Dr' Scott Schaefer

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Differential Geometry for Curves and Surfaces

• Dr. Scott Schaefer

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Intrinsic Properties of Curves
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Intrinsic Properties of Curves
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Intrinsic Properties of Curves
Identical curves but different derivatives!!!
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Arc Length

• s(t)t implies arc-length parameterization
• Independent under parameterization!!!

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Fernet Frame

• Unit-length tangent

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Fernet Frame

• Unit-length tangent
• Unit-length normal

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Fernet Frame

• Unit-length tangent
• Unit-length normal
• Binormal

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Fernet Frame

• Provides an orthogonal frame anywhere on curve

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Fernet Frame

• Provides an orthogonal frame anywhere on curve

Trivial due to cross-product
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Fernet Frame

• Provides an orthogonal frame anywhere on curve

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Fernet Frame

• Provides an orthogonal frame anywhere on curve

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Fernet Frame

• Provides an orthogonal frame anywhere on curve

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Uses of Fernet Frames

• Animation of a camera
• Extruding a cylinder along a path

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Uses of Fernet Frames

• Animation of a camera
• Extruding a cylinder along a path
• Problems The Fernet frame becomes unstable at
  inflection points or even undefined when

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Osculating Plane

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

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Curvature

• Measure of how much the curve bends

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Curvature

• Measure of how much the curve bends

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Curvature

• Measure of how much the curve bends

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Curvature

• Measure of how much the curve bends

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Curvature

• Measure of how much the curve bends

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Curvature

• Measure of how much the curve bends

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Torsion

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

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Fernet Equations

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Fernet Frames

• Unit-length tangent
• Unit-length normal
• Binormal

Problem!
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Rotation Minimizing Frames
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Rotation Minimizing Frames
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Rotation Minimizing Frames

• Build minimal rotation to next tangent

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Rotation Minimizing Frames

• Build minimal rotation to next tangent

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Rotation Minimizing Frames
Fernet Frame
Rotation Minimizing Frame
Image taken from Computation of Rotation
Minimizing Frames
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Rotation Minimizing Frames
Image taken from Computation of Rotation
Minimizing Frames
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Surfaces

• Consider a curve r(t)(u(t),v(t))

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Surfaces

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

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Surfaces

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

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Surfaces

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

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Surfaces

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

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Surfaces

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

First fundamental form
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First Fundamental Form

• Given any curve in parameter space
  r(t)(u(t),v(t)), arc length of curve on surface
  is

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First Fundamental Form

• The infinitesimal surface area at u, v is given by

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First Fundamental Form

• The infinitesimal surface area at u, v is given by

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First Fundamental Form

• The infinitesimal surface area at u, v is given by

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First Fundamental Form

• The infinitesimal surface area at u, v is given by

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First Fundamental Form

• The infinitesimal surface area at u, v is given by

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First Fundamental Form

• The infinitesimal surface area at u, v is given by

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First Fundamental Form

• Surface area over U is given by

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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

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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

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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)

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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)

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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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Meusniers Theorem

• Assume , is called the
  normal curvature
• Meusniers Theorem states that all curves on
  p(u,v) passing through a point x having the same
  tangent, have the same normal curvature

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Lines of Curvature

• We can parameterize all tangents through x using
  a single parameter

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Principle Curvatures
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Principle Curvatures
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Principle Curvatures
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Gaussian and Mean Curvature

• Gaussian Curvature
• Mean Curvature

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Gaussian and Mean Curvature

• Gaussian Curvature
• Mean Curvature
• elliptic
• hyperbolic
• parabolic
• flat