What is the usage of an osculating plane?

[Desikosh]

so i joined this forum almost 2 weeks ago i was wandering in its vast halls till now and still feel a bit lost,as an student couldn't let myself to get into many things i couldn't understand but i enjoyed this huge amount of knowledge being shared here :)
sorry if that was much off topic now the real question:
so i was listening to professor when she was talking about osculating plane and its equations when i failed to communicate with the topic properly, then i simply asked her "what is the usage of an osculating plane?" she just smiled and said" is there a usage?" ,"yes there MUST be a use for it !" i replied.then another smile and she said"oh great, then you'll find that reason and present it to us next week"..
i looked where ever i could, although i don't know many sources. but i couldn't find the USAGE of osculating plane. now i may not be able to prepare it for presentation but at least I'd appreciate if one of you could lead me somewhere i can find the answer .

thank you :)

 

[micromass]

Worst professor ever.. She just tries to avoid that her students ask her perfectly reasonable questions. Seriously, this is awful.

Anyway, the osculating plane is the plane which best approximates the given space curve. One possible application of the osculating plane is to find the "equation" of the osculating circle. The osculating circle has a clear physical interpretation: let's say you travel in a spaceship and your path is the given space curve. So in traveling the space curve, you are given a steering wheel which you turn in different directions. Let's say that at a certain point you just hold your steering wheel fixed, then your path will be described by the osculating circle at that point.

The osculating circle also gives an incredibly neat interpretation of the curvature. Indeed, the radius of the osculating circle at a point $\alpha(s)$ is exactly $1/\kappa(s)$. In order to prove that (and to find the equation of the osculating circle), it is easiest to work with the osculating plane.

A rather nice treatment is given in Spivak's "A Comprehensive Introduction to Differential Geometry, Vol. 2" (don't worry, you don't need to know Vol. 1 to follow his treatment). See page 24 (it starts right before prop. 10) and you will need the discussion on page 3,4,5,6 too.

See here for an alternative discussion: http://theronhitchman.blogspot.be/2013/02/an-interpretation-of-curvature.html I recommend reading this discussion first. It's well-organized, but not completely rigorous. After that you should look things up in Spivak to make it more rigorous.

 

[HallsofIvy]

I would not consider asking students to think for themselves "awful" teaching.

 

[micromass]

I would not consider asking students to think for themselves "awful" teaching.

It's clear she didn't know the answer. So instead of admitting it and perhaps looking it up herself, she transfers her job to the student.

Well, that this is awful teaching is probably a matter of opinion. So let's just agree to disagree

 

[Desikosh]

thank you very much that will surely help a lot :)
plus the spaceship thingy was an amazing example !