11.4.1 Curves

Farouki and Sakkalis [108] introduced a class of special planar polynomial curves called Pythagorean hodograph ( ) curves , whose hodograph (derivative) components , and a polynomial form a Pythagorean triple . Thus, the curve has polynomial parametric speed ; accordingly its offset is a rational curve and its arc length is a polynomial function of the parameter . The Pythagorean condition is satisfied by

where , and are polynomials satisfying and [108,114] where denotes greatest common divisor. This condition is only a sufficient condition for a polynomial curve to have rational offset. For most applications is chosen to be 1. The lowest degree curve occurs when the polynomials and are linear and thus from (11.98) its degree is cubic. However, the resulting curve cannot possess an inflection point and hence is not practical. When and are quadratic, the curve will be a quintic and is the lowest degree curve to have enough flexibility for practical use. The quintics can inflect and can interpolate arbitrary first-order Hermite data [103]. The degree of the offset is for a degree curve. Therefore, the lowest degree of the offset of a curve for practical use is nine.

Farouki and Shah [112] developed a real-time interpolator for curves using the fact that the arc length of a curve is a polynomial function. As a consequence the generation of reference points along a curve is reduced to a sequence of polynomial rootfinding problems.

The planar curves can be easily generalized to space curves [110] by setting the four real polynomials , , , in the form

which leads to a polynomial parametric speed . A more thorough review of curves can be found in [114].

Pottmann [324] generalized the concept of curves to the full class of rational curves with rational offsets, by utilizing the projective dual representation. A rational planar curve is obtained as the envelope of its tangent line which is described as

where is the signed distance of the tangent line from the origin and is a rational function. The vector is a rational unit normal of the tangent line g(t) and is given by

where (11.98) is used so that the unit normal vector becomes rational. The envelope of the one-parameter family of can be obtained by solving a linear system consisting of (11.100) and its derivative for and as a function of , resulting in:

where

Here the rational function is replaced by . The offset to (11.102) is obtained by simply replacing by or equivalently by , and thus the degree of the offset remains the same as that of (11.102), which is an advantage over curves. The rational Bézier representation can be easily derived by prescribing the polynomials , , and and expressing the resulting polynomials , and in Bernstein form.

The form of (11.102) and (11.103) becomes simpler if the dual Bézier representation is used [324]. A plane dual Bézier curve is defined by a family of tangent lines which has the form

where are the Bézier lines (constant line vectors) and is the - Bernstein polynomial of degree . A line vector determines a straight line . From the homogeneous representation of (11.100) in the form, , the dual representation in terms of projective geometry is given by

When has a factor , there exists a common divisor in the dual representation, thus it is convenient to set which leads to

The control lines in (11.104) are easily obtained by expressing (11.106) in Bernstein form.

Lü [238] showed that the offset to a parabola is rational; its singular point at infinity was studied by Farouki and Sederberg [111]. In [238] Lü proved that although the offset (to a parabola) is not rational in the parameter , it may be expressed as a rational form in a new parameter, say , via a parameter transformation. The reparametrizing function is a rational function of the form where is a quadratic polynomial in . The transformed curve , is not parametrized properly, since there are two values of , which are the roots of the quadratic equation , for each corresponding point . While the curve is traced twice in opposite directions as increases from to 0 and from 0 to , defines a two-sided offset, i.e. the inward offset for and the outward offset for . The resulting rational curve is of degree 6. Lü [239] further derives a necessary and sufficient condition for a polynomial or more generally rational planar parametric curve to have rational parametric speed.