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Spring returns a list of x coordinates and a list of y coordinates corresponding to the "nodes" of a sawtooth-shaped spring given two endpoints located anywhere in 2-dimensional space, making it easy to dynamically plot a spring.

Click here to watch a video of a script that uses to animate a spring stretching and compressing

See the sections below or this blog post for a more detailed explanation.


x_coords, y_coords = spring(start, end, nodes, width)


start: array-like

A container or iterable with two elements representing the x and y coordinates of the first endpoint.

end: array-like

A container or iterable with two elements representing the x and y coordinates of the second endpoint.

nodes: integer

The number of nodes n. n ≥ 1. Defaults to 1 if a number less than 1 is provided.

width: float

The width w of the spring. If w is not large enough for the distance between the endpoints (see below), it will be adjusted automatically and the spring will simply be displayed as a line.


x_coords, y_coords: list, list

Two lists containing the x coordinates and y coordinates, respectively, of the spring nodes, including the endpoints. This makes it easy to feed the output of the function directly to methods in matplotlib.


import matplotlib.pyplot as plt
from spring import spring

point_a = (1, 1.5)
point_b = (8.1, 6)

fig, ax = plt.subplots()
ax.plot(*spring(point_a, point_b, 12, 1.2), c="black")
ax.set_aspect("equal", "box")


The number of nodes between the two endpoints is defined as n, with the nodes being numbered from 0 (first endpoint) to n+1 (second endpoint):

Numbering of spring nodes

The length of the spring, i.e., the distance between the two endpoints, is defined as l, illustrated by the following figure:

Length of the spring

Then, for any number of nodes n and length l, the "straight-line" distance between nodes can be obtained from the following pattern (only n=1 and n=2 are shown below):

Node spacing

Expressed mathematically, the distance of the ith node along (parallel to) the centerline between the two endpoints is

Equation 1

for 0 < i < n+1. Next, I define the width w as the length of one "link" in the spring, based on the idea that this would be its width if it were fully compressed:

Spring width

Then the perpendicular distance of a node from the imaginary centerline connecting the endpoints is found as follows:

Perpendicular distance to node

Accordingly, the perpendicular distance of the ith node can be expressed as follows:

Equation 2

for 0 < i < n+1. Observe that the spring can attain a maximum length of nw, at which point the nodes would be collinear, i.e., the spring would be a straight line. If l > nw, the quantity in the square root will be negative, indicating that the spring cannot physically stretch to the desired length.

Finally, the unit tangent and unit normal vectors are computed (tangent/parallel to and normal/perpendicular to the imaginary centerline between the endpoints, respectively), allowing the node coordinates to be easily obtained for endpoints located anywhere in the xy plane.

Unit vectors

Putting it all together, the location of the ith node relative to the first endpoint (node 0) is given by the following equation:

Final equation