Straight-line linkages: Watt, Chebyshev, Peaucellier-Lipkin

Before precise guideways, a straight motion had to come from links and pins. The example is three linkages that make one: Watt’s linkage and Chebyshev’s give a nearly straight line, and in 1864, Peaucellier and Lipkin found a linkage whose point moves on an exactly straight line. Every position is a chain of circle intersections, and the deviation from the straight line is measured.

import math
import matplotlib.pyplot as plt
from matplotlib.animation import FuncAnimation
from IPython.display import HTML
import planex as px

meet gives the points where two circles meet, a circle of radius r around p and one of radius s around q:

def meet(p, r, q, s):
    return [(v.x, v.y) for v in px.Circle(p, r).Intersections(px.Circle(q, s))]

Peaucellier-Lipkin

In the Peaucellier-Lipkin linkage, two long arms L from the fixed pivot O hold a rhombus of side s at its corners A and B. The rhombus’s corner C is driven on a circle through O, and the opposite corner P lies at the distance (L² − s²) / OC from O on the same ray. This is an inversion, which turns the circle through O into a straight line at x = (L² − s²) / (2 r). Here, the driving circle has a radius r = 2, the arms are L = 5 long and the rhombus has sides of s = 2 (Figure 1).

r = 2.0
L = 5.0
s = 2.0
bars = [("Q", "C"), ("O", "A"), ("O", "B"), ("A", "C"), ("B", "C"), ("A", "P"), ("B", "P")]


def peaucellier(angle):
    t = math.radians(angle)
    C = (r * (1 + math.cos(t)), r * math.sin(t))
    A, B = meet((0, 0), L, C, s)
    P = max(meet(A, s, B, s), key=lambda q: q[0])
    return {"O": (0, 0), "Q": (r, 0), "C": C, "A": A, "B": B, "P": P}


cell = peaucellier(40)
sketch = px.QuickSketch()
for u, w in bars:
    sketch.Move(cell[u]).Line(cell[w], id=u + w)

Figure 1: The Peaucellier-Lipkin linkage

Watt and Chebyshev

Two linkages give a nearly straight line from four bars:

• Watt’s linkage has two equal bars of 2 pointing at each other from two pivots, joined by a coupler of 1 whose middle moves nearly straight;

• Chebyshev’s linkage has two crossed bars of 2.5 on a base of 2, joined by a coupler of 1, whose middle also moves nearly straight.

def watt(angle, bar=2.0, coupler=1.0):
    t = math.radians(angle)
    left = (-bar, coupler / 2)
    right = (bar, -coupler / 2)
    A = (left[0] + bar * math.cos(t), left[1] + bar * math.sin(t))
    B = min(meet(right, bar, A, coupler), key=lambda q: q[0])
    return {"left": left, "right": right, "A": A, "B": B, "P": ((A[0] + B[0]) / 2, (A[1] + B[1]) / 2)}


def chebyshev(angle):
    t = math.radians(angle)
    A = (2.5 * math.cos(t), 2.5 * math.sin(t))
    B = min(meet((2.0, 0.0), 2.5, A, 1.0), key=lambda q: q[0])
    return {"left": (0.0, 0.0), "right": (2.0, 0.0), "A": A, "B": B, "P": ((A[0] + B[0]) / 2, (A[1] + B[1]) / 2)}

How straight

For each linkage, over a comparable stroke of its point, the deviation is how far the point strays across the stroke from where it started. The Peaucellier-Lipkin linkage and Watt’s move along y, Chebyshev’s along x.

def deviation(points, across):
    return max(abs(q[across] - points[0][across]) for q in points)


strokes = {
    "Peaucellier-Lipkin": ([peaucellier(angle)["P"] for angle in range(-60, 61, 2)], 0),
    "Watt": ([watt(angle)["P"] for angle in range(-20, 21)], 0),
    "Chebyshev": ([chebyshev(angle)["P"] for angle in range(42, 91)], 1),
}
for name, (points, across) in strokes.items():
    length = max(q[1 - across] for q in points) - min(q[1 - across] for q in points)
    print(name, "stroke", round(length, 3), "deviation", deviation(points, across))

Peaucellier-Lipkin stroke 6.062 deviation 7.105427357601002e-15
Watt stroke 1.371 deviation 0.010938477390811704
Chebyshev stroke 1.484 deviation 0.004401804672808485

Check: the Peaucellier-Lipkin linkage’s line lies at x = (L² − s²) / (2 r):

print("Peaucellier line at x", strokes["Peaucellier-Lipkin"][0][0][0], "formula", (L**2 - s**2) / (2 * r))

Peaucellier line at x 5.25 formula 5.25

In motion

The three linkages, with the paths of their points: