modeling_thearts.sagews

Some interesting graphs

(I) Graph each of the following as instructed:

(1) $x=1.5 \cos(t)-\cos(30t)$, $y=1.5\sin(t)-\sin(30t)$,     for $t\in(0,50)$

(2) $x = 4 \sin (t + \cos (100t))$,   $y = 4 \cos (t + \sin (100t))$,     for $t\in(0,10)$

(3) $(x^2+y^2-1)^3 = x^2 y^3$,     with $x \in [-1.5,1.5]$, $y\in[-1.5,1.5]$

(II) Uncomment all lines in the following and see what you get:

# y = var('y')
# mouth = implicit_plot( (-y+2.4)^2*(2.2-x^2)==(x^2+3*(-y+2.4)-2)^2, (x,-6,6), (y,-2,8), linewidth=3, color='red' )
# head = implicit_plot( 10*x^2==y^3*(6-y), (x,-6,6), (y,-2,8), linewidth=2 )
# rear = implicit_plot( ((x-3.6)^2 + (y-4)^2)^2==(x-3.6)^2 - (y-4)^2, (x,-6,6), (y,-2,8), linewidth=2, color='green' )
# lear = implicit_plot( ((x+3.6)^2 + (y-4)^2)^2==(x+3.6)^2 - (y-4)^2, (x,-6,6), (y,-2,8), linewidth=2, color='green' )
# t = var('t')
# reye = parametric_plot( ( 0.02*t*cos(t)+1.5, 0.02*t*sin(t)+4 ), (t,0,20), color='black' )
# leye = parametric_plot( ( 0.02*t*cos(-t+pi)-1.5, 0.02*t*sin(-t+pi)+4 ), (t,0,20), color='black' )
#
# show(mouth+head+lear+rear+reye+leye)

(III) Plot each of the following 3D graphs as intructed:

(1) $x^2+y^2=z^2$,     on $x, y, z \in [-3,3]$

(2) $(x^2+(9/4)y^2+z^2-1)^3 -x^2z^3 - (9/80)y^2 z^3 = 0$,     on $x, y, z \in [-1.5,1.5]$, with red color.

(IV) Uncomment all lines in the following and see what you get:

# def cf(u,ph): return 0.25*sin(u+ph)^2
# t = var('t')
# curve = (t+sin(2*t), t+cos(t))
# revolution_plot3d(curve, (t,-pi,4*pi), parallel_axis='z', color=(cf, colormaps.PiYG), opacity=0.8)

# And here is a very interesting example I found from sagemath.sfasu.edu/home/pub/43/
# It defines a piecewise curve, rotates it about the x-axis, and then computes 
# the volume of the resulting shape by integration.
x = var('x')
f1 = -(1/pi*(x-pi))^2+3
f2 = 0.125*sin(x) + 2.875
f3 = 1.875/(6*pi-22)*(x-22) + 1
f4 = 1
f = piecewise( [ ((0,3*pi/2),f1), ((3*pi/2,6*pi),f2), ((6*pi,22),f3), ((22,23),f4) ] )
plot(f, (x,0,23), aspect_ratio=1)
a = revolution_plot3d(f1, (x,0,3*pi/2), parallel_axis='x', aspect_ratio=1, color='chartreuse')
b = revolution_plot3d(f2, (x,3*pi/2,6*pi), parallel_axis='x', color='violet')
c = revolution_plot3d(f3, (x,6*pi,22), parallel_axis='x', color='aqua')
d = revolution_plot3d(f4, (x,22,23), parallel_axis='x', color='brown')
show(a+b+c+d, frame=False)
# Compute the volume (exclude cap):
vol = integral(pi*(f1)^2, x,0,3*pi/2) + integral(pi*(f2)^2, x,3*pi/2,6*pi) + integral(pi*(f3)^2, x, 6*pi, 22)
print "\n Volume = %f" % vol

3D rendering not yet implemented

Volume = 518.314223