Game Development Reference
In-Depth Information
and with it we can compute a point on the curve for a given t without
knowing the polynomial for the curve. Lagrange interpolation works by
creating basis functions that depend only on the knot vector. We can view
the use of the basis polynomials in two ways. Either we can think about
scaling each basis polynomial by the corresponding coordinate value and
then adding them all together, or we can think about the polynomials as
functions that compute barycentric coordinates that are used as blending
weights in a simple weighted average of the coordinate points.
Both methods yield the same curve when given the same data. Further-
more, this polynomial is unique—no other polynomial of the same degree
interpolates the data points. An informal argument for why this is true
goes like this: A polynomial of degree n has n + 1 degrees of freedom,
corresponding to the n + 1 coe cients in monomial form. Therefore, the
degree n polynomial that interpolates n+ 1 control points must be unique.
(Farin [20, Section 6.2] gives a more rigorous argument.)
For purposes of curve design, polynomial interpolation is not ideal, pri-
marily because of our inability to control the overshoot. The overshoot is
guaranteed by the fact that the underlying Lagrange basis polynomials are
not restricted to the unit interval [0,1], and the curve escapes the convex
hull of the control points.
Direct polynomial interpolation finds limited application in video games,
but our study has introduced the themes of repeated linear interpolation
and basis polynomials. We've also seen a bit of the beautiful duality be-
tween the two techniques.
13.3
Hermite Curves
Polynomial interpolation tries to control the interior of the curve by thread-
ing the curve through specified knots. This doesn't work as well as we would
like, because of the tendency to oscillate and overshoot, so let's try a differ-
ent approach. We're still going to want to specify the endpoint positions,
of course. But instead of specifying the interior positions to interpolate,
let's control the shape of the curve through the tangents at the endpoints.
A curve thus specified is said to be a Hermite curve or a curve in Hermite
form, named in honor of Charles Hermite 8 (1822-1901).
The Hermite form specifies a curve by listing its starting and ending
positions and derivatives. A cubic curve has only four coe cients, which
allows for the specification of just the first derivatives, the velocities at the
8 He's another French guy, and his mother probably pronounced his name “air-
MEET.” But many English speakers, even some we know with PhDs, pronounce it
“HUR-mite,” so you can probably do the same.
 
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