13-3 Arc Length and Curvature.sagews

var('t')
r = vector((3*cos(t), 2*sin(t)))
@interact
def _(a = slider(0.01, 2*pi, 0.01, 0.01, label = 't')):
    v = r.diff(t)      # velocity
    s = v.norm()       # speed
    T = v/s            # unit tangent
    K = T.diff(t)/s    # curvature vector
    k = K.norm()       # curvature
    c = r(t = a) + (1 / (k(t = a))^2) * K(t = a) # center of osculating circle
    G1 = parametric_plot(r, (t, 0, 2*pi), color = 'blue')
    G2 = circle(c, 1/k(t = a), color = 'green')
    P1 = point(r(t = a), size = 30, color = 'black')
    P2 = point(c, size = 30, color = 'black')
    L = line([r(t = a), c], color = 'black')
    show(G1 + G2 + P1 + P2 + L, xmin = -5, xmax = 5, ymin = -7, ymax = 7)

t

Interact: please open in CoCalc

var('t', 'u')
r = vector((5*cos(t), 5*sin(t), t))
@interact
def _(a = slider(0.01, 4*pi, 0.01, 0.01, label = 't')):
    v = r.diff(t)      # velocity
    s = v.norm()       # speed
    T = v/s            # unit tangent
    K = T.diff(t)/s    # curvature vector
    k = K.norm()       # curvature
    N = K/k            # unit normal
    c = r(t = a) + (1 / (k(t = a))^2) * K(t = a) # center of osculating circle
    G1 = parametric_plot3d(r, (t, 0, 4*pi), color = 'blue')
    G2 = parametric_plot3d(c + (cos(u) / k(t = a))*N(t = a) + (sin(u) / k(t = a))*T(t = a), (u, 0, 2*pi), color = 'green')
    P1 = point(r(t = a), size = 10, color = 'black')
    P2 = point(c, size = 10, color = 'black')
    L = line([r(t = a), c], color = 'black')
    show(G1 + G2 + P1 + P2 + L)

(t, u)

Interact: please open in CoCalc