Draw a curve connecting two points instead of a straight line

Question

I want to do something like this:

I have the points but don't know how to plot the curves instead of straight lines.

Thank you.

Answer 1

For people interested in this question, I followed Matthew's suggestion and came up with this implementation:

def hanging_line(point1, point2):
    import numpy as np

    a = (point2[1] - point1[1])/(np.cosh(point2[0]) - np.cosh(point1[0]))
    b = point1[1] - a*np.cosh(point1[0])
    x = np.linspace(point1[0], point2[0], 100)
    y = a*np.cosh(x) + b

    return (x,y)

Here is what the result looks like:

import matplotlib.pyplot as plt

point1 = [0,1]
point2 = [1,2]
x,y = hanging_line(point1, point2)

plt.plot(point1[0], point1[1], 'o')
plt.plot(point2[0], point2[1], 'o')
plt.plot(x,y)

Answer 2

You are going to need some expression for the curve you want to plot, then you can make the curve out of many line segments.

Here's a parabola:

x = np.linspace(-1, 1, 100)
y = x*x
plt.plot(x, y)

Here's a sin curve:

x = np.linspace(-2*np.pi, 2*np.pi, 100)
y = np.sin(x)
plt.plot(x, y)

Each of these looks smooth, but is actually made up of many small line segments.

To get a collection of curves like you showed, you are going to need some expression for a curve you want to plot in terms of its two endpoints. The ones in your picture look like catenarys which are (approximately) the shape a hanging chain assumes under the force of gravity:

x = np.linspace(-2*np.pi, 2*np.pi, 100)
y = 2*np.cosh(x/2)
plt.plot(x, y)

You will have to find a way of parameterizing this curve in terms of its two endpoints, which will require you substituting your values of y and x into:

y = a*cosh(x/a) + b

and solving the resulting pair of equations for a and b.

Answer 3

There is a cool (at least for me) way to draw curve lines between two points, using Bezier curves . Just with some simple code you can create lists with dots connecting points and chart them with matplotlib, for example:

def recta(x1, y1, x2, y2):
    a = (y1 - y2) / (x1 - x2)
    b = y1 - a * x1
    return (a, b)

def curva_b(xa, ya, xb, yb, xc, yc):
    (x1, y1, x2, y2) = (xa, ya, xb, yb)
    (a1, b1) = recta(xa, ya, xb, yb)
    (a2, b2) = recta(xb, yb, xc, yc)
    puntos = []

    for i in range(0, 1000):
        if x1 == x2:
            continue
        else:
            (a, b) = recta(x1, y1, x2, y2)
        x = i*(x2 - x1)/1000 + x1
        y = a*x + b
        puntos.append((x,y))
        x1 += (xb - xa)/1000
        y1 = a1*x1 + b1
        x2 += (xc - xb)/1000
        y2 = a2*x2 + b2
    return puntos

Then, just run the function for some starting, mid and ending points, and use matplotlib:

lista1 = curva_b(1, 2, 2, 1, 3, 2.5)
lista2 = curva_b(1, 2, 2.5, 1.5, 3, 2.5)
lista3 = curva_b(1, 2, 2.5, 2, 3, 2.5)
lista4 = curva_b(1, 2, 1.5, 3, 3, 2.5)

fig, ax = plt.subplots()
ax.scatter(*zip(*lista1), s=1, c='b')
ax.scatter(*zip(*lista2), s=1, c='r')
ax.scatter(*zip(*lista3), s=1, c='g')
ax.scatter(*zip(*lista4), s=1, c='k')

This should be the results:

several Bezier quadratic curves

By extending the code a little more, you can get forms like this:

Bezier quartic curve