G^1 Hermite Interpolation with Pythagorean-Hodograph Cubic Curves

curve

speed: σ(t) = 3.752

-2.431t+4.062

A polynomial curve

M(t)=(x(t),y(t))

is a Pythagorean-hodograph curve if

is the square of another polynomial. The lowest-degree curves satisfying this condition are PH-cubics. They are represented here in Bézier form. The degrees of freedom of such a curve allow using it to solve a partial Hermite interpolation problem: the boundary points and the tangent directions can be specified, but not the speeds at these points. Some situations have no solutions.

Details

Consider a polynomial parametric curve

M(t)=(x(t),y(t))

. By definition, its hodograph is its derivative

M'(t)=(x'(t),y'(t))

. The curve is called Pythagorean if there exists another polynomial

c(t)

such that

(x')

(y')

. The curve is then said to have a Pythagorean hodograph or to be a PH curve. Therefore its speed

σ(t)=M'(t)

is also a polynomial function of

. The lowest degree allowing this property is three.

Hence we illustrate here how cubic curves, represented in Bézier form (see Related Link below) by their control polygons

(

)

, can be used for a

Hermite interpolation. Specifying the boundary points

and

and the two associated unit tangent vector directions, defined by the angles

and

, we determine the cubic interpolatory PH-curve by its control points

. In certain cases, such a curve cannot exist, because a cubic (PH) curve does not have an inflexion point, so some values of

and

do not give a solution.

References

[1] G. Jaklic, J. Kozak, M. Krajnc, V. Vitrih, and E. Zaga, "On Interpolation by Planar Cubic

Pythagorean-Hodograph Spline Curves," Mathematics of Computation, 79(269), 2010 pp. 305–326.

[2] R. T. Farouki, Pythagorean-Hodograph Curves: Algebra and Geometry Inseparable, Berlin: Springer, 2008.

External Links

Bézier Curve (Wolfram MathWorld)

Pythagorean-Hodograph (PH) Cubic Curves

Permanent Citation

Isabelle Cattiaux-Huillard

"G^1 Hermite Interpolation with Pythagorean-Hodograph Cubic Curves"
http://demonstrations.wolfram.com/G1HermiteInterpolationWithPythagoreanHodographCubicCurves/
Wolfram Demonstrations Project
Published: June 3, 2014