13.3 The Normal and  Binormal VectorsAt a given point on a space curve   ( ),   theunit tangent vector is                 ...
We define the principal unit normal vector as                         ()                   ()=                       | ( )...
The plane determined by the normal andbinormal vectors at point  is called thenormal plane at .The plane determined by the...
Ex: Find the unit normal and binormal vectorsand the normal and osculating plane of thehelix ( ) =      ,   , at the point...
Ex: Find the unit normal and binormal vectorsand the normal and osculating plane of thehelix ( ) =      ,   , at the point...
Ex: Find the unit normal and binormal vectorsand the normal and osculating plane of thehelix ( ) =      ,   , at the point...
Ex: Find the unit normal and binormal vectorsand the normal and osculating plane of thehelix ( ) =      ,   , at the point...
Ex: Find the unit normal and binormal vectorsand the normal and osculating plane of thehelix ( ) =      ,   , at the point...
Ex: Find the unit normal and binormal vectorsand the normal and osculating plane of thehelix ( ) =      ,   , at the point...
() ()=        =        ,   ,     | ( )|       () ()=        =    ,       ,     | ( )|()= ()     ()=       ,       ,
()       ()=        =                       ,   ,           | ( )|              ()        ()=        =              ,     ...
( )=        , , ( )=   , ,( )=    ,         ,
( )=           , ,                 ( )=     , ,               ( )=        ,         ,The normal vector of the normal plane...
( )=          , ,                  ( )=     , ,                 ( )=      ,         ,The normal vector of the normal plane...
( )=          , ,                   ( )=    , ,                  ( )=      ,         ,The normal vector of the normal plan...
( )=             , ,                   ( )=        , ,                  ( )=         ,         ,The normal vector of the n...
Upcoming SlideShare
Loading in …5
×

Caculus II - 37

951 views

Published on

Stewart Calculus 13.3

0 Comments
1 Like
Statistics
Notes
  • Be the first to comment

No Downloads
Views
Total views
951
On SlideShare
0
From Embeds
0
Number of Embeds
1
Actions
Shares
0
Downloads
33
Comments
0
Likes
1
Embeds 0
No embeds

No notes for slide
  • \n
  • \n
  • \n
  • \n
  • \n
  • \n
  • \n
  • \n
  • \n
  • \n
  • \n
  • \n
  • \n
  • \n
  • \n
  • \n
  • Caculus II - 37

    1. 1. 13.3 The Normal and Binormal VectorsAt a given point on a space curve ( ), theunit tangent vector is () ()= | ( )|Since ( )· ( )=we have ( )· ( )=so () ()We define the principal unit normal vector as () ()= | ( )|
    2. 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 TNBframe at point ( ) .
    3. 3. The plane determined by the normal andbinormal vectors at point is called thenormal plane at .The plane determined by the tangent andnormal vectors at point is called theosculating plane at .The circle that lies in the osculating planetowards the direction of , has the sametangent at and has radius / ( ) iscalled the osculating circle at .This circle describes the behavior of thecurve at : it shares the same tangent,normal, curvature and osculating plane.
    4. 4. Ex: Find the unit normal and binormal vectorsand the normal and osculating plane of thehelix ( ) = , , at the point ( , , ).
    5. 5. Ex: Find the unit normal and binormal vectorsand the normal and osculating plane of thehelix ( ) = , , at the point ( , , ). ()= , ,
    6. 6. Ex: Find the unit normal and binormal vectorsand the normal and osculating plane of thehelix ( ) = , , at the point ( , , ). ()= , , () ()= = , , | ( )|
    7. 7. Ex: Find the unit normal and binormal vectorsand the normal and osculating plane of thehelix ( ) = , , at the point ( , , ). ()= , , () ()= = , , | ( )| ()= , ,
    8. 8. Ex: Find the unit normal and binormal vectorsand the normal and osculating plane of thehelix ( ) = , , at the point ( , , ). ()= , , () ()= = , , | ( )| ()= , , () ()= = , , | ( )|
    9. 9. Ex: Find the unit normal and binormal vectorsand the normal and osculating plane of thehelix ( ) = , , at the point ( , , ). ()= , , () ()= = , , | ( )| ()= , , () ()= = , , | ( )| ()= () ()= , ,
    10. 10. () ()= = , , | ( )| () ()= = , , | ( )|()= () ()= , ,
    11. 11. () ()= = , , | ( )| () ()= = , , | ( )| ()= () ()= , ,At the point ( , , ), = . ( )= , , ( )= , , ( )= , ,
    12. 12. ( )= , , ( )= , ,( )= , ,
    13. 13. ( )= , , ( )= , , ( )= , ,The normal vector of the normal plane is ( ).
    14. 14. ( )= , , ( )= , , ( )= , ,The normal vector of the normal plane is ( ).The normal plane is ( )+ ( )+ ( )= or + =
    15. 15. ( )= , , ( )= , , ( )= , ,The normal vector of the normal plane is ( ).The normal plane is ( )+ ( )+ ( )= or + =The normal vector of the osculating plane is ( ).
    16. 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 =

    ×