def __init__(self, normal_vector=None, constant_term=None):
self.dimension = 3
if not normal_vector:
all_zeros = ['0'] * self.dimension
normal_vector = Vector(all_zeros)
self.normal_vector = Vector(normal_vector)
if not constant_term:
constant_term = Decimal('0')
self.constant_term = Decimal(constant_term)
self.set_basepoint()
def __init__(self,
normal_vector=None,
constant_term=None): #normal_vector：法向量 constant_term：参数
self.dimension = 2
if not normal_vector:
all_zeros = ['0'] * self.dimension
normal_vector = Vector(all_zeros)
self.normal_vector = Vector(normal_vector)
#法向量的向量形式存储
if not constant_term:
constant_term = Decimal('0')
self.constant_term = Decimal(constant_term)
self.set_basepoint()
def set_basepoint(self):
try:
n = self.normal_vector
c = self.constant_term
basepoint_coords = ['0'] * self.dimension
initial_index = Line.first_nonzero_index(n)
initial_coefficient = n[initial_index]
basepoint_coords[initial_index] = c / initial_coefficient
self.basepoint = Vector(basepoint_coords)
except Exception as e:
if str(e) == Line.NO_NONZERO_ELTS_FOUND_MSG:
self.basepoint = None
else:
raise e
def calc_cross(self, l): #在2个向量既不平行也不重合的前提下，计算向量交点
try:
result = self.is_parallel_to(l)
if (not result):
#获取2个法向量的坐标
a1 = self[0]
b1 = self[1]
a2 = l[0]
b2 = l[1]
#获取2个常参数
k1 = self.constant_term
k2 = l.constant_term
x = (b2 * k1 - b1 * k2) / (a1 * b2 - a2 * b1)
y = (a1 * k2 - a2 * k1) / (a1 * b2 - a2 * b1)
return Vector((x, y))
else:
raise Exception(self.LINE_IS_PARALLEL)
except Exception as e:
if str(e) == self.LINE_IS_PARALLEL:
raise Exception(self.LINE_IS_PARALLEL)
else:
raise e
class Line(object):
NO_NONZERO_ELTS_FOUND_MSG = 'No nonzero elements found'
LINE_IS_PARALLEL = 'Line is parallel'
def __init__(self,
normal_vector=None,
constant_term=None): #normal_vector：法向量 constant_term：参数
self.dimension = 2
if not normal_vector:
all_zeros = ['0'] * self.dimension
normal_vector = Vector(all_zeros)
self.normal_vector = Vector(normal_vector)
#法向量的向量形式存储
if not constant_term:
constant_term = Decimal('0')
self.constant_term = Decimal(constant_term)
self.set_basepoint()
def set_basepoint(self):
try:
n = self.normal_vector
c = self.constant_term
basepoint_coords = ['0'] * self.dimension
initial_index = Line.first_nonzero_index(n)
initial_coefficient = n[initial_index]
basepoint_coords[initial_index] = c / initial_coefficient
self.basepoint = Vector(basepoint_coords)
except Exception as e:
if str(e) == Line.NO_NONZERO_ELTS_FOUND_MSG:
self.basepoint = None
else:
raise e
def __str__(self):
num_decimal_places = 3
def write_coefficient(coefficient, is_initial_term=False):
coefficient = round(coefficient, num_decimal_places)
if coefficient % 1 == 0:
coefficient = int(coefficient)
output = ''
if coefficient < 0:
output += '-'
if coefficient > 0 and not is_initial_term:
output += '+'
if not is_initial_term:
output += ' '
if abs(coefficient) != 1:
output += '{}'.format(abs(coefficient))
return output
n = self.normal_vector
try:
initial_index = Line.first_nonzero_index(n)
terms = [
write_coefficient(n[i], is_initial_term=(i == initial_index)) +
'x_{}'.format(i + 1) for i in range(self.dimension)
