Quick definition of an osculating plane

[kickthatbike]

I was absent to a calculus III lecture last monday. I didn't miss much, and have gone over what I did. I understand the work and know how to do the problems, the only thing I'm having trouble with is actually picturing what an osculating plane is. I just need a simple explanation of what it does, or what exactly it means. I googled the definition but still didn't really understand it... Thanks in advance.

 

[lurflurf]

Have you no books? An osculating plane also known as a tangent plane can many times be defined as a plane the intersects a function at one point or we could also say it is perpendicular to the normal line of its point of intersection. It is the analog of a tangent line. Informally we can say that a small patch of a sufficiently well behaved function looks like a plane and the osculating plane of a point in that small patch is the plane that it looks like.

 

[tiny-tim]

welcome to pf!

hi kickthatbike! welcome to pf!

at a particular point of a curve, the osculating plane is the plane that most closely fits the curve at that point

it is the tangent plane that contains the normal

the normal is the direction of the centre of the circle of curvature (the osculating circle), the circle that most closely fits the curve at that point

so it is the tangent plane that contains the circle of curvature

if you want a physics motivation, the osculating plane is the tangent plane which contains the direction of the force ​