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Differential Geometry for Curves and Surfaces

Differential Geometry for Curves and Surfaces. Dr. Scott Schaefer. Intrinsic Properties of Curves. Intrinsic Properties of Curves. Intrinsic Properties of Curves. Identical curves but different derivatives!!!. Arc Length. s ( t )= t implies arc-length parameterization

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Differential Geometry for Curves and Surfaces

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  1. Differential Geometry for Curves and Surfaces Dr. Scott Schaefer

  2. Intrinsic Properties of Curves

  3. Intrinsic Properties of Curves

  4. Intrinsic Properties of Curves Identical curves but different derivatives!!!

  5. Arc Length • s(t)=t implies arc-length parameterization • Independent under parameterization!!!

  6. Frenet Frame • Unit-length tangent

  7. Frenet Frame • Unit-length tangent • Unit-length normal

  8. Frenet Frame • Unit-length tangent • Unit-length normal • Binormal

  9. Frenet Frame • Provides an orthogonal frame anywhere on curve

  10. Frenet Frame • Provides an orthogonal frame anywhere on curve Trivial due to cross-product

  11. Frenet Frame • Provides an orthogonal frame anywhere on curve

  12. Frenet Frame • Provides an orthogonal frame anywhere on curve

  13. Frenet Frame • Provides an orthogonal frame anywhere on curve

  14. Uses of Frenet Frames • Animation of a camera • Extruding a cylinder along a path

  15. Uses of Frenet Frames • Animation of a camera • Extruding a cylinder along a path • Problems: The Frenet frame becomes unstable at inflection points or even undefined when

  16. Osculating Plane • Plane defined by the point p(t) and the vectors T(t) and N(t) • Locally the curve resides in this plane

  17. Curvature • Measure of how much the curve bends

  18. Curvature • Measure of how much the curve bends

  19. Curvature • Measure of how much the curve bends

  20. Curvature • Measure of how much the curve bends

  21. Curvature • Measure of how much the curve bends

  22. Curvature • Measure of how much the curve bends

  23. Torsion • Measure of how much the curve twists or how quickly the curve leaves the osculating plane

  24. Frenet Equations

  25. Frenet Frames • Unit-length tangent • Unit-length normal • Binormal Problem!

  26. Rotation Minimizing Frames

  27. Rotation Minimizing Frames

  28. Rotation Minimizing Frames • Build minimal rotation to next tangent

  29. Rotation Minimizing Frames • Build minimal rotation to next tangent

  30. Rotation Minimizing Frames Frenet Frame Rotation Minimizing Frame Image taken from “Computation of Rotation Minimizing Frames”

  31. Rotation Minimizing Frames Image taken from “Computation of Rotation Minimizing Frames”

  32. Surfaces • Consider a curve r(t)=(u(t),v(t))

  33. Surfaces • Consider a curve r(t)=(u(t),v(t)) • p(r(t)) is a curve on the surface

  34. Surfaces • Consider a curve r(t)=(u(t),v(t)) • p(r(t)) is a curve on the surface

  35. Surfaces • Consider a curve r(t)=(u(t),v(t)) • p(r(t)) is a curve on the surface

  36. Surfaces • Consider a curve r(t)=(u(t),v(t)) • p(r(t)) is a curve on the surface

  37. Surfaces • Consider a curve r(t)=(u(t),v(t)) • p(r(t)) is a curve on the surface First fundamental form

  38. First Fundamental Form • Given any curve in parameter space r(t)=(u(t),v(t)), arc length of curve on surface is

  39. First Fundamental Form • The infinitesimal surface area at u, v is given by

  40. First Fundamental Form • The infinitesimal surface area at u, v is given by

  41. First Fundamental Form • The infinitesimal surface area at u, v is given by

  42. First Fundamental Form • The infinitesimal surface area at u, v is given by

  43. First Fundamental Form • The infinitesimal surface area at u, v is given by

  44. First Fundamental Form • The infinitesimal surface area at u, v is given by

  45. First Fundamental Form • Surface area over U is given by

  46. Second Fundamental Form • Consider a curve p(r(s)) parameterized with respect to arc-length where r(s)=(u(s),v(s)) • Curvature is given by

  47. Second Fundamental Form • Consider a curve p(r(s)) parameterized with respect to arc-length where r(s)=(u(s),v(s)) • Curvature is given by

  48. Second Fundamental Form • Consider a curve p(r(s)) parameterized with respect to arc-length where r(s)=(u(s),v(s)) • Curvature is given by • Let n be the normal of p(u,v)

  49. Second Fundamental Form • Consider a curve p(r(s)) parameterized with respect to arc-length where r(s)=(u(s),v(s)) • Curvature is given by • Let n be the normal of p(u,v)

  50. Second Fundamental Form • Consider a curve p(r(s)) parameterized with respect to arc-length where r(s)=(u(s),v(s)) • Curvature is given by • Let n be the normal of p(u,v) Second Fundamental Form

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