def _to_extra(head: Expansion, coordinate: Coordinate,
coordinate_tail: Coordinate, left_coordinate: Coordinate,
left_squared_length: Expansion, right_coordinate: Coordinate,
right_squared_length: Expansion) -> Expansion:
second_addend = scale_expansion(
scale_expansion(right_squared_length, coordinate_tail),
left_coordinate)
first_addend = scale_expansion(head, 2 * coordinate)
minuend = sum_expansions(first_addend, second_addend)
subtrahend = scale_expansion(
scale_expansion(left_squared_length, coordinate_tail),
-right_coordinate)
return sum_expansions(subtrahend, minuend)
def centroid(point_cls: Type[Point],
vertices: Sequence[Point],
inverse: Callable[[int], Fraction] = Fraction(1).__truediv__
) -> Point:
double_area = x_numerator = y_numerator = (0,)
prev = vertices[-1]
prev_x, prev_y = prev.x, prev.y
for vertex in vertices:
x, y = vertex.x, vertex.y
area_component = to_cross_product(prev_x, prev_y, x, y)
double_area = sum_expansions(double_area, area_component)
x_numerator = sum_expansions(x_numerator,
scale_expansion(area_component,
prev_x + x))
y_numerator = sum_expansions(y_numerator,
scale_expansion(area_component,
prev_y + y))
prev_x, prev_y = x, y
inverted_denominator = inverse(3 * double_area[-1])
return point_cls(x_numerator[-1] * inverted_denominator,
y_numerator[-1] * inverted_denominator)
def _to_dy_extras(expansion: Expansion, dy_head: Coordinate,
dy_tail: Coordinate, rest_crossed_tails_head: Expansion,
rest_crossed_tails_tail: Expansion) -> Iterable[Expansion]:
dy_tail_rest_crossed_tails = scale_expansion(rest_crossed_tails_head,
dy_tail)
yield sum_expansions(
scale_expansion(expansion, dy_tail),
scale_expansion(dy_tail_rest_crossed_tails, 2 * dy_head))
first_addend = scale_expansion(dy_tail_rest_crossed_tails, dy_tail)
dy_tail_rest_crossed_tails_tail = scale_expansion(rest_crossed_tails_tail,
dy_tail)
second_addend = sum_expansions(
scale_expansion(dy_tail_rest_crossed_tails_tail, 2 * dy_head),
scale_expansion(dy_tail_rest_crossed_tails_tail, dy_tail))
yield sum_expansions(first_addend, second_addend)
def _adjusted_determinant(first_x: Coordinate, first_y: Coordinate,
second_x: Coordinate, second_y: Coordinate,
third_x: Coordinate, third_y: Coordinate,
fourth_x: Coordinate, fourth_y: Coordinate,
upper_bound: Coordinate) -> Coordinate:
first_dx_head, first_dy_head = first_x - fourth_x, first_y - fourth_y
second_dx_head, second_dy_head = second_x - fourth_x, second_y - fourth_y
third_dx_head, third_dy_head = third_x - fourth_x, third_y - fourth_y
first_second_cross_product = to_cross_product(first_dx_head, first_dy_head,
second_dx_head,
second_dy_head)
second_third_cross_product = to_cross_product(second_dx_head,
second_dy_head,
third_dx_head, third_dy_head)
third_first_cross_product = to_cross_product(third_dx_head, third_dy_head,
first_dx_head, first_dy_head)
result_expansion = sum_expansions(
sum_expansions(
_multiply_by_squared_length(second_third_cross_product,
first_dx_head, first_dy_head),
_multiply_by_squared_length(third_first_cross_product,
second_dx_head, second_dy_head)),
_multiply_by_squared_length(first_second_cross_product, third_dx_head,
third_dy_head))
result = sum(result_expansion)
error_bound = bounds.to_point_point_point_second_error(upper_bound)
if result >= error_bound or -result >= error_bound:
return result
first_dx_tail = two_sub_tail(first_x, fourth_x, first_dx_head)
first_dy_tail = two_sub_tail(first_y, fourth_y, first_dy_head)
second_dx_tail = two_sub_tail(second_x, fourth_x, second_dx_head)
second_dy_tail = two_sub_tail(second_y, fourth_y, second_dy_head)
third_dx_tail = two_sub_tail(third_x, fourth_x, third_dx_head)
