Example finding the osculating plane and TNB, Multivariable Calculus
In this exercise, we start with a parametric curve r(t) and find various vector quantities for the curve geometry, finishing with the osculating plane at t=1. (Not all of this necessary to find just the osculating plane.) We begin with the velocity vector r'(t), followed by calculating the speed, which is the magnitude of the velocity vector. We continue by determining the tangent vector and subsequently the normal vector. The cross product of these vectors yields the binormal vector, which is orthogonal to the osculating plane.
Notes:
Getting the TNB vectors positions us for the osculating circle, see the sequel to this video here: • Example finding the osculating circle...
You can just cross r' and r'' to get a normal vector for the plane, see this video for that: • Osculating plane and circle, Multivar... .
Throughout the computation, we aim for efficiency, particularly by substituting t=1 wherever appropriate to streamline our calculations. This approach simplified our expressions significantly.
We finish by expressing the osculating plane equation in vector form. Expanding the vector dot product gives the general form of the plane.
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