Linkage geometry · Similarity

Pantograph Drawing Copier

How can four bars copy a drawing at a different size? The hinged parallelogram keeps the fixed pivot, tracer and pen on one line, with a constant ratio of their distances from the pivot. Every point of the drawing is copied at that ratio, so the shape stays the same. Doubling the scale doubles lengths and pen speed, and makes each enclosed area four times as large.

Guide the blue tracer around a circle, an ellipse or a flower. Four hinged bars keep the coral pen on an enlarged copy of the same shape. Change the scale and compare distances, speeds and areas.

Pen horizontal position $0$

Pen vertical position $0$

Length and speed scale $0$

Enclosed area scale $0$

Tracer speed $0$

Pen speed $0$

Bar-length residual $0$

Similarity error $0$

Move the linkage

Let a parallelogram copy your drawing.

Pattern phase

\varphi

220.00\,{}^\circ

Sweep rate

\Omega

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

Short link length

L

1.40\,\mathrm m

Drawing scale

k

2.00\,

Pattern size / link length

a/L

0.20\,

Right and up are positive. All motion lies in the front plane; depth separates the rods and their pins. Controls use degrees, and calculations and CSV use radians, metres, seconds, newtons and joules.

Four bars make a parallelogram. The blue bar from the fixed pivot and the blue pen bar have equal lengths. Their two intermediate pins set the scale. The tracer, fixed pivot and pen remain collinear as the joints move. The short link and long link lengths change when the scale changes.

The tracer follows one prescribed closed path. Dragging the blue handle moves it along that path. Circle, ellipse, figure eight and flower presets change the path. The faint output is a complete reference; the bright line shows how much has been drawn.

Lengths and speeds scale by the drawing ratio; the area enclosed by a simple loop scales by its square. Every figure-eight lobe scales the same way. These are exact rigid-link kinematics, with no actuator or contact-force calculation.

The positive similarity scale enlarges without rotation or reflection. One full phase interval closes the drawing. Samples and CSV are calculated model values. Link widths, thicknesses, pin heads and pen size are schematic; the pin-center lengths and drawing coordinates use one consistent scale.

The selected paths stay away from a straightened parallelogram. Real pin bores and separate depth layers show how the mechanism joins, but manufacturing tolerances, rubbing, inertia and bending are omitted.

Rigid bars and frictionless pins impose exact planar geometry. The tracer follows a prescribed pattern; forces, inertia, joint play, bending and pen friction are omitted. Changing the scale rebuilds the bar lengths and pin positions.

Start with the essentials

Focus question: Does doubling every distance also double the enclosed area?
One-sentence intuition: The fixed pivot, tracer and pen stay collinear at a constant distance ratio. Lengths scale by that ratio and enclosed areas by its square.

Core mathematical model

The parallelogram construction

\mathbf D=\mathbf A+\mathbf v,\quad \mathbf B=k\mathbf A,\quad \mathbf C=\mathbf B+\mathbf v,\quad \mathbf Q=\mathbf B+k\mathbf v

The fixed pivot is the origin. The two basic vectors have the short-link length; extending them by the same ratio makes the tracer and pen collinear with the origin.

Length, velocity and area scale

\mathbf Q=k\mathbf D,\qquad \dot{\mathbf Q}=k\dot{\mathbf D},\qquad A_Q=k^2 A_D

The area relation applies to the region inside a simple closed loop, or to each lobe separately. The scale is positive and fixed during each drawing.

Common difficulties

A parallelogram needs correctly placed pivots

Typical misconceptionAny four bars in a parallelogram will copy at any chosen scale.

Better mental modelThe two long bars, their intermediate pin locations and the two shorter bars must be matched. The scale control changes this geometry together.

Area grows faster than length

Typical misconceptionA double-size circle has double the area.

Better mental modelBoth horizontal and vertical distances double. Every small patch becomes four times as large, so a simple enclosed area quadruples.

Run the experiment

01
Trace a circle
Choose Circle and move the blue tracer around its path.
What to observe: The coral pen follows a larger circle. The three points stay on the same ray from the fixed pivot.
02
Change the whole shape
Try Ellipse, Figure eight and Five-lobe flower without changing the scale.
What to observe: The same bar arrangement copies every bend. Changing the path does not change the length or speed ratio.
03
Compare length and area
Select Circle and change Drawing scale from two to three.
What to observe: The copied radius becomes three times the original and the area nine times as large. The pen moves three times as fast as the tracer.

Sources: US Army Engineer School · Engineering Drawing II, §6 · He et al. · Scissor lift virtual work · MIT · Mechanical advantage