Objective

2

The offset curve

also known as the parallel curve

refers to the locus of points along the normal vector direction with distance

d

. In recent years

the offset curve has played an important role in many fields and is widely applied in computer-aided geometric design (CAGD). In general

the arc length and offset curve of the polynomial curve have no rational form

and the offset-rational (OR) curve is a special polynomial parameter curve with exactly rational offset curves. The special properties of the curve have attracted the attention of many researchers. In recent years

the interpolation problem of OR curves has been widely studied. The problem of curve interpolation is widely used in many modern industrial fields

such as robot design

machinery industry

and space industry. The Hermite interpolation of given endpoint is a common method to construct a curve in CAGD. The

$${{\rm{C}}^{\rm{2}}} $$

Hermite interpolation problem of the quartic parabolic-PH curve

which is an even order of offset rational curve is discussed in this paper.

Method

2

Based on the parameters introduced by M bius transformation

a bijective linear fractional transformation

the

$${{\rm{C}}^{\rm{2}}} $$

Hermite interpolation of quartic rational parabolic-PH curve is constructed through complex analysis. The data

$$ H_C^2 = \{ {R_0}

{R_1}

{T_0}

{T_1}

{V_0}

{V_1}\} $$

to be interpolated is given with

R

0

and

R

1

referring to the two end points

T

0

and

T

1

for the tangent vectors at

R

0

and

R

1

and

V

0

and

V

1

for the second tangent vectors at

R

0

and

R

1

. By appropriate transformation

rotation

and scaling

making

R

0

=0 and

R

1

=1

we can further obtain the interpolation conditions for

$${{\rm{C}}^{\rm{2}}} $$

curves after M bius transformation. This paper shows a concrete construction method of quartic rational parabolic-PH curves for

$${{\rm{C}}^{\rm{2}}} $$

Hermite interpolation

whose tangents have three orders. By supposing the expression of

$$r\left( t \right) $$

$$F\left( t \right) $$

$$G\left( t \right) $$

the first-and second-order derivative of the curve can be obtained. The corresponding expression of the control points and the Bézier curve can be obtained by using the integral relation formula. The exact value of the parameter are calculated by the

$${{\rm{C}}^{\rm{2}}} $$

Hermite interpolation condition of the curve. Then

the quartic rational parabolic-PH curves formed by the M bius transformation are finally constructed.

Result

2

By providing a set of "reasonable" endpoints to be interpolated

we can obtain 12

$${{\rm{C}}^{\rm{2}}} $$

Hermite interpolation curves from the transformed quartic polynomial parabolic-PH curve under the initial interpolation condition and further obtain the

$${{\rm{C}}^{\rm{2}}} $$

Hermite interpolation curves of the 12 quartic rational parabolic-PH curves under the initial interpolation condition. Numerical examples show the effectiveness of the algorithm. It is not clear and convenient to choose the appropriate interpolation curve from the 12 interpolation curves. We need to select the curves that satisfy the interpolated condition and can elastically handle the inflection points. Other interpolation curves may have cusp points

node points

closed loops

or obviously inconsistent with geometric design requirements. By combining the minimum absolute rotation number and the elastic bending energy minimization

the selection method for determining the optimal curve satisfying the interpolation condition is put forward. When the absolute rotation number and the elastic bending energy of the interpolated curve are minimized

the optimal curve is often obtained

which has better smoothness and natural shape that meet the needs of geometric design. The examples illustrate that the traditional quartic parabolic-PH curve can construct

$$ {{\rm{C}}^1}$$

Hermite interpolation curves. However

the constraint of interpolation condition does not allow the direct construction of a curve with higher continuity. For traditional quintic PH curve

we cannot directly construct a curve with a continuity higher than

$$ {{\rm{C}}^1}$$

whereas through M bius transformation

we can achieve

$${{\rm{C}}^{\rm{2}}} $$

Hermite interpolation

which has a higher continuity than the traditional method. For the same set of given data

we construct the

$${{\rm{C}}^{\rm{2}}} $$

Hermite interpolation curve from quintic rational PH curve and quartic rational parabolic-PH curve. Compared with the 18 quintic rational PH curves

we can achieve the optimal curve from the 12 quartic rational parabolic-PH curves with lower elastic bending energy. Hence

the quartic rational parabolic-PH curves constructed by our method have more natural geometry than the traditional quintic rational PH curves. Although parabolic-PH curves with eight degree can be used to construct

$${{\rm{C}}^{\rm{2}}} $$

Hermite interpolation curves

the solution is complex

and the computation is large. Hence

through analysis and comparison

the quartic rational parabolic-PH curve presented in this paper has a simpler computation than quintic PH curves and parabolic-PH curves with eight degree. The interpolation results of the quartic rational parabolic-PH curve is more obvious

and the optimal curve best meets the requirement for the geometric design.

Conclusion

2

The use of

$${{\rm{C}}^{\rm{2}}} $$

Hermite interpolation of quartic rational parabolic-PH curves constructed by the introduction of M bius transformation not only ensures low degree of interpolation curve but also ensures a higher continuity of interpolation conditions. It makes the calculation simpler and the interpolation effect more obvious compared with the traditional PH curve with odd number of order. Related research on the sub PH curve is of certain significance. This report is significant for the study of PH curves with even number of degree.