mat-273 final project

mat-273-final.sage

def dog():
    """Plot the word DOG using six parametric graphs"""
    u, v = var('u, v')
    
    # D
    x(u, v) = u
    y(u, v) = -5
    z(u, v) = v
    D_plane = parametric_plot3d( (x, y, z), (u, -7, 7), (v, -10, 10) )
    y(u, v) = -(.1*(u^2))
    D_parabola = parametric_plot3d( (x, y, z), (u, -7, 7), (v, -10, 10) )
    
    # O
    radius = 6
    x(u, v) = radius * cos(v)
    y(u, v) = 7 + radius * sin(v)
    z(u, v) = u
    O_cylinder = parametric_plot3d( (x, y, z), (u, -10, 10), (v, 0, 2*pi) )
    
    # G
    radius = 6
    x(u, v) = radius * cos(v)
    y(u, v) = 20 + (radius * sin(v))
    z(u, v) = u
    G_halfcyl = parametric_plot3d( (x, y, z), (u, -10, 10), (v, pi, 2*pi) )
    G_quartercyl = parametric_plot3d( (x, y, z), (u, -10, 10), (v, 0, pi / 2) )
    
    x(u, v) = 0
    y(u, v) = u
    z(u, v) = v
    G_plane = parametric_plot3d( (x, y, z), (u, 20, 26), (v, -10, 10) )
    
    # plot
    dogplot = D_plane + D_parabola + O_cylinder + G_halfcyl + G_quartercyl + G_plane
    print '\nThe plot of dog using six parametric equations is'
    show(dogplot)

def vector_field():
    """Plot a vector field"""
    x, y = var('x,y')
    vfield = plot_vector_field((sin(x), cos(y)), (x,-5,5), (y,-5,5), color='blue')
    
    print "\n\n2d vector field:"
    show(vfield)
    
def ice_cream():
    """
    Plot an 'ice cream cone' as the composite graph of a cone and a sphere and
    calculate its volume and surface area
    """
    # cone
    u, v = var('u, v')
    height = 5
    radius = 2
    x(u, v) = ((height - u) / height) * radius * cos(v)
    y(u, v) = ((height - u) / height) * radius * sin(v)
    z(u, v) = -u
    # plot of the cone
    cone = parametric_plot3d( (x, y, z), (u, 0, height), (v, 0, 2*pi), color='brown')
    # volume of the cone
    cone_volume = pi * radius^2 * (height / 3)
    # surface area of the cone
    x_prime_u = derivative(x, u)
    y_prime_u = derivative(y, u)
    z_prime_u = derivative(z, u)
    x_prime_v = derivative(x, v)
    y_prime_v = derivative(y, v)
    z_prime_v = derivative(z, v)
    i = (y_prime_u * z_prime_v) - (z_prime_u * y_prime_v)
    j = -(x_prime_u * z_prime_v) - (z_prime_u * x_prime_v)
    k = (x_prime_u * y_prime_v) - (x_prime_v * y_prime_u)
    integrand = sqrt(i^2 + j^2 + k^2)
    cone_surface_area = integrate( integrate(integrand, (v, 0, 2*pi)), (u, 0, 5))
    
    # sphere
    phi, theta = var('phi, theta')
    radius = 2
    x(phi, theta) = radius * cos(theta) * sin(phi)
    y(phi, theta) = radius * sin(theta) * sin(phi)
    z(phi, theta) = .8 + radius * cos(phi)
    # plot of sphere
    sphere = parametric_plot3d( (x, y, z), (phi, 0, 2 * pi), (theta, 0, pi), color='gray')
    # volume of sphere
    sphere_volume = (4/3) * pi * radius^3
    # surface area of sphere
    sphere_surface_area = 4 * pi * radius^2
    
    # print results
    print "Plot of an ice cream cone:"
    show(cone + sphere)
    print "Volume of cone:"
    show(cone_volume)
    print "Surface area of cone:"
    show(cone_surface_area)
    print "Volume of sphere:"
    show(sphere_volume)
    print "Surface area of sphere:"
    show(sphere_surface_area)
    print "Volume of ice cream cone:"
    show(cone_volume + sphere_volume)
    print "Surface area of ice cream cone:"
    show(cone_surface_area + (sphere_surface_area / 2))

def line_integral():
    """Plot a helical space curve and take the line integral of f(x, y, z) = xyz over it"""
    # curve
    t = var('t')
    radius = 3
    stretchiness = .5
    x(t) = radius * cos(t)
    y(t) = radius * sin(t)
    z(t) = stretchiness * t
    curve = parametric_plot3d( (x, y, z), (t, -10, 10) )
    
    # function
    function = x*y*z
    
    # line integral of function over curve
    x_prime = derivative(x, t)
    y_prime = derivative(y, t)
    z_prime = derivative(z, t)
    ds = sqrt(x_prime ^ 2 + y_prime ^2 + z_prime ^2)
    result = integrate(function * ds, t)
    numeric_result = integrate(function * ds, (t, -10, 10))
    
    # and print the results
    print '\nThe line integral of the function'
    show("f(x, y, z) = xyz")
    print 'over the helical space curve defined by parametric equations'
    show("x(t) = 3cos(t)")
    show("y(t) = 3sin(t)")
    show("z(t) = .5t")
    print 'which looks like this'
    show(curve)
    print 'in the range'
    show("-10 \leq t \leq 10")
    print 'is'
    show(numeric_result)
    show("=")
    show(n(numeric_result))
    print 'and its general form is'
    show(result)


# MAT-273 Final Project 0WN3D /)
print '\n' * 5 + '=' * 40 + "MAT 273 Final Project" + '=' * 40

# -plot the word 'dog'
dog()

# -create a vector field in sage
vector_field()

# -plot an 'ice cream cone'
# -find the area inside the ice cream cone
# -find the surface area of the ice cream cone
ice_cream()

# -draw a 3D line and find the value of the line integral
line_integral()

print """
                            ___      _____     ____  
                           / _ \    | ____|   |  _ \ 
                          | | | |   |  _|     | | | |
                          | |_| |_  | |___ _  | |_| |
                           \__\_(_) |_____(_) |____/ 
                                                     
"""