third_dy_tail = two_sub_tail(third_y, fourth_y, third_dy_head)
if not (first_dx_tail or first_dy_tail or second_dx_tail or second_dy_tail
or third_dx_tail or third_dy_tail):
return result
error_bound = (bounds.to_point_point_point_third_error(upper_bound) +
bounds.to_point_point_point_determinant_error(result))
result += (_to_addend(first_dx_head, first_dx_tail, first_dy_head,
first_dy_tail, second_dx_head, second_dx_tail,
second_dy_head, second_dy_tail, third_dx_head,
third_dx_tail, third_dy_head, third_dy_tail) +
_to_addend(second_dx_head, second_dx_tail, second_dy_head,
second_dy_tail, third_dx_head, third_dx_tail,
third_dy_head, third_dy_tail, first_dx_head,
first_dx_tail, first_dy_head, first_dy_tail) +
_to_addend(third_dx_head, third_dx_tail, third_dy_head,
third_dy_tail, first_dx_head, first_dx_tail,
first_dy_head, first_dy_tail, second_dx_head,
second_dx_tail, second_dy_head, second_dy_tail))
if result >= error_bound or -result >= error_bound:
return result
first_squared_length = (_to_squared_length(first_dx_head, first_dy_head) if
(second_dx_tail or second_dy_tail or third_dx_tail
or third_dy_tail) else (0, ) * 4)
second_squared_length = (
_to_squared_length(second_dx_head, second_dy_head) if
(first_dx_tail or first_dy_tail or third_dx_tail or third_dy_tail) else
(0, ) * 4)
third_squared_length = (_to_squared_length(third_dx_head, third_dy_head) if
(first_dx_tail or first_dy_tail or second_dx_tail
or second_dy_tail) else (0, ) * 4)
if first_dx_tail or first_dy_tail:
if second_dx_tail or second_dy_tail or third_dx_tail or third_dy_tail:
second_third_crossed_tails_tail, second_third_crossed_tails = (
_to_crossed_tails(second_dx_head, second_dx_tail,
second_dy_head, second_dy_tail,
third_dx_head, third_dx_tail, third_dy_head,
third_dy_tail))
else:
second_third_crossed_tails_tail = second_third_crossed_tails = (
0, )
if first_dx_tail:
first_dx_tail_second_third_cross_product = scale_expansion(
second_third_cross_product, first_dx_tail)
result_expansion = reduce(
sum_expansions,
_to_dx_extras(first_dx_tail_second_third_cross_product,
first_dx_head, first_dx_tail, second_dy_tail,
second_squared_length, third_dy_tail,
third_squared_length, second_third_crossed_tails,
second_third_crossed_tails_tail),
sum_expansions(
result_expansion,
_to_extra(first_dx_tail_second_third_cross_product,
first_dx_head, first_dx_tail, second_dy_head,
second_squared_length, third_dy_head,
third_squared_length)))
if first_dy_tail:
first_dy_tail_second_third_cross_product = scale_expansion(
second_third_cross_product, first_dy_tail)
result_expansion = reduce(
sum_expansions,
_to_dy_extras(first_dy_tail_second_third_cross_product,
first_dy_head, first_dy_tail,
second_third_crossed_tails,
second_third_crossed_tails_tail),
sum_expansions(
result_expansion,
_to_extra(first_dy_tail_second_third_cross_product,
first_dy_head, first_dy_tail, third_dx_head,
third_squared_length, second_dx_head,
second_squared_length)))
if second_dx_tail or second_dy_tail:
if first_dx_tail or first_dy_tail or third_dx_tail or third_dy_tail:
third_first_crossed_tails_tail, third_first_crossed_tails = (
_to_crossed_tails(third_dx_head, third_dx_tail, third_dy_head,
third_dy_tail, first_dx_head, first_dx_tail,
first_dy_head, first_dy_tail))
else:
third_first_crossed_tails_tail = third_first_crossed_tails = (0, )
if second_dx_tail:
second_dx_tail_third_first_cross_product = scale_expansion(
third_first_cross_product, second_dx_tail)
result_expansion = reduce(
sum_expansions,
_to_dx_extras(second_dx_tail_third_first_cross_product,
second_dx_head, second_dx_tail, third_dy_tail,
third_squared_length, first_dy_tail,
