Peaucellier-Lipkin linkage

The Peaucellier–Lipkin linkage (Wikipedia) was the first planar linkage to produce exact straight line motion from a rotary motion of a hand. To demonstrate how it works, we declaratively perform the following steps:

Setting up scalar values

Running these commands does not show anything on the canvas.

• radius of rotation of the hand: r1 = \scalar 0.9
• angle of rotation of the hand: theta = \scalar 0
• convert theta to radians: thetaRad = \deg2rad theta
• distance from origin to fixed point: r2 = \scalar 2.6
• origin point: p0 = \point 0 0
• point at distance r1 from origin: p1 = \point r1 0
• cosine of theta: cosTheta = \cos thetaRad
• sine of theta: sinTheta = \sin thetaRad
• x-component of rotation: r1cosTheta = \mul r1 cosTheta
• y-component of rotation: r1sinTheta = \mul r1 sinTheta
• x-coordinate of rotating point: p2x = \add r1 r1cosTheta
• y-coordinate of rotating point: p2y = \add 0 r1sinTheta

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Setting up the visible apparatus

Now we create the points and lines:

• the rotating point: p2 = \point p2x p2y
• line from origin to p1: line01 = \line p0 p1

This is the rotating hand:

• line from p1 to rotating point: line12 = \line p1 p2

You can also:

• highlight the hand to avoid any confusion about which line it is.

• define a new color and set it to be the line's color

• radius for circle around p2: r3 = \mul 1.2 r1

The fun part begins here: we create a circle of radius r3 centered at p2 and another circle of radius r2 centered at the origin.

• circle centered at p2: c0 = \circle p2 r3
• circle centered at origin: c1 = \circle p0 r2

... and draw a line joining their points of intersection:

• get intersection points: intersection-points = \intersect-circles c0 c1
• Get the first point and the second point
• Join them with a line

Finally the point p2 is reflected across the line to create the final point:

• reflected point tracing straight line: pFinal = \reflect-point p2 line0

Now as you rotate this point, you'll notice that the reflected point traces a straight line!

If you update the value of theta, by setting it to 45 degrees, the canvas reacts to it immediately. You can also animate it to -45 degrees. Notice how the rotation of the hand is translated into a linear (vertical) motion of this point.

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Trying out different angles

• Create an array of angle values

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Run the \peaucellier-lipkin-linkage command in the geomatic editor and modify the variables to play with it.

You can do things like:

• removing unwanted shapes. Eg you can hide the circles.
• explore the geometry a bit more. Eg the fact that these points are always collinear.
  • you can join them with a line