Newton's Method

• suppose that ，and is differentiable near
  
  

• take an arbitrary near as the initial value, and recursively compute:
  
  

Example

• find a solution of
  
  

• let ，then
  
  

Example

• 
  
  

• if ，then
  
  

• take as the initial value we get
  
  

Finite differences

• 
  
  

• let , we have
  forward difference:
  backward difference:
  
  
  central difference:

Second Order Difference

Motivations for Curvature

• need a scalar to describe how ''flat'' a curve is
  
  
• lines are flat, they should be assigned the value
  
  
• the larger the radius of a circle, the flatter it is, hence the value should have a negative dependence on the radius
  
  
• consider particle motions on a curve, the less angles of tangent lines change, the flatter the curve should be

Arc Length Derivatives

• suppose that is continuously differentiable on
  
  

• for each ，let be the arc length of the curve for
  
  

• being differentiabl implies that it locally changes almost linearly as
  
  

Definition of the Curvature

• suppose that a particle moves on a smooth curve at constant speed, starting from
  
  

• after time it traveled distances , the change of angles of tangent lines is
  
  

• the average curvature within time is
  
  

• the curvature at is

Example

• find the curvature at each point on a circle of radius
  
  

• if the angular velocity is ，then the distance it travled is ，the change of angles is
  
  

Computation of Curvatures

• given a smooth curve , find the curvature at each point
  
  

• suppose that the angle of the tangent line at is ，then ，and . Thus , the arc length derivative is
  
  

Computation of Curvatures

• if the curve is given instead by
  
  

• details omitted and left as an exercise
  
  

Example

• find the curvature of the hyperbola at
  
  

• 
  
  

Summary

• Newton's method and its convergence
  
  
• Finite differences
  
  
• Curvature