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Arc Length and Curvature

Arc Length and Curvature. Chapter 14.3. We develop …. A natural extension of arc length via parameterization Introduce the concept of curvature. example…. A surgeon studies the x-ray of the spine of an adolescent male There is a clear indication of scoliosis – but how do you measure this?

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Arc Length and Curvature

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  1. Arc Length and Curvature Chapter 14.3

  2. We develop … • A natural extension of arc length via parameterization • Introduce the concept of curvature

  3. example… • A surgeon studies the x-ray of the spine of an adolescent male • There is a clear indication of scoliosis – but how do you measure this? • By taking a series of x-rays from different positions a spacecurve can be generated that represents the spine

  4. Curvature and Arclength • We now that something is curving because its tangent vector is changing direction! The more it changes in a given distance the greater the curvature. We can define curvature as: curvature = rate of change of unit tangent vector wrt length, or K = |dT/ds|

  5. Arc Length • This has a very simple “intuitive” idea – set a bunch of meter sticks along the trace of the curve!

  6. Different ways to define Arclength…

  7. example 2 pg 900: • Parameterize wrt arc length • try 14.3#10

  8. Curvature • There are several different ways to determine the curvature:

  9. Examples: • Pg 900 #3 • Pg 902 #4 • Pg 902 #5

  10. Tangents, Normals and Binormals • Tangents T • Normals N • Binormals B

  11. Curvature and Torsion • Curvature and torsion are ways of describing how a curve can “bend”

  12. Example pg 907 #55 or …How long are YOUR genes? Can you model this with a parametric equation?

  13. The Snowbirds! Case II: The Snowbirds fly in tightening spiral path beginning 2.5 km overhead and descending to 500 m and described by: What do the path and velocity and acceleration vectors look like? Case I: The Snowbirds fly in a circular path given as What do the path and velocity and acceleration vectors look like?

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