SerretFrenet Equations

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Serret-Frenet Equations

• Greg Angelides
• December 6, 2006
• Math Methods and Modeling

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Serret-Frenet Equations

• Given curve parameterized by arc length
  s
• Tangent vector
• Normal vector
• Binormal vector X
• Curvature
• Torsion -
• Serret-Frenet equations fully describe
• differentiable curves in

Serret-Frenet Equations

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Outline

• Serret-Frenet Equations
• Curve Analysis
• Modeling with the Serret-Frenet Frame
• Summary

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Fundamental Theorem of Space Curves

• Let , a,b R be continuous with gt
  0 on a,b. Then there is a curve ca,b
  R3 parameterized by arc length whose curvature
  and torsion functions are and
• Suppose c1, c2 are curves parameterized by arc
  length and c1, c2 have the same curvature and
  torsion functions. Then there exists a rigid
  motion f such that c2 f(c1)

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Curves with Constant Curvature and Torsion
Serret-Frenet Equations

• ( ) 0
• Solving with Laplace transform yields
• (cos(rs) - cos(rs))
  ( sin(rs))
• ( - cos(rs))
• Where r
• ( sin(rs) s - sin(rs))
  ( - cos(rs)) ( s -
  sin(rs))

-

-
gt 0 0 sin(
s) - cos( s)
0 0 s
0 gt 0 s
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Curves with Constant Curvature and Torsion

• For gt 0 gt 0
• ( sin(rs) s - sin(rs))
  ( - cos(rs)) ( s -
  sin(rs))
• Helix parameterized by arc length has form
• f(s) (a cos( ), a sin(
  ), )
• a is the radius of the helix
• b is the pitch of the helix
• Solving the Serret-Frenet equations yields

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Modeling with Serret-Frenet Frame

• Given a differentiable curve with normal
  , a ribbon can be constructed with the
  parrallel curve f(s) ? , ?ltlt1
• Given a differentiable curve with normal
  , binormal , a tube can be
  constructed with circles orthogonal to the
  tangent vector.
• f(s, ?) ?( cos(?)
  sin(?)), ?ltlt1, 0 ?lt 2?

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Modeling Seashells

• A structural curve defines the general shape of
  the seashell
• E.g. a(s) (ae-sc1cos(s), ae-sc2sin(s),
  be-sc3)
• A generating curve follows the structural curve
  and defines the seashell
• E.g. g(s,?) a(s)
• Small perturbations along with the wide variety
  of structural and generating curves allows
  accurate modeling of the diversity of seashells

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Summary

• Serret-Frenet equations and frame greatly
  simplify the study of complex differentiable
  curves
• Functions for curvature and torsion and the
  Serret-Frenet equations fully describe a curve up
  to a Euclidean movement
• Serret-Frenet framework used extensively in a
  wide variety of modeling studies