if round(n[i], num_decimal_places) != 0
]
output = ' '.join(terms)
except Exception as e:
if str(e) == self.NO_NONZERO_ELTS_FOUND_MSG:
output = '0'
else:
raise e
constant = round(self.constant_term, num_decimal_places)
if constant % 1 == 0:
constant = int(constant)
output += ' = {}'.format(constant)
return output
@staticmethod
def first_nonzero_index(iterable):
for k, item in enumerate(iterable):
if not MyDecimal(item).is_near_zero():
return k
raise Exception(Line.NO_NONZERO_ELTS_FOUND_MSG)
# 以下代码均由本人编写
#返回法向量的坐标值
def __getitem__(self, item):
return self.normal_vector.coordinates[item]
#判断两直线是否重合
def __eq__(self, l):
if (self.normal_vector.is_parallel_to(l.normal_vector)):
minus_vector = self.basepoint.minus(l.basepoint)
if (minus_vector.is_zero() or minus_vector.is_orthogonal_to(
l.normal_vector)): #判断该向量是否与当前法向量正交
return True
else:
return False
else:
return False
def is_parallel_to(self, l):
return (self.normal_vector.is_parallel_to(l.normal_vector))
def calc_cross(self, l): #在2个向量既不平行也不重合的前提下，计算向量交点
try:
result = self.is_parallel_to(l)
if (not result):
#获取2个法向量的坐标
a1 = self[0]
b1 = self[1]
a2 = l[0]
b2 = l[1]
#获取2个常参数
k1 = self.constant_term
k2 = l.constant_term
x = (b2 * k1 - b1 * k2) / (a1 * b2 - a2 * b1)
y = (a1 * k2 - a2 * k1) / (a1 * b2 - a2 * b1)
return Vector((x, y))
else:
raise Exception(self.LINE_IS_PARALLEL)
except Exception as e:
if str(e) == self.LINE_IS_PARALLEL:
raise Exception(self.LINE_IS_PARALLEL)
else:
raise e
def test3(): #线性方程组化简为行简化阶梯方程组
p1 = Plain(normal_vector=Vector(['1', '1', '1']), constant_term='1')
p2 = Plain(normal_vector=Vector(['0', '1', '1']), constant_term='2')
s = LinearSystem([p1, p2])
r = s.compute_rref()
if not (r[0] == Plain(normal_vector=Vector(['1', '0', '0']),
constant_term='-1') and r[1] == p2):
print('test case 1 failed')
else:
print('test case 1 successed')
LinearSystem_print(s, '原始方程')
LinearSystem_print(r, '行简化阶梯化后方程')
p1 = Plain(normal_vector=Vector(['1', '1', '1']), constant_term='1')
p2 = Plain(normal_vector=Vector(['1', '1', '1']), constant_term='2')
s = LinearSystem([p1, p2])
r = s.compute_rref()
if not (r[0] == p1 and r[1] == Plain(constant_term='1')):
print('test case 2 failed')
else:
print('test case 2 successed')
LinearSystem_print(s, '原始方程')
LinearSystem_print(r, '行简化阶梯化后方程')
p1 = Plain(normal_vector=Vector(['1', '1', '1']), constant_term='1')
p2 = Plain(normal_vector=Vector(['0', '1', '0']), constant_term='2')
p3 = Plain(normal_vector=Vector(['1', '1', '-1']), constant_term='3')
p4 = Plain(normal_vector=Vector(['1', '0', '-2']), constant_term='2')
s = LinearSystem([p1, p2, p3, p4])
r = s.compute_rref()
if not (r[0] == Plain(normal_vector=Vector(['1', '0', '0']),
constant_term='0') and r[1] == p2
and r[2] == Plain(normal_vector=Vector(['0', '0', '-2']),
constant_term='2') and r[3] == Plain()):
print('test case 3 failed')
else:
print('test case 3 successed')
LinearSystem_print(s, '原始方程')
LinearSystem_print(r, '行简化阶梯化后方程')
p1 = Plain(normal_vector=Vector(['0', '1', '1']), constant_term='1')
p2 = Plain(normal_vector=Vector(['1', '-1', '1']), constant_term='2')
p3 = Plain(normal_vector=Vector(['1', '2', '-5']), constant_term='3')
s = LinearSystem([p1, p2, p3])
