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
08/30/00
Dinesh 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
ï
[0,1]
• Cubic curve defined by 12 parameters
• Hermite curve: Specified using endpoints and tangent
directions at these points
08/30/00
Dinesh 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 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
08/30/00
Dinesh 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
08/30/00
Dinesh 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
â
F = [u3 u2 u 1]
2
-3
0
1
-2
3
0
0
1
-2
1
0
1
-1
0
0
ã
This is the 4 X 4 Hermite basis transformation matrix.
P(u) = U Mf B, where
U = [u3 u2 u 1]
08/30/00
Dinesh 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
08/30/00
Dinesh 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
08/30/00
Dinesh Manocha, COMP258