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Hermite Curves. A mathematical representation as a link between the algebraic & geometric form Defined by specifying the end points and tangent vectors at the end points Use of control points Geometric points that control the shape

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hermite curves
Hermite Curves
  • A mathematical representation as a link between the algebraic & geometric form
  • Defined by specifying the end points and tangent vectors at the end points
  • Use of control points
    • Geometric points that control the shape
    • Algebraically: used for linear combination of basis functions

Dinesh Manocha, COMP258

cubic parametric curves
Cubic Parametric Curves
  • Power basis:

X(u)= ax u3 + bx u2 + cx u + dx

Y(u)= ay u3 + by u2 + cy u + dy

Z(u)= az u3 + bz u2 + cz u + dz

P(u) = (X(u) Y(u) Z(u)), u [0,1]

  • Cubic curve defined by 12 parameters
  • Hermite curve: Specified using endpoints and tangent directions at these points

Dinesh Manocha, COMP258

hermite cubic curves
Hermite Cubic Curves

P(u) = F1(u) P(0) + F2(u) P(1) + F3(u) Pu(0) + F4(u) Pu(1)

where

F1(u) = 2u3 – 3u2 + 1

F2(u) = -2u3 + 3u2

F3(u) = u3 – 2u2 + u

F4(u) = u3 – u2,

The Fi(u) are the Hermite basis functions and

P(0), P(1), Pu(0) and Pu(1) are the geometric coefficients

  • The coefficients are specified to maintain continuity between different segments

Dinesh Manocha, COMP258

hermite basis functions
Hermite Basis Functions

Important Characteristics

  • Universality – hold for all cubic Hermite curves
  • Dimensional independence: extend to higher dimension
  • Separation of Boundary Condition Effects: constituent boundary condition coefficients are decoupled from each other (i.e P(0) & P(1))
    • Local Control: can modify a single specific boundary condition to alter the shape of the curve locally
  • Can be extended to higher degree curves

Dinesh Manocha, COMP258

cubic hermite curve matrix representation
Cubic Hermite Curve: Matrix Representation

Let B = [P(0) P(1) Pu(0) Pu(1)]

F = [F1(u) F2(u) F3(u) F4(u)] or

F = [u3 u2 u 1]

  • -2 1 1

-3 3 -2 -1

0 0 1 0

1 0 0 0

This is the 4 X 4 Hermite basis transformation matrix.

P(u) = UMfB, where

U = [u3 u2 u 1]

Dinesh Manocha, COMP258

composing parametric curves
Composing Parametric Curves
  • Given a large collection of data points, compute a curve representation that approximates or interpolates
  • Higher degree curves (say more than 4 or 5) can result in numerical problems (evaluation, intersection, subdivision etc.)
  • Need to multiple segments and compose them with appropriate continuity

Dinesh Manocha, COMP258

parametric geometric continuity
Parametric & Geometric Continuity
  • Parametric Continuity (or Cn): Two curves have nth order parametric continuity, Cn, if their 0th to nth derivatives match at the end points
  • Geometric Continuity (or Gn): Less restrictive than parametric continuity. Two curves have nth order geometric continuity, Gn, if there is a reparametrization of the curve, so that the reparametrized curves have Cn continuity.
    • G1:Unit tangent vectors at the end point are continuous
    • G2:Relates the curvature of the curves at the endpoints
    • Geometric continuity results in more degrees of freedom

Dinesh Manocha, COMP258