Stewart Calculus 13.3

1. 13.3 The Normal and
Binormal Vectors
At a given point on a space curve ( ), the
unit tangent vector is
()
()=
| ( )|
Since ( )· ( )=
we have ( )· ( )=
so () ()
We define the principal unit normal vector as
()
()=
| ( )|

2. We define the principal unit normal vector as
()
()=
| ( )|
We define the binormal vector as
()= () ()
it is also unit.
( ), ( ), ( ) are three unit vectors,
perpendicular to each other. They form a TNB
frame at point ( ) .

3. The plane determined by the normal and
binormal vectors at point is called the
normal plane at .
The plane determined by the tangent and
normal vectors at point is called the
osculating plane at .
The circle that lies in the osculating plane
towards the direction of , has the same
tangent at and has radius / ( ) is
called the osculating circle at .
This circle describes the behavior of the
curve at : it shares the same tangent,
normal, curvature and osculating plane.

4. Ex: Find the unit normal and binormal vectors
and the normal and osculating plane of the
helix ( ) = , , at the point ( , , ).

5. Ex: Find the unit normal and binormal vectors
and the normal and osculating plane of the
helix ( ) = , , at the point ( , , ).
()= , ,

6. Ex: Find the unit normal and binormal vectors
and the normal and osculating plane of the
helix ( ) = , , at the point ( , , ).
()= , ,
()
()= = , ,
| ( )|

7. Ex: Find the unit normal and binormal vectors
and the normal and osculating plane of the
helix ( ) = , , at the point ( , , ).
()= , ,
()
()= = , ,
| ( )|
()= , ,

8. Ex: Find the unit normal and binormal vectors
and the normal and osculating plane of the
helix ( ) = , , at the point ( , , ).
()= , ,
()
()= = , ,
| ( )|
()= , ,
()
()= = , ,
| ( )|

9. Ex: Find the unit normal and binormal vectors
and the normal and osculating plane of the
helix ( ) = , , at the point ( , , ).
()= , ,
()
()= = , ,
| ( )|
()= , ,
()
()= = , ,
| ( )|
()= () ()= , ,

10. ()
()= = , ,
| ( )|
()
()= = , ,
| ( )|
()= () ()= , ,

11. ()
()= = , ,
| ( )|
()
()= = , ,
| ( )|
()= () ()= , ,
At the point ( , , ), = .
( )= , ,
( )= , ,
( )= , ,

12. ( )= , ,
( )= , ,
( )= , ,

13. ( )= , ,
( )= , ,
( )= , ,
The normal vector of the normal plane is ( ).

14. ( )= , ,
( )= , ,
( )= , ,
The normal vector of the normal plane is ( ).
The normal plane is
( )+ ( )+ ( )= or + =

15. ( )= , ,
( )= , ,
( )= , ,
The normal vector of the normal plane is ( ).
The normal plane is
( )+ ( )+ ( )= or + =
The normal vector of the osculating plane is ( ).

16. ( )= , ,
( )= , ,
( )= , ,
The normal vector of the normal plane is ( ).
The normal plane is
( )+ ( )+ ( )= or + =
The normal vector of the osculating plane is ( ).
The osculating plane is
( ) ( )+ ( )= or =