circle passes through the fixed inversion pivot, and the output is an exact straight line. Other ratios give circular arcs. The deviation plot compares every path with that same straight reference, not with a fitted line. The dashed input circle is drawn complete as a construction; the crank cannot actually make a full turn.

Playback traces the displayed phase range once and stops. The pale path is the full geometric reference, and the coral ink shows the phases up to the current cursor. Dragging backward simply redraws this view; it does not erase a physical recording. The CSV holds 721 kinematic samples, not measured data.

The input motion is prescribed. These models solve the geometry and its 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 not modeled.

The links are ideal, rigid and planar, and the crank angle is prescribed. Only a working arc away from singular positions is shown. Thicknesses and pin stacking are schematic, and friction, inertia, collisions, contact loads and tolerances are not modeled.

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 point and the pen on one line through the fixed pivot, and the product of their distances from that pivot stays constant.

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. The two long links and the four equal sides of the rhombus enforce this relation.

The input circle

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

Its center is the other fixed pivot. When the radius equals the distance between the pivots, the circle passes through the origin.

When the path is exactly straight

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

Only a safe arc of crank angles is shown. Driving the crank through the inversion center leads to a singular position.

The output circle in general

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

When the coefficient of the quadratic term vanishes, the same equation describes a line.

Common difficulties

The pale curve is not a measurement

Typical misconceptionThe pale curve is an independent experimental check.

Better mental modelIt is the full path of the ideal model. The constraint residuals and independent geometric tests check that the model is consistent; no real data are acquired.

The motion is set directly

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 need a separate analysis. The thicknesses of the three-dimensional parts only show how the pieces are assembled.

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, and the pen, with no guide of its own, moves along a vertical line.
02
Bend the straight path
Change the crank radius ratio, keeping the two fixed pivots in place.
What to observe: The pen now follows a circular arc, although the inversion product is still constant.
03
Separate geometry from speed
Increase the angular rate and export the phase sweep.
What to observe: The pen covers the same path faster. Velocities grow in proportion to the angular rate, and accelerations grow with its square.

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