Tangent, Normal, and Binormal

Here, I will plot a curve with unit tangent, normal, and binormal vectors at a point, and show the osculating plane with the osculating circle. We consider [ \mathbf{r}(t) = \langle \cos t, \sin t, t \rangle ] at the point $t = \pi$ from the notes.

Set up position, unit tangent, and unit normal vectors at $t = \pi$

r = vector([-1, 0, pi]);
unitTan = vector([0, -1/sqrt(2), 1/sqrt(2)]);
unitNorm = vector([1, 0, 0]);

Now the binormal:

unitTan.cross_product(unitNorm)

(0, 1/2*sqrt(2), 1/2*sqrt(2))

Plot the curve and vectors:

var("t")
p1 = parametric_plot3d((cos(t), sin(t), t), (t, 0, 2*pi));
unitTanArrow = arrow((-1, 0, pi), (-1, -1/sqrt(2), pi+1/sqrt(2)), color="black");
unitNormArrow = arrow((-1, 0, pi), (0, 0, pi), color="green");
binormalArrow = arrow((-1, 0, pi), (-1, 1/sqrt(2), pi + 1/sqrt(2)), color = "red");
p1 + unitTanArrow + unitNormArrow + binormalArrow

t

3D rendering not yet implemented

To plot the circle, set up the center of the circle, then parametrize circle around that point in the plane shared by unitTan and unitNorm, with radius $2$ . Then plot the plane, and combine with the original curve.

oscCircCenter = r + 2*unitNorm;
pCircle = parametric_plot3d(oscCircCenter + 2*cos(t)*unitNorm + 2*sin(t)*unitTan, (t, 0, 2*pi), color = "red")
var("y"); 
pPlane = plot3d(pi-y, (x, -1, 3), (y, -1.5, 1.5), color="gray", opacity=.6);
p1 + pCircle + pPlane

y

3D rendering not yet implemented

︠4b53298b-f036-43b9-bd8b-db0b4ba53a44︠