WOLFRAM|DEMONSTRATIONS PROJECT

Pythagorean-Hodograph Quintic Curves

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curve
σ(t) = 4.600
4
t
-23.53
3
t
+49.43
2
t
-38.76t+18.03
A polynomial curve
M(t)=(x(t),y(t))
is a Pythagorean-hodograph (PH) curve if
2
dx
dt
+
2
dy
dt
is the square of another polynomial. The lowest degree curves satisfying this condition are PH cubics, but such a curve cannot inflect. Quintic PH curves are needed to get an inflexion point. The degrees of freedom of such a curve allow the choice of four of the six control polygon points. The PH property determines the two remaining points.