2316-A Vector-Valued Function in $\mathbb{R}^3$.sagews

A Vector-Valued Function in $\mathbb{R}^3$

$r(t) = (\cos t, \sin t, t)$.

t = var('t')
parametric_plot3d((cos(t), sin(t), t), (t,0,4*pi), color='purple', aspect_ratio=1)

A Vector-Valued Function in $\mathbb{R}^3$, with Thicker Plot

$r(t) = (t, t^2, t^3)$.

t = var('t')
parametric_plot3d((t, t^2, t^3), (t,-2,2), thickness=2, color='purple', aspect_ratio=1)

Two Surfaces in $\mathbb{R}^3$ that Intersect in a Curve

The graph of $x^2 + y^2 = 1$ is in blue.

The graph of $x^2 - z = 0$ is in yellow.

The intersection, in purple, is the curve given by the vector-valued function $r(t) = (\sin t, \cos t, {\sin}^2 t)$.

x, y, z, t = var('x,y,z,t')
P = parametric_plot3d((sin(t), cos(t), (sin(t))^2), (t,-pi,pi), color='purple', thickness=2, aspect_ratio=1)
Q = implicit_plot3d(x^2 - z==0, (x, -4, 4), (y, -4, 4), (z, -4, 4), color='yellow', opacity=0.7, aspect_ratio=1)
R = implicit_plot3d(x^2 + y^2==1, (x, -4, 4), (y, -4, 4), (z, -4, 4), opacity=0.7, aspect_ratio=1)
P + Q + R