Osculating Circle of a Parabola + Summary of TNB Vectors and Curvature Formulas

mes (67)in DTube • 17 hours ago

In this video, I go over the osculating circle, which lies on the same osculating plane and has the same tangent and normal vector as a given point on a space curve. I illustrate this by determining the osculating circle of a parabola at the origin. I also graph out the general osculating circle (with help from Grok AI) of the parabola in the Desmos 2D graphing calculator. Since the curvature of a circle is 1/radius, the radius is thus 1/curvature. This means that as the curve gets flatter, the curvature decreases, but the osculating circle gets bigger!

Grok AI formula: https://grok.com/share/c2hhcmQtMg_38dc330d-fddc-4436-87b3-a3dcbbb2a2da
Desmos graph: https://www.desmos.com/calculator/tzb5bjsxkl

I also go over a summary of the formulas for the tangent, normal, and binormal vectors as well as the 3 formulas for the curvature.

#math #vectors #calculus #desmos #grok

Timestamps:

Osculating circle lies in the osculating plane and is tangent to the curve – 0:00
Example 8: Osculating circle of a parabola at the origin – 1:26
- Equation of the osculating circle – 3:11
- Deriving the parametric equations of a circle – 5:21
- Graphing the circle and parabola – 7:24
- Graphing osculating circle with Desmos, while using Grok AI for the general osculating circle equation: https://www.desmos.com/calculator/tzb5bjsxkl and https://grok.com/share/c2hhcmQtMg_38dc330d-fddc-4436-87b3-a3dcbbb2a2da – 8:57
- Tangent vector has same curvature, normal vector, and tangent vector – 11:01
Summary of T, N, B, and k formulas – 13:40

Notes and playlists: