The TNB Frame: Normal and Osculating Planes + Example on a Circular Helix

In this video, I go over the TNB Frame for space curves and their associated Normal and Osculating Planes, which have applications in the motion of spacecraft. The TNB Frame is named after the Tangent, Normal, and Binormal vectors at every point on the curve. The normal plane contains the normal and binormal vectors and is orthogonal to the tangent vector. The osculating plane (derived from Latin osculum meaning "kiss") contains the normal and tangent vector and is orthogonal to the binormal plane. I illustrate these planes for our earlier example on the Circular Helix and have graphed it out on GeoGebra. https://www.geogebra.org/3d/wyr5mysb

#math #vectors #calculus #geogebra #education

Timestamps

• Tangent, Normal, Binormal (TNB) Frame – 0:00
• Normal plane contains vectors N and B and is orthogonal to T – 1:01
• Osculating plane contains vectors T and N and is orthogonal to B. Name stems from Latin osculum meaning "kiss" – 1:36
  • For a plane curve, the osculating plane contains the whole curve – 2:30
• Example 7: Normal and Osculating Planes of a Circular Helix – 2:49
  • Tangent vector is normal to the Normal Plane – 4:48
  • Recap on the equation of the plane – 9:55
  • Equation of the Normal Plane – 11:40
  • Equation of the Osculating Plane – 13:11
  • Binormal Vector is orthogonal to osculating plane – 14:57
  • Illustration of the circular helix and osculating plane + graphing in GeoGebra: https://www.geogebra.org/3d/wyr5mysb – 19:33

Notes and playlists

• Summary: @mes/re-leothreads-eytzh6f4
• Notes: @mes/koxharrd
• Sections playlist: https://www.youtube.com/playlist?list=PLai3U8-WIK0Gd5OieYiZMaSKw1QfyHbiG
• Vector Functions playlist: https://www.youtube.com/playlist?list=PLai3U8-WIK0HQl7xTQBS_O8Te8PcpNm4U .

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