1 / 9

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 =

wayde
Download Presentation

Serret-Frenet Equations

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  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

More Related