r = s.compute_rref()
if not (r[0] == Plain(normal_vector=Vector(['1', '0', '0']),
constant_term=Decimal('23') / Decimal('9'))
and r[1] == Plain(normal_vector=Vector(['0', '1', '0']),
constant_term=Decimal('7') / Decimal('9'))
and r[2] == Plain(normal_vector=Vector(['0', '0', '1']),
constant_term=Decimal('2') / Decimal('9'))):
print('test case 4 failed')
else:
print('test case 4 successed')
LinearSystem_print(s, '原始方程')
LinearSystem_print(r, '行简化阶梯化后方程')
def test2(): #线性方程组阶梯化
p1 = Plain(normal_vector=Vector(['1', '1', '1']), constant_term='1')
p2 = Plain(normal_vector=Vector(['0', '1', '1']), constant_term='2')
s = LinearSystem([p1, p2])
t = s.compute_triangular_form()
print('*******************************')
if not (t[0] == p1 and t[1] == p2):
print('test case 1 failed')
else:
print('test case 1 successed')
LinearSystem_print(s, '原始方程')
LinearSystem_print(t, '阶梯化后方程')
print('*******************************')
p1 = Plain(normal_vector=Vector(['1', '1', '1']), constant_term='1')
p2 = Plain(normal_vector=Vector(['1', '1', '1']), constant_term='2')
s = LinearSystem([p1, p2])
t = s.compute_triangular_form()
print('*******************************')
if not (t[0] == p1 and t[1] == Plain(constant_term='1')):
print('test case 2 failed')
else:
print('test case 2 successed')
LinearSystem_print(s, '原始方程')
LinearSystem_print(t, '阶梯化后方程')
print('*******************************')
p1 = Plain(normal_vector=Vector(['1', '1', '1']), constant_term='1')
p2 = Plain(normal_vector=Vector(['0', '1', '0']), constant_term='2')
p3 = Plain(normal_vector=Vector(['1', '1', '-1']), constant_term='3')
p4 = Plain(normal_vector=Vector(['1', '0', '-2']), constant_term='2')
s = LinearSystem([p1, p2, p3, p4])
t = s.compute_triangular_form()
print('*******************************')
if not (t[0] == p1 and t[1] == p2
and t[2] == Plain(normal_vector=Vector(['0', '0', '-2']),
constant_term='2') and t[3] == Plain()):
print('test case 3 failed')
else:
print('test case 3 successed')
LinearSystem_print(s, '原始方程')
LinearSystem_print(t, '阶梯化后方程')
print('*******************************')
p1 = Plain(normal_vector=Vector(['0', '1', '1']), constant_term='1')
p2 = Plain(normal_vector=Vector(['1', '-1', '1']), constant_term='2')
p3 = Plain(normal_vector=Vector(['1', '2', '-5']), constant_term='3')
s = LinearSystem([p1, p2, p3])
t = s.compute_triangular_form()
print('*******************************')
if not (t[0] == Plain(normal_vector=Vector(['1', '-1', '1']),
constant_term='2') and t[1]
== Plain(normal_vector=Vector(['0', '1', '1']), constant_term='1')
and t[2] == Plain(normal_vector=Vector(['0', '0', '-9']),
constant_term='-2')):
print('test case 4 failed')
else:
print('test case 4 successed')
LinearSystem_print(s, '原始方程')
LinearSystem_print(t, '阶梯化后方程')
print('*******************************')
def test1(): #测试线性方程组的基本操作
#********************************************
#判断两直线位置关系，如果相交，输出交点
p0 = Plain(normal_vector=Vector(['1', '1', '1']), constant_term='1')
p1 = Plain(normal_vector=Vector(['0', '1', '0']), constant_term='2')
p2 = Plain(normal_vector=Vector(['1', '1', '-1']), constant_term='3')
p3 = Plain(normal_vector=Vector(['1', '0', '-2']), constant_term='2')
