tangentplane-parametric.jl

# Julia
begin
    using Plots; plotly()

    # radius of the sphere
    r = 2
    # point to find the tangent at
    x_0 = 1
    y_0 = 1
    # working with square roots is a little finicky... 
    z_0 = -sqrt(2)

    # parametric equations of the sphere in terms of theta and phi
	x_sphere(theta, phi) = r*cos(theta) * sin(phi) 
	y_sphere(theta, phi) = r*cos(theta) * cos(phi) 
	z_sphere(theta, phi) = r*sin(theta)
    # the corresponding theta and phi to the point (x_0, y_0, z_0)
    theta_param = asin(z_0/r)
    phi_param = acos(y_0 / (r * cos(theta_param)))
	
    # range of theta and phi to plot the sphere
	theta_range =range(-2pi, stop=2pi, length=100)
	phi_range = range(-2pi, stop=2pi, length=100)
    # plot ranges for sphere
	x_range_sphere = [x_sphere(t, p) for t in theta_range, p in phi_range]
	y_range_sphere = [y_sphere(t, p) for t in theta_range, p in phi_range]
	z_range_sphere = [z_sphere(t, p) for t in theta_range, p in phi_range]
    # sphere plot
	surface(
        x_range_sphere, 
        y_range_sphere, 
        z_range_sphere, 
        label="Sphere", 
        color=RGBA(172/255, 39/255, 245/255, 1),
        legend=:none
    )

    # derivatives the parametric equations of the sphere:
    # in terms of theta
    d_theta_x(theta, phi) = -r*sin(theta) * sin(phi)
    d_theta_y(theta, phi) = -r*sin(theta) * cos(phi)
    d_theta_z(theta, phi) = r*cos(theta)
    # in terms of phi
    d_phi_x(theta, phi) = r*cos(theta) * cos(phi)
    d_phi_y(theta, phi) = -r*cos(theta) * sin(phi)
    d_phi_z(theta, phi) = 0

    # a vector of "d_theta" of the sphere at the point (x_0, y_0, z_0)
    d_theta_vec = [
        d_theta_x(theta_param, phi_param), 
        d_theta_y(theta_param, phi_param), 
        d_theta_z(theta_param, phi_param)
    ]
    # a vector of "d_phi" of the sphere at the point (x_0, y_0, z_0)
    d_phi_vec = [
        d_phi_x(theta_param, phi_param), 
        d_phi_y(theta_param, phi_param), 
        d_phi_z(theta_param, phi_param)
    ]
    # plotting the respective tangent vectors
    plot!(
        [x_0-d_theta_vec[1], x_0], 
        [y_0-d_theta_vec[2], y_0], 
        [-z_0+d_theta_vec[3], -z_0],
        label="Tangent Vector, theta", 
        color=RGBA(0/255, 255, 0/255, 1)
    )
    plot!(
        [x_0, x_0 + d_phi_vec[1]], 
        [y_0, y_0 + d_phi_vec[2]], 
        [-z_0, -z_0 + d_phi_vec[3]], 
        label="Tangent Vector, phi", 
        color=RGBA(255, 0/255, 0/255, 1)
    )

    # the cross product of the two vectors above
    # this returns the line normal to the sphere at the point (x_0, y_0, z_0)
    d_cross = cross( d_theta_vec, d_phi_vec )
    plot!(
        [x_0, x_0+d_cross[1]], 
        [y_0,  y_0+d_cross[2]], 
        [-z_0,  -z_0-d_cross[3]], 
        label="Normal Vector", 
        color=RGBA(0/255, 0/255, 255, 1)
    )
    # the tangent plane: dot product of the cross product
    tangent_plane(x,y) = (
        d_cross[1]*(x-x_0) + 
        d_cross[2]*(y-y_0) - 
        d_cross[3]*z_0
    )/d_cross[3]
    # range of the tangent to plot
	x_range_tangent = range(-3, stop=3, length=100)
	y_range_tangent = range(-3, stop=3, length=100)
    z_range_tangent = [tangent_plane(a,b) for a in x_range_tangent, b in y_range_tangent]
    # plot tangent
	surface!(
        x_range_tangent, 
        y_range_tangent, 
        z_range_tangent, 
        label="Tangent Plane",
        color=RGBA(0, 0, 0, 0.3)
    )
end