bpf4

Welcome to the bpf4 documentation!

bpf4 is a python library to operate with curves in 2D space. Curves can be defined via breakpoints (break-point functions, hence the name), using functions or curves can be used to build other curves. It can be used to perform curve fitting, data analysis, plotting, etc. It is mainly programmed in cython for efficiency.

https://github.com/gesellkammer/bpf4

Installation

pip install bpf4

Quick Introduction

A BPF (Break-Point-Function) is defined by points in 2D space. Each different BPF defines a specific interpolation type: linear, exponential, spline, etc. Operations on BPFs are lazy and result in new BPFs representing these operations. A BPF can be evaluated at a specific x coord or an array of such coords or rendered with a given sampling period.

All curve types implement the standard arithmetic methods (+, -, *, /, **, etc) plus all standard mathematical functions (sin, cos, asin, log, sqrt). Each operation results in a new curve representing such operation and together they build a pipeline representing a given function. When a curve is evaluated, each child curve is itself evaluated to generate the numerical result.

Example 1

>>> from bpf4 import *
>>> a = linear((0, 0), (1.5, 10), (3.4, 15))
>>> b = linear((1, 1), (4, 10))

# Construct a third BPF representing the average
>>> avgbpf = (a + b) / 2
>>> avgbpf
_BpfLambdaDivConst[0.0:4.0]

# Sample the bpf at a regular interval, generate 30 elements. This first
# samples curve `a` and `b`, then performs the binary operations
# on the resulting arrays, avoiding any extra allocation
>>> avgbpf.map(30)
array([ 0.5       ,  0.95977011,  2.67816092,  4.51724138,  6.35632184,
        7.29885057,  7.75862069,  8.2183908 ,  8.67816092,  9.13793103,
        9.59770115, 10.02268603, 10.20417423, 10.38566243, 10.56715064,
       10.74863884, 10.93012704, 11.11161525, 11.29310345, 11.47459165,
       11.65607985, 11.83756806, 12.01905626, 12.20054446, 12.38203267,
       12.5       , 12.5       , 12.5       , 12.5       , 12.5       ])

avgbpf is a BPF representing a set of mathematical operations. It is defined over a given range (the bounds of the BPF). If not indicated otherwise outside of the bounds of the BPF its values remain constant:

>>> a(3.4)
15
>>> a(10)
15

Example 2: Intersection between two curves

from bpf4 import *  
import matplotlib.pyplot as plt
a = spline((0, 0), (1, 5), (2, 3), (5, 10))  
b = expon((0, -10), (2,15), (5, 3), exp=3)
# plots are performed using matplotlib
a.plot() 
b.plot() 
zeros = (a - b).zeros()
plt.plot(zeros, a.map(zeros), 'o')

Features

Many interpolation and curve-fitting types:

spline
univariate splie
pchip (hermite)
cosine
exponential
logarithmic
etc.

With the exception of curve-fitting bpfs, interpolation types can be mixed, so that each segment has a different interpolation. Following from the example above:

c = (a + b).sin().abs()
# plot only the range (1.5, 4)
c[1.5:4].plot()

Syntax support for shifting, scaling and slicing a bpf

from bpf4 import *
a = spline((0, 0), (1, 5), (2, 3), (5, 10))  
b = expon((0, -10), (2,15), (5, 3), exp=3)

shifted = a >> 2        # a shifted to the right
scaled = (a * 2) ^ 3   # scale the x coord by 3, scale the y coord by 2
cropped = a[2:2.5]      # slice only a portion of the bpf
sampled = a[::0.5]     # sample the bpf with an interval of 0.5

import matplotlib.pyplot as plt
fig, axs = plt.subplots(1, 4, figsize=(16, 4), sharey=True, tight_layout=True)
for curve, ax in zip((shifted,scaled, cropped, sampled), axs):
    curve.plot(axes=ax, show=False)
plt.show()

Derivation / Integration

from bpf4 import *
a = spline((0, 0), (1, 5), (2, 3), (5, 10))
deriv = a.derivative()
integr = a.integrated()

import matplotlib.pyplot as plt 
fig, axs = plt.subplots(3, 1, sharex=True, figsize=(16, 8), tight_layout=True)
a.plot(axes=axs[0], show=False)
deriv.plot(axes=axs[1], show=False)
integr.plot(axes=axs[2])

Mathematical operations

Max / Min

a = linear(0, 0, 1, 0.5, 2, 0)
b = expon(0, 0, 2, 1, exp=3)
a.plot(show=False, color="red", linewidth=4, alpha=0.3)
b.plot(show=False, color="blue", linewidth=4, alpha=0.3)
core.Max((a, b)).plot(color="black", linewidth=4, alpha=0.8, linestyle='dotted')

a = linear(0, 0, 1, 0.5, 2, 0)
b = expon(0, 0, 2, 1, exp=3)
a.plot(show=False, color="red", linewidth=4, alpha=0.3)
b.plot(show=False, color="blue", linewidth=4, alpha=0.3)
core.Min((a, b)).plot(color="black", linewidth=4, alpha=0.8, linestyle='dotted')

`+, -, *, /`

a = linear(0, 0, 1, 0.5, 2, 0)
b = expon(0, 0, 2, 1, exp=3)
a.plot(show=False, color="red", linewidth=4, alpha=0.3)
b.plot(show=False, color="blue", linewidth=4, alpha=0.3)
(a*b).plot(color="black", linewidth=4, alpha=0.8, linestyle='dotted')

a = linear(0, 0, 1, 0.5, 2, 0)
b = expon(0, 0, 2, 1, exp=3)
a.plot(show=False, color="red", linewidth=4, alpha=0.3)
b.plot(show=False, color="blue", linewidth=4, alpha=0.3)
(a**b).plot(color="black", linewidth=4, alpha=0.8, linestyle='dotted')

a = linear(0, 0, 1, 0.5, 2, 0)
b = expon(0, 0, 2, 1, exp=3)
a.plot(show=False, color="red", linewidth=4, alpha=0.3)
b.plot(show=False, color="blue", linewidth=4, alpha=0.3)
((a+b)/2).plot(color="black", linewidth=4, alpha=0.8, linestyle='dotted')

Building functions

A bpf can be used to build complex formulas

Fresnel's Integral: $S(x) = \int_0^x {sin(t^2)} dt$

t = slope(1)
f = (t**2).sin()[0:10:0.001].integrated()
f.plot()

Polar plots

Any kind of matplotlib plot can be used. For example, polar plots are possible by creating an axes with polar=True

Cardiod: $\rho = 1 + sin(-\theta)$

from math import *
theta = slope(1, bounds=(0, 2*pi))
r = 1 + (-theta).sin()

ax = plt.axes(polar=True)
ax.set_rticks([0.5, 1, 1.5, 2]); ax.set_rlabel_position(38)
r.plot(axes=ax)

Flower 5: $\rho = 3 + cos(5 * \theta)$

theta = core.Slope(1, bounds=(0, 2*pi))
r = 3 + (5*theta).cos()

ax = plt.axes(polar=True)
r.plot(axes=ax)