Wikipedia / Parallel curve
Wikipedia / Parallel curve
A parallel curve of a given (progenitor) curve is the envelope of a family of congruent (equal-radius) circles centered on the curve. It generalises the concept of parallel (straight) lines. It can also be defined as a curve whose points are at a constant normal distance from a given curve. 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. (In other geometric contexts, the term "offset" can also refer to a translation; however, a parallel curve may have a different shape than its progenitor.) Offset curves are important, for example, in numerically controlled (NC) 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 (approximate) computation of parallel curves is involved in one of the fundamental drawing operations, called stroking, which is typically applied to polylines or polybeziers (themselves called paths) 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 surfacess, where it is called an offset surface or parallel surface. Increasing a solid volume by a (constant) distance offset is sometimes called dilation (similar to the dilation image operation). The opposite operation is sometimes called shelling. Offset surfaces are important in NC, 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
:
.
In cartesian coordinates:
may be negative. In this case, one gets a parallel curve on the opposite side of the curve (see diagram on the parallel curves of a circle). One can easily check that 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
Source:
that means: the tangent vectors for a fixed parameter are parallel.
the curvature.
the radius of curvature.
When they exist, the osculating circles to parallel curves at corresponding points are concentric.
As for parallel lines, a normal line to a curve is also normal to its parallels.
When parallel curves are constructed they will have cuspss when the distance from the curve matches the radius of curvature. These are the points where the curve touches the evolute.
If the progenitor curve is a boundary of a planar set and its parallel curve is without self-intersections, then the latter is the boundary of the Minkowski sum of the planar set and the disk of the given radius.
are polynomials), then the parallel curves are usually not polynomial. In the 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
Not all implicit curves have parallel curves with analytic representations, but this is possible in some special cases. For instance, the Pythagorean hodograph curves are rational curves with rational parallel curves, which can be converted to implicit representations. Another class of implicit rational curves with rational parallel curves is the parabolas. For the simpler cases of lines and circles the parallel curves can be described easily. For example:
Line (Hesse normalform)
Circle
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:
The parallel curve for distance d is the level set.
Properties of the distance function
Source:
Example: Remark: is not true in the area of interest.
Further examples
The involutes of a given curve are a set of parallel curves. For example: the involutes of a circle are parallel spirals (see diagram).
And:
A parabola has as (two-sided) offsets rational curves of degree 6.
A hyperbola or an ellipse has as (two-sided) offsets an algebraic curve of degree 8.
A Bézier curve of degree has as (two-sided) offsets algebraic curves of degree . In particular, a cubic Bézier curve has as (two-sided) offsets algebraic curves of degree 10.
Parallel curve to a curve with a corner
When determining the cutting path of part with a sharp corner for machining, one must define the parallel (offset) curve to a given curve that has a discontinuous 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 cuspss when the distance from the curve matches the radius of curvature at the sharp corner.
As described above is:
.
is discontinuous, meaning the one-sided limit. Mathematically,
.
However, we can define a normal fan that provides an interpolant at the sharp corner:
.
provides the desired behavior:
Algorithms
In general, the parallel curve of a Bézier curve is not another Bézier curve, a result proved by Tiller and Hanson in 1984. Thus, in practice, approximation techniques are used. Any desired level of accuracy is possible by repeatedly subdividing the curve, though better techniques require fewer subdivisions to attain the same level of accuracy. A 1997 survey by Elber, Lee and Kim is widely cited, though better techniques have been proposed more recently. A modern technique based on curve fitting, with references and comparisons to other algorithms, as well as open source JavaScript source code, was published in a blog post in September 2022.
Another efficient algorithm for offsetting is the level approach described by Kimmel and Bruckstein (1993).
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.:
.
may be negative, too. In this case one gets a parallel surface on the opposite side of the surface (see similar diagram on the parallel curves of a circle). 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
Source:
that means: the tangent vectors for fixed parameters are parallel.
that means: the normal vectors for fixed parameters match direction.
are the shape operators, respectively.
The principal curvatures are the eigenvalues of the shape operator, the principal curvature directions are its eigenvectors, the Gaussian curvature is its determinant, and the mean curvature is half its trace.
are the inverses of the shape operators, respectively.
The principal radii of curvature are the eigenvalues of the inverse of the shape operator, the principal curvature directions are its eigenvectors, the reciprocal of the Gaussian curvature is its determinant, and the mean radius of curvature is half its trace.
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 (even in 2D) to consider a variable distance, e.g. parametrized by another curve. One can for example stroke (envelope) 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
is:
.
, gives you ordinary parallel (aka, offset) curves.
Geometric properties
Source:
that means: the tangent vectors for a fixed parameter are parallel.
As for parallel lines, a normal to a curve is also normal to its general offsets.
the curvature.
the radius of curvature.
When general offset curves are constructed they will have cuspss when the curvature of the curve matches curvature of the offset. These are the points where the curve touches the evolute.
General offset surfaces
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. (this parameterization by normal exists for surfaces whose Gaussian curvature is:
.
, gives you ordinary parallel (aka, offset) surfaces.
Geometric properties
Source:
As for parallel lines, the tangent plane of a surface is parallel to the tangent plane of its general offsets.
As for parallel lines, a normal to a surface is also normal to its general offsets.
are the shape operators, respectively.
The principal curvatures are the eigenvalues of the shape operator, the principal curvature directions are its eigenvectors, the Gaussian curvature is its determinant, and the mean curvature is half its trace.
are the inverses of the shape operators, respectively.
The principal radii of curvature are the eigenvalues of the inverse of the shape operator, the principal curvature directions are its eigenvectors, the reciprocal of the Gaussian curvature is its determinant, and the mean radius of curvature is half its trace.
Note the similarity to the geometric properties of general offset curves.
Derivation of geometric properties for general offsets
(this parameterization by normal exists for surfaces whose Gaussian curvature is:
, gives you ordinary parallel surfaces.)
, which gives us its tangent vectors spanning its tangent plane.
, which share the same unit normal. Thus, the general offset surface shares the same tangent plane and normal with. That aligns with the nature of envelopes.
We now consider the Weingarten equations for the shape operator. 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.
, we get:
.
Next, we use the Weingarten equations:
.
to get back to the Weingarten equations:
.
See also
References
Josef Hoschek: Offset curves in the plane. In: CAD. 17 (1985), S. 77–81.
Takashi Maekawa: An overview of offset curves and surfaces. In: CAD. 31 (1999), S. 165–173.
Further reading
Pages listed are the general and introductory material.
External links