project_euler.problem_587.sol1
==============================

.. py:module:: project_euler.problem_587.sol1

.. autoapi-nested-parse::

   Project Euler Problem 587: https://projecteuler.net/problem=587

   A square is drawn around a circle as shown in the diagram below on the left.
   We shall call the blue shaded region the L-section.
   A line is drawn from the bottom left of the square to the top right
   as shown in the diagram on the right.
   We shall call the orange shaded region a concave triangle.

   It should be clear that the concave triangle occupies exactly half of the L-section.

   Two circles are placed next to each other horizontally,
   a rectangle is drawn around both circles, and
   a line is drawn from the bottom left to the top right as shown in the diagram below.

   This time the concave triangle occupies approximately 36.46% of the L-section.

   If n circles are placed next to each other horizontally,
   a rectangle is drawn around the n circles, and
   a line is drawn from the bottom left to the top right,
   then it can be shown that the least value of n
   for which the concave triangle occupies less than 10% of the L-section is n = 15.

   What is the least value of n
   for which the concave triangle occupies less than 0.1% of the L-section?



Functions
---------

.. autoapisummary::

   project_euler.problem_587.sol1.circle_bottom_arc_integral
   project_euler.problem_587.sol1.concave_triangle_area
   project_euler.problem_587.sol1.solution


Module Contents
---------------

.. py:function:: circle_bottom_arc_integral(point: float) -> float

   Returns integral of circle bottom arc y = 1 / 2 - sqrt(1 / 4 - (x - 1 / 2) ^ 2)

   >>> circle_bottom_arc_integral(0)
   0.39269908169872414

   >>> circle_bottom_arc_integral(1 / 2)
   0.44634954084936207

   >>> circle_bottom_arc_integral(1)
   0.5


.. py:function:: concave_triangle_area(circles_number: int) -> float

   Returns area of concave triangle

   >>> concave_triangle_area(1)
   0.026825229575318944

   >>> concave_triangle_area(2)
   0.01956236140083944


.. py:function:: solution(fraction: float = 1 / 1000) -> int

   Returns least value of n
   for which the concave triangle occupies less than fraction of the L-section

   >>> solution(1 / 10)
   15