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Serret-Frenet Equations

Serret-Frenet Equations. Greg Angelides December 6, 2006 Math Methods and Modeling. Serret-Frenet Equations. Given curve parameterized by arc length s Tangent vector = Normal vector = Binormal vector = X Curvature =

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Serret-Frenet Equations

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  1. Serret-Frenet Equations Greg Angelides December 6, 2006 Math Methods and Modeling

  2. 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 = = - + = -

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

  4. Fundamental Theorem of Space Curves • Let , :[a,b] R be continuous with > 0 on [a,b]. Then there is a curve c:[a,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)

  5. 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)) = = - + = - > 0 = 0 = + sin( s) - cos( s) = 0 = 0 = + s = 0 > 0 = + s

  6. Curves with Constant Curvature and Torsion • For > 0 > 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 • = • =

  7. Modeling with Serret-Frenet Frame • Given a differentiable curve with normal , a ribbon can be constructed with the parrallel curve f(s) = + , <<1 • Given a differentiable curve with normal , binormal , a tube can be constructed with circles orthogonal to the tangent vector. • f(s, ) = + ( cos() + sin()), <<1, 0 ≤ < 2

  8. Modeling Seashells • A structural curve defines the general shape of the seashell • E.g. a(s) = (a*e-sc1cos(s), a*e-sc2sin(s), b*e-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

  9. 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

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