2415.sagews

# Rotation anticlockwise through angle t
def rotate(v,t,show_matrix=False):
    M= matrix([[cos(t),-sin(t)],[sin(t),cos(t)]])
    if show_matrix:
        return M
    return M*vector(v)
    
# Reflection through line of angle t from positive x-axis
def reflect(v,t,show_matrix=False):
    M= matrix([[cos(2*t),sin(2*t)],[sin(2*t),-cos(2*t)]])
    if show_matrix:
        return M
    return M*vector(v)

A= (4,0)
B= (4,4)
C= (6,3)
D= (4,2)
points= [A,B,C,D]

# Rotate through 2*pi/3
t= 2*pi/3
points_rot= [rotate(v,t) for v in points]
M= rotate(A,t,show_matrix=True)
#print M
print M.n()
print [v.n() for v in point_rot]
p= line(points)
p+= line(points_rot,color='red')
show(p,aspect_ratio=1)

[-0.500000000000000 -0.866025403784439] [ 0.866025403784439 -0.500000000000000] [(-2.00000000000000, 3.46410161513775), (-5.46410161513775, 1.46410161513775), (-5.59807621135332, 3.69615242270663), (-3.73205080756888, 2.46410161513775)]

# Reflect through y=2*x 
var('x')
s= arccos(1/sqrt(5))
#s= pi/3
#print (s*180/pi).n()
points_ref= [reflect(v,s) for v in points]
N= reflect(A,s,show_matrix=True)
#print N
print N.n()
print [v.n() for v in points_ref]
q= plot(2*x,(x,0,4),color='black')
q+= line(points)
q+= line(points_ref,color='green')
show(q,aspect_ratio=1)

[-0.500000000000000 0.866025403784439] [ 0.866025403784439 0.500000000000000] [(-2.00000000000000, 3.46410161513775), (1.46410161513775, 5.46410161513775), (-0.401923788646684, 6.69615242270663), (-0.267949192431123, 4.46410161513775)]

# Rotate then reflect as above
points_rr= [reflect(rotate(v,t),s) for v in points]
#print N*M
print (N*M).n()
print [v.n() for v in points_rr]
r= plot(2*x,(x,0,4),color='black')
r+= line(points)
r+= line(points_rr,color='green')
show(r,aspect_ratio=1)

[ 1.00000000000000 0.000000000000000] [0.000000000000000 -1.00000000000000] [(4.00000000000000, 0.000000000000000), (4.00000000000000, -4.00000000000000), (6.00000000000000, -3.00000000000000), (4.00000000000000, -2.00000000000000)]