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Caculus II - 37

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Stewart Calculus 13.3

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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 =

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