s = LinearSystem([p0, p1, p2, p3])
print(s.indices_of_first_nonzero_terms_in_each_row())
print('{},{},{},{}'.format(s[0], s[1], s[2], s[3]))
print(len(s))
print(s)
s[0] = p1
print(s)
print(MyDecimal('1e-9').is_near_zero())
print(MyDecimal('1e-11').is_near_zero())
p0 = Plain(normal_vector=Vector(['1', '1', '1']), constant_term='1')
p1 = Plain(normal_vector=Vector(['0', '1', '0']), constant_term='2')
p2 = Plain(normal_vector=Vector(['1', '1', '-1']), constant_term='3')
p3 = Plain(normal_vector=Vector(['1', '0', '-2']), constant_term='2')
s = LinearSystem([p0, p1, p2, p3])
LinearSystem_print(s, '原始方程组')
#交换等式
s.swap_rows(0, 1)
print(s[2].normal_vector.is_parallel_to(p2.normal_vector))
if not (s[0] == p1 and s[1] == p0 and s[2] == p2 and s[3] == p3):
print('test case 1 failed')
else:
print('test case 1 successed')
LinearSystem_print(s, '交换1-2行')
s.swap_rows(1, 3)
if not (s[0] == p1 and s[1] == p3 and s[2] == p2 and s[3] == p0):
print('test case 2 failed')
else:
print('test case 2 successed')
LinearSystem_print(s, '交换2-4行')
s.swap_rows(3, 1)
if not (s[0] == p1 and s[1] == p0 and s[2] == p2 and s[3] == p3):
print('test case 3 failed')
else:
print('test case 3 successed')
LinearSystem_print(s, '交换4-2行')
s.multiply_coefficient_and_row(1, 0)
if not (s[0] == p1 and s[1] == p0 and s[2] == p2 and s[3] == p3):
print('test case 4 failed')
else:
print('test case 4 successed')
LinearSystem_print(s, '方程1乘1')
s.multiply_coefficient_and_row(-1, 2)
if not (s[0] == p1 and s[1] == p0
and s[2] == Plain(normal_vector=Vector(['-1', '-1', '1']),
constant_term='-3') and s[3] == p3):
print('test case 5 failed')
else:
print('test case 5 successed')
LinearSystem_print(s, '方程3乘-1')
s.multiply_coefficient_and_row(10, 1)
if not (s[0] == p1 and s[1] == Plain(
normal_vector=Vector(['10', '10', '10']), constant_term='10')
and s[2] == Plain(normal_vector=Vector(['-1', '-1', '1']),
constant_term='-3') and s[3] == p3):
print('test case 6 failed')
else:
print('test case 6 successed')
LinearSystem_print(s, '方程2乘10')
s.add_multiple_times_row_to_row(0, 0, 1)
if not (s[0] == p1 and s[1] == Plain(
normal_vector=Vector(['10', '10', '10']), constant_term='10')
and s[2] == Plain(normal_vector=Vector(['-1', '-1', '1']),
constant_term='-3') and s[3] == p3):
print('test case 7 failed')
else:
print('test case 7 successed')
LinearSystem_print(s, '方程1乘0，加到方程2')
s.add_multiple_times_row_to_row(1, 0, 1)
if not (s[0] == p1 and s[1] == Plain(
normal_vector=Vector(['10', '11', '10']), constant_term='12')
and s[2] == Plain(normal_vector=Vector(['-1', '-1', '1']),
constant_term='-3') and s[3] == p3):
print('test case 8 failed')
else:
print('test case 8 successed')
LinearSystem_print(s, '方程1乘1，加到方程2')
s.add_multiple_times_row_to_row(-1, 1, 0)
if not (s[0] == Plain(normal_vector=Vector(['-10', '-10', '-10']),
constant_term='-10')
and s[1] == Plain(normal_vector=Vector(['10', '11', '10']),
constant_term='12')
and s[2] == Plain(normal_vector=Vector(['-1', '-1', '1']),
constant_term='-3') and s[3] == p3):
print('test case 9 failed')
else:
print('test case 9 successed')
LinearSystem_print(s, '方程2乘-1，加到方程1')
class Plain(object):
