line length Algorithm

The line length algorithm is a simple yet effective technique used in various applications such as text processing, computer graphics, and signal processing. It primarily focuses on calculating the length of a line segment, which could either be a straight line connecting two points or a more complex curve represented by a series of points. This algorithm is particularly useful in determining the optimal line breaks in text formatting, measuring distances between points in geospatial analysis, and analyzing waveform signals in biomedical engineering.

In its most basic form, the line length algorithm uses the Cartesian coordinate system to calculate the distance between two points using the Pythagorean theorem. Given two points (x1, y1) and (x2, y2), the line length can be computed as the square root of the sum of the squares of the differences of their corresponding coordinates, i.e., sqrt((x2-x1)^2 + (y2-y1)^2). For more complex curves, the algorithm computes the length of each line segment connecting consecutive points and sums them up to obtain the total length of the curve. The line length algorithm is computationally efficient and straightforward to implement, making it a popular choice for various applications that require distance or length calculations.

 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

line length Algorithm

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	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