   Chapter 13, Problem 7RCC

Chapter
Section
Textbook Problem

# (a) Write formulas for the unit normal and binormal vectors of a smooth space curve r(t).(b) What is the normal plane of a curve at a point? What is the osculating plane? What is the osculating circle?

(a)

To determine

To Write: The formula for the unit normal vector of a smooth space curve r(t) .

Explanation

A point P on a smooth space curve r(t) can consist of many vectors which are orthogonal to the unit tangent vector T(t) .

As T(t)T(t)=0 , that is T(t) is orthogonal to T(t) .

At any point where, k0 , it can define the unit normal vector [N(t)] .

N(t)=T(t)|T(t)|

Thus, the formula for the unit normal vector of a smooth space curve r(t) is N(t)=T(t)|T(t)|_

To Write: The formula for the binormal vector of a smooth space curve r(t)

(b)

To determine

To explain: The normal plane of a curve at a given point, osculating plane and osculating circle.

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