first_squared_length, third_first_crossed_tails,
third_first_crossed_tails_tail),
sum_expansions(
result_expansion,
_to_extra(second_dx_tail_third_first_cross_product,
second_dx_head, second_dx_tail, third_dy_head,
third_squared_length, first_dy_head,
first_squared_length)))
if second_dy_tail:
second_dy_tail_third_first_cross_product = scale_expansion(
third_first_cross_product, second_dy_tail)
result_expansion = reduce(
sum_expansions,
_to_dy_extras(second_dy_tail_third_first_cross_product,
second_dy_head, second_dy_tail,
third_first_crossed_tails,
third_first_crossed_tails_tail),
sum_expansions(
result_expansion,
_to_extra(second_dy_tail_third_first_cross_product,
second_dy_head, second_dy_tail, first_dx_head,
first_squared_length, third_dx_head,
third_squared_length)))
if third_dx_tail or third_dy_tail:
if first_dx_tail or first_dy_tail or second_dx_tail or second_dy_tail:
first_second_crossed_tails_tail, first_second_crossed_tails = (
_to_crossed_tails(first_dx_head, first_dx_tail, first_dy_head,
first_dy_tail, second_dx_head,
second_dx_tail, second_dy_head,
second_dy_tail))
else:
first_second_crossed_tails_tail = first_second_crossed_tails = (
0, )
if third_dx_tail:
third_dx_tail_first_second_cross_product = scale_expansion(
first_second_cross_product, third_dx_tail)
result_expansion = reduce(
sum_expansions,
_to_dx_extras(third_dx_tail_first_second_cross_product,
third_dx_head, third_dx_tail, first_dy_tail,
first_squared_length, second_dy_tail,
second_squared_length,
first_second_crossed_tails,
first_second_crossed_tails_tail),
sum_expansions(
result_expansion,
_to_extra(third_dx_tail_first_second_cross_product,
third_dx_head, third_dx_tail, first_dy_head,
first_squared_length, second_dy_head,
second_squared_length)))
if third_dy_tail:
third_dy_tail_first_second_cross_product = scale_expansion(
first_second_cross_product, third_dy_tail)
result_expansion = reduce(
sum_expansions,
_to_dy_extras(third_dy_tail_first_second_cross_product,
third_dy_head, third_dy_tail,
first_second_crossed_tails,
first_second_crossed_tails_tail),
sum_expansions(
result_expansion,
_to_extra(third_dy_tail_first_second_cross_product,
third_dy_head, third_dy_tail, second_dx_head,
second_squared_length, first_dx_head,
first_squared_length)))
return result_expansion[-1]
def _multiply_by_squared_length(expansion: Expansion, dx_head: Coordinate,
dy_head: Coordinate) -> Expansion:
return sum_expansions(
scale_expansion(scale_expansion(expansion, dx_head), dx_head),
scale_expansion(scale_expansion(expansion, dy_head), dy_head))
def _to_dx_extras(
head: Expansion, dx_head: Coordinate, dx_tail: Coordinate,
left_dy_tail: Coordinate, left_squared_length: Expansion,
right_dy_tail: Coordinate, right_squared_length: Expansion,
left_right_crossed_tails_head: Expansion,
left_right_crossed_tails_tail: Expansion) -> Iterable[Expansion]:
dx_tail_left_right_crossed_tails = scale_expansion(
left_right_crossed_tails_head, dx_tail)
yield sum_expansions(
scale_expansion(head, dx_tail),
scale_expansion(dx_tail_left_right_crossed_tails, 2 * dx_head))
if left_dy_tail:
yield scale_expansion(scale_expansion(right_squared_length, dx_tail),
left_dy_tail)
if right_dy_tail:
yield scale_expansion(scale_expansion(left_squared_length, -dx_tail),
right_dy_tail)
first_addend = scale_expansion(dx_tail_left_right_crossed_tails, dx_tail)
dx_tail_left_right_crossed_tails_tail = scale_expansion(
left_right_crossed_tails_tail, dx_tail)
second_addend = sum_expansions(
scale_expansion(dx_tail_left_right_crossed_tails_tail, 2 * dx_head),
scale_expansion(dx_tail_left_right_crossed_tails_tail, dx_tail))
yield sum_expansions(first_addend, second_addend)