Parallel curve


A parallel of a curve is the
It generalises the concept of parallel lines. It can also be defined as a
These two definitions are not entirely equivalent as the latter assumes smoothness, whereas the former does not.
In computer-aided design the preferred term for a parallel curve is offset curve. Offset curves are important for example in numerically controlled machining, where they describe for example the shape of the cut made by a round cutting tool of a two-axis machine. The shape of the cut is offset from the trajectory of the cutter by a constant distance in the direction normal to the cutter trajectory at every point.
In the area of 2D computer graphics known as vector graphics, the computation of parallel curves is involved in one of the fundamental drawing operations, called stroking, which is typically applied to polylines or polybeziers in that field.
Except in the case of a line or circle, the parallel curves have a more complicated mathematical structure than the progenitor curve. For example, even if the progenitor curve is smooth, its offsets may not be so; this property is illustrated in the top figure, using a sine curve as progenitor curve. In general, even if a curve is rational, its offsets may not be so. For example, the offsets of a parabola are rational curves, but the offsets of an ellipse or of a hyperbola are not rational, even though these progenitor curves themselves are rational.
The notion also generalizes to 3D surfaces, where it is called an offset surface. Increasing a solid volume by a distance offset is sometimes called dilation. The opposite operation is sometimes called shelling. Offset surfaces are important in numerically controlled machining, where they describe the shape of the cut made by a ball nose end mill of a three-axis machine. Other shapes of cutting bits can be modelled mathematically by general offset surfaces.

Parallel curve of a parametrically given curve

If there is a regular parametric representation of the given curve available, the second definition of a parallel curve leads to the following parametric representation of the parallel curve with distance :
In cartesian coordinates:
Distance parameter may be negative, too. In this case one gets a parallel curve on the opposite side of the curve. One easily checks: a parallel curve of a line is a parallel line in the common sense and the parallel curve of a circle is a concentric circle.

Geometric properties:E. Hartmann: http://www.mathematik.tu-darmstadt.de/~ehartmann/cdgen0104.pdf ''Geometry and Algorithms for [COMPUTER AIDED DESIGN.''] S. 30.

If the given curve is polynomial, then the parallel curves are usually not polynomial. In CAD area this is a drawback, because CAD systems use polynomials or rational curves. In order to get at least rational curves, the square root of the representation of the parallel curve has to be solvable. Such curves are called pythagorean hodograph curves and were investigated by R.T. Farouki.

Parallel curves of an implicit curve

Generally the analytic representation of a parallel curve of an implicit curve is not possible. Only for the simple cases of lines and circles the parallel curves can be described easily.
For example:
In general, presuming certain conditions, one can prove the existence of an oriented distance function. In practice one has to treat it numerically. Considering parallel curves the following is true:
Example:

The diagram shows parallel curves of the implicit curve with equation
Remark:
The curves are not parallel curves, because is not true in the area of interest.

Further examples

And:
When determining the cutting path of part with a sharp corner for machining, you must define the parallel curve to a given curve that has a discontinous normal at the corner. Even though the given curve is not smooth at the sharp corner, its parallel curve may be smooth with a continuous normal, or it may have may have cusps when the distance from the curve matches the radius of curvature at the sharp corner.

Normal fans

As described [|above], the parametric representation of a parallel curve,, to a given curver,, with distance is:
At a sharp corner, the normal to given by is discontinuous, meaning the one-sided limit of the normal from the left is unequal to the limit from the right. Mathematically,
However, we can define a normal fan that provides an interpolant between and, and use in place of at the sharp corner:
The resulting definition of the parallel curve provides the desired behavior:

Algorithms

An efficient algorithm for offsetting is the level approach described by
Kimel and Bruckstein.
There are numerous approximation algorithms for this problem. For a 1997 survey, see Elber, Lee and Kim's "Comparing Offset Curve Approximation Methods".

Parallel (offset) surfaces

Offset surfaces are important in numerically controlled machining, where they describe the shape of the cut made by a ball nose end mill of a three-axis mill. If there is a regular parametric representation of the given surface available, the second definition of a parallel curve generalizes to the following parametric representation of the parallel surface with distance :
Distance parameter may be negative, too. In this case one gets a parallel surface on the opposite side of the surface. One easily checks: a parallel surface of a plane is a parallel plane in the common sense and the parallel surface of a sphere is a concentric sphere.

Geometric properties:

Note the similarity to the geometric properties of parallel curves.

Generalizations

The problem generalizes fairly obviously to higher dimensions e.g. to offset surfaces, and slightly less trivially to pipe surfaces. Note that the terminology for the higher-dimensional versions varies even more widely than in the planar case, e.g. other authors speak of parallel fibers, ribbons, and tubes. For curves embedded in 3D surfaces the offset may be taken along a geodesic.
Another way to generalize it is to consider a variable distance, e.g. parametrized by another curve. One can for example stroke with an ellipse instead of circle as it is possible for example in METAFONT.
More recently Adobe Illustrator has added somewhat similar facility in version CS5, although the control points for the variable width are visually specified. In contexts where it's important to distinguish between constant and variable distance offsetting the acronyms CDO and VDO are sometimes used.

General offset curves

Assume you have a regular parametric representation of a curve,, and you have a second curve that can be parameterized by its unit normal,, where the normal of . The parametric representation of the general offset curve of offset by is:
Note that the trival offset,, gives you ordinary parallel curves.

Geometric properties:

General offset surfaces describe the shape of cuts made by a variety of cutting bits used by three-axis end mills in numerically controlled machining. Assume you have a regular parametric representation of a surface,, and you have a second surface that can be parameterized by its unit normal,, where the normal of . The parametric representation of the general offset surface of offset by is:
Note that the trival offset,, gives you ordinary parallel surfaces.

Geometric properties:

Note the similarity to the geometric properties of general offset curves.

Derivation of geometric properties for general offsets

The geometric properties listed above for general offset curves and surfaces can be derived for offsets of arbitrary dimension. Assume you have a regular parametric representation of an n-dimensional surface,, where the dimension of is n-1. Also assume you have a second n-dimensional surface that can be parameterized by its unit normal,, where the normal of . The parametric representation of the general offset surface of offset by is:
First, notice that the normal of the normal of by definition. Now, we'll apply the differential w.r.t. to, which gives us its tangent vectors spanning its tangent plane.
Notice, the tangent vectors for are the sum of tangent vectors for and its offset, which share the same unit normal. Thus, the general offset surface shares the same tangent plane and normal with and. That aligns with the nature of envelopes.
We now consider the Weingarten equations for the shape operator, which can be written as. If is invertable,. Recall that the principal curvatures of a surface are the eigenvalues of the shape operator, the principal curvature directions are its eigenvectors, the Gauss curvature is its determinant, and the mean curvature is half its trace. The inverse of the shape operator holds these same values for the radii of curvature.
Substituting into the equation for the differential of, we get:
Next, we use the Weingarten equations again to replace :
Then, we solve for and multiple both sides by to get back to the Weingarten equations, this time for :
Thus,, and inverting both sides gives us,.