def is_hexagonal_number(n):
'''
tests whether number is a hexagonal number using quadratic formula
problem 45
'''
roots = solve_quadratic(2,-1,-n)
return True if [r for r in roots if r > 0 and r == int(r)] else False
def is_triangle_number(n):
'''
tests whether number is a triangle number using quadratic formula
problem 45
'''
roots = solve_quadratic(1,1,-2*n)
return True if [r for r in roots if r > 0 and r == int(r)] else False
def has_triple(p):
m_max = int(p**.5)
m_min = int(max(solve_quadratic(2,-1,-p)))
for m in range(m_min,m_max+1):
for n in range(1 if m % 2 == 0 else 2,m,2):
if gcd(m,n) > 1: continue
if m*(m+n) == p: return 1
return 0
def intersect(self, ray_direction, ray_origin):
'''Contruct and solve a quadratic equation to find the intersection of the given ray and this sphere'''
a = ray_direction.length**2
to_sphere = ray_origin - self.position
b = (ray_direction*to_sphere).product*2
c = (to_sphere.length**2) - (self.radius**2)
ts = solve_quadratic(a, b, c)
return ts
def points_of_reflection(a_x = 0,a_y = 10.1,b_x=1.4,b_y=-9.6):
'''
problem 144
'''
points = []
while b_x < -0.01 or b_x > 0.01 or b_y < 0:
ab_slope = (b_y - a_y)/(b_x - a_x) #slope of incoming line AB
points.append((b_x,b_y)) #current reflection at bx,by
tang_slope = -4*b_x/b_y #slope of tang at bx,by
#find bc slope (angle of reflection)
trig_ident = (ab_slope - tang_slope)/(1+ab_slope*tang_slope)
bc_slope = (tang_slope - trig_ident) / (1+trig_ident*tang_slope) #trig identities (slope = tan) gives us outgoing bc_slope
#use bc_slope to find new points cx,cy
bc_intercept = b_y-(bc_slope*b_x) #use known points on BC (bx,by) and slope BC to solve b = y-mx
new_xs = solve_quadratic(4+bc_slope*bc_slope,2*bc_intercept*bc_slope,bc_intercept*bc_intercept-100) #find intersection by plugging values into equation for ellipse
c_x = new_xs[0] if abs(b_x-new_xs[0]) > abs(b_x-new_xs[1]) else new_xs[1] #pick value which is not current one
#move everything along (b_y = mx+b or b_slope*c_x+intercept)
a_x,a_y,b_x,b_y = b_x,b_y,c_x,bc_slope*c_x + bc_intercept
return points