NO_NONZERO_ELTS_FOUND_MSG = 'No nonzero elements found'
def __init__(self, normal_vector=None, constant_term=None):
self.dimension = 3
if not normal_vector:
all_zeros = ['0'] * self.dimension
normal_vector = Vector(all_zeros)
self.normal_vector = Vector(normal_vector)
if not constant_term:
constant_term = Decimal('0')
self.constant_term = Decimal(constant_term)
self.set_basepoint()
def set_basepoint(self):
try:
n = self.normal_vector
c = self.constant_term
basepoint_coords = ['0'] * self.dimension
initial_index = Plain.first_nonzero_index(n)
initial_coefficient = n[initial_index]
basepoint_coords[initial_index] = c / initial_coefficient
self.basepoint = Vector(basepoint_coords)
except Exception as e:
if str(e) == Plain.NO_NONZERO_ELTS_FOUND_MSG:
self.basepoint = None
else:
raise e
def __str__(self):
num_decimal_places = 3
def write_coefficient(coefficient, is_initial_term=False):
coefficient = round(coefficient, num_decimal_places)
if coefficient % 1 == 0:
coefficient = int(coefficient)
output = ''
if coefficient < 0:
output += '-'
if coefficient > 0 and not is_initial_term:
output += '+'
if not is_initial_term:
output += ' '
if abs(coefficient) != 1:
output += '{}'.format(abs(coefficient))
return output
n = self.normal_vector
try:
initial_index = Plain.first_nonzero_index(n)
terms = [
write_coefficient(n[i], is_initial_term=(i == initial_index)) +
'x_{}'.format(i + 1) for i in range(self.dimension)
if round(n[i], num_decimal_places) != 0
]
output = ' '.join(terms)
except Exception as e:
if str(e) == self.NO_NONZERO_ELTS_FOUND_MSG:
output = '0'
else:
raise e
constant = round(self.constant_term, num_decimal_places)
if constant % 1 == 0:
constant = int(constant)
output += ' = {}'.format(constant)
return output
@staticmethod
def first_nonzero_index(iterable):
for k, item in enumerate(iterable):
if not MyDecimal(item).is_near_zero():
return k
raise Exception(Plain.NO_NONZERO_ELTS_FOUND_MSG)
# 以下代码均由本人编写
#返回法向量的坐标值
def __getitem__(self, item):
return self.normal_vector.coordinates[item]
#判断两平面是否重合
def __eq__(self, l):
if (self.normal_vector.is_zero()
and l.normal_vector.is_zero()): #都是零向量
return True
elif (self.normal_vector.is_zero()
or l.normal_vector.is_zero()): #其中之一是零向量
return False
elif (self.normal_vector.is_parallel_to(l.normal_vector)):
minus_vector = self.basepoint.minus(l.basepoint)
if (minus_vector.is_zero() or minus_vector.is_orthogonal_to(
l.normal_vector)): #判断该向量是否与当前法向量正交
return True
else:
return False
else:
return False
#平面方程倍乘函数 法向量和常数均乘以同一常数
def times_scalar(self, c):
product_vector = self.normal_vector.times_scalar(c)
product_constant = self.constant_term * c
return Plain(product_vector, product_constant)
#平面方程相加，法向量、常数均相加
def plus(self, v):
sum_vector = self.normal_vector.plus(v.normal_vector)
sum_constant = self.constant_term + v.constant_term
return Plain(sum_vector, sum_constant)
#平面方程相减，法向量、常数均相减
def minus(self, v):
sum_vector = self.normal_vector.minus(v.normal_vector)
sum_constant = self.constant_term - v.constant_term
return Plain(sum_vector, sum_constant)
#判断两平面是否平行
def is_parallel_to(self, l):
return (self.normal_vector.is_parallel_to(l.normal_vector))