4

I couldn't understand the exact geometrical meaning of the osculating plane definition. Can any one help me with this? Thanks advance.

Osculating plane: Let $\gamma$ be a smooth curve and P and Q be two neighboring points on $\gamma$. The limiting position of the plane that contains the tangential line at P and passes through the point Q as Q $\to$ P is defined as the osculating plane at P.

In another definition, osculating plane is a plane spanned by the tangent and normal line. But, I couldn't understand how we can find normal line geometrical?

Selvakumar A
  • 617
  • The first definition seems perfectly clear. What's your question about it? As for the "normal" in the second definition, look up TNB frame - the normal vector of a curve is (proportional to) the derivative of the tangent vector (naturally parametrized). – anon Jan 23 '19 at 07:20
  • Is it mean that limiting of the planes spanned by the tangent line and the line joining P and Q? – Selvakumar A Jan 23 '19 at 07:24
  • Is it possible to visualise that derivative of tangent vector with our calculations? – Selvakumar A Jan 23 '19 at 07:26
  • Yes to your first comment. Not sure what you're asking in the second comment. – anon Jan 23 '19 at 07:27
  • Without taking derivative by looking at the curve itself can possible to see that line? – Selvakumar A Jan 23 '19 at 07:29

1 Answers1

1

https://youtu.be/dJ6q3ZV_kjE https://youtu.be/nLJy0B6CNMs Please see these videos, it may help you. The language is not English, but I hope you can understand it by the 3D presentation. However in simple language we can tell that, an oscillating plane is the plane containing curve's principal normal and tangent.

Rajat Dash
  • 91