Added an algorithm that approximates line lengths (#1692)

* A recursive insertion sort

* added doctests and typehints

* Added arc length and numerical integration calculators

* fixed doc test

* Fixed some conversion errors

* Fixed some commenting

* Deleted numerical integration to allow 1 file per push

* Changed string formatting method

maths/line_length.py

@@ -0,0 +1,61 @@
+from typing import Callable, Union
+import math as m
+
+def line_length(fnc: Callable[[Union[int, float]], Union[int, float]],
+                x_start: Union[int, float],
+                x_end: Union[int, float],
+                steps: int = 100) -> float:
+
+    """
+    Approximates the arc length of a line segment by treating the curve as a
+    sequence of linear lines and summing their lengths
+    :param fnc: a function which defines a curve
+    :param x_start: left end point to indicate the start of line segment
+    :param x_end: right end point to indicate end of line segment
+    :param steps: an accuracy gauge; more steps increases accuracy
+    :return: a float representing the length of the curve
+
+    >>> def f(x):
+    ...    return x
+    >>> f"{line_length(f, 0, 1, 10):.6f}"
+    '1.414214'
+
+    >>> def f(x):
+    ...    return 1
+    >>> f"{line_length(f, -5.5, 4.5):.6f}"
+    '10.000000'
+
+    >>> def f(x):
+    ...    return m.sin(5 * x) + m.cos(10 * x) + x * x/10
+    >>> f"{line_length(f, 0.0, 10.0, 10000):.6f}"
+    '69.534930'
+    """
+
+    x1 = x_start
+    fx1 = fnc(x_start)
+    length = 0.0
+
+    for i in range(steps):
+
+        # Approximates curve as a sequence of linear lines and sums their length
+        x2 = (x_end - x_start) / steps + x1
+        fx2 = fnc(x2)
+        length += m.hypot(x2 - x1, fx2 - fx1)
+
+        # Increment step
+        x1 = x2
+        fx1 = fx2
+
+    return length
+
+if __name__ == "__main__":
+
+    def f(x):
+        return m.sin(10*x)
+
+    print("f(x) = sin(10 * x)")
+    print("The length of the curve from x = -10 to x = 10 is:")
+    i = 10
+    while i <= 100000:
+        print(f"With {i} steps: {line_length(f, -10, 10, i)}")
+        i *= 10

maths/numerical_integration.py

@@ -0,0 +1,63 @@
+"""
+Approximates the area under the curve using the trapezoidal rule
+"""
+
+from typing import Callable, Union
+
+def trapezoidal_area(fnc: Callable[[Union[int, float]], Union[int, float]],
+                     x_start: Union[int, float],
+                     x_end: Union[int, float],
+                     steps: int = 100) -> float:
+
+    """
+    Treats curve as a collection of linear lines and sums the area of the
+    trapezium shape they form
+    :param fnc: a function which defines a curve
+    :param x_start: left end point to indicate the start of line segment
+    :param x_end: right end point to indicate end of line segment
+    :param steps: an accuracy gauge; more steps increases the accuracy
+    :return: a float representing the length of the curve
+
+    >>> def f(x):
+    ...    return 5
+    >>> '%.3f' % trapezoidal_area(f, 12.0, 14.0, 1000)
+    '10.000'
+
+    >>> def f(x):
+    ...    return 9*x**2
+    >>> '%.4f' % trapezoidal_area(f, -4.0, 0, 10000)
+    '192.0000'
+
+    >>> '%.4f' % trapezoidal_area(f, -4.0, 4.0, 10000)
+    '384.0000'
+    """
+    x1 = x_start
+    fx1 = fnc(x_start)
+    area = 0.0
+
+    for i in range(steps):
+
+        # Approximates small segments of curve as linear and solve
+        # for trapezoidal area
+        x2 = (x_end - x_start)/steps + x1
+        fx2 = fnc(x2)
+        area += abs(fx2 + fx1) * (x2 - x1)/2
+
+        # Increment step
+        x1 = x2
+        fx1 = fx2
+    return area
+
+
+if __name__ == "__main__":
+
+    def f(x):
+        return x**3
+
+    print("f(x) = x^3")
+    print("The area between the curve, x = -10, x = 10 and the x axis is:")
+    i = 10
+    while i <= 100000:
+        area = trapezoidal_area(f, -5, 5, i)
+        print("with {} steps: {}".format(i, area))
+        i*=10