Hermite spline ( named after ~the French ~

mathematician Charles ~Hermite) is an

interpolating piecewise cubic polynomial with a specified tangent at each control

point.

Unlike the natural cubic splines, Hermite splints can be adjusted locally

because each curve section is only dependent on its endpoint constraints.

If P(L)represents a parametric cubic point function for the curve section be-

tween control points pi and pk, a s shown in Fig. then the boundary

conditions that define this Hermite spline is

with Dpk and Dpk+1,

specifying the values for the parametric derivatives (slope of

the curve) a t control points pk and p k + respectively.

We can write the vector equivalent of above equation for hermite also:

where the x component of P is,

and similarly for the

y and z components

The matrix equivalent of above equation

and the derivative of thin point function can be expressed as

Hermite polynomials can be useful for some digitizing applications where

it may not be too difficult to specify or approximate the curve slopes. But for

most problems in computer graphics, it is more useful to generate spline curves

without requiring input values for curve slopes or other geometric information,

in addition to control-point coordinates. Cardinal splines and Kochanek-Bartels

splines, discussed in the following two sections, are variations on the Hermite

splines that d o not require input values for the curve derivatives at the control

points. Procedures for these splines compute parametric derivatives from the co-

ordinate positions of the control points.

Substituting endpoint values and 1 for parameter u Into the previous two equations, we can express the Hermite boundary conditions

Hermite polynomials can be useful for some digitizing applications where

it may not be too difficult to specify or approximate the curve slopes. But for

most problems in computer graphics, it is more useful to generate spline curves

without requiring input values for curve slopes or other geometric information,

in addition to control-point coordinates. Cardinal splines and Kochanek-Bartels

splines, discussed in the following two sections, are variations on the Hermite

splines that d o not require input values for the curve derivatives at the control

points. Procedures for these splines compute parametric derivatives from the co-

ordinate positions of the control points.

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