08/30/00Dinesh Manocha, COMP258

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

08/30/00Dinesh Manocha, COMP258

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

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08/30/00Dinesh Manocha, COMP258

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 functionsHermite 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

08/30/00Dinesh Manocha, COMP258

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

08/30/00Dinesh Manocha, COMP258

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 -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) = U Mf B, where

U = [u3 u2 u 1]

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08/30/00Dinesh Manocha, COMP258

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

08/30/00Dinesh Manocha, COMP258

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