A parallel of a
curve
In mathematics, a curve (also called a curved line in older texts) is an object similar to a line, but that does not have to be straight.
Intuitively, a curve may be thought of as the trace left by a moving point. This is the definition that ...
is the
envelope
An envelope is a common packaging item, usually made of thin, flat material. It is designed to contain a flat object, such as a letter or card.
Traditional envelopes are made from sheets of paper cut to one of three shapes: a rhombus, a sho ...
of a family of
congruent circle
A circle is a shape consisting of all points in a plane that are at a given distance from a given point, the centre. Equivalently, it is the curve traced out by a point that moves in a plane so that its distance from a given point is con ...
s 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 In geometry, the perpendicular distance between two objects is the distance from one to the other, measured along a line that is perpendicular to one or both.
The distance from a point to a line is the distance to the nearest point on that line. Th ...
'' 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
Computer-aided design (CAD) is the use of computers (or ) to aid in the creation, modification, analysis, or optimization of a design. This software is used to increase the productivity of the designer, improve the quality of design, improve c ...
the preferred term for a parallel curve is offset curve.
(In other geometric contexts,
the term offset can also refer to
translation.
) 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 (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.
[http://www.slideshare.net/Mark_Kilgard/22pathrender, p. 28]
Except in the case of a line or
circle
A circle is a shape consisting of all points in a plane that are at a given distance from a given point, the centre. Equivalently, it is the curve traced out by a point that moves in a plane so that its distance from a given point is con ...
, the parallel curves have a more complicated mathematical structure than the progenitor curve.
For example, even if the progenitor curve is
smooth
Smooth may refer to:
Mathematics
* Smooth function, a function that is infinitely differentiable; used in calculus and topology
* Smooth manifold, a differentiable manifold for which all the transition maps are smooth functions
* Smooth algebraic ...
, its offsets may not be so; this property is illustrated in the top figure, using a
sine curve
A sine wave, sinusoidal wave, or just sinusoid is a mathematical curve defined in terms of the '' sine'' trigonometric function, of which it is the graph. It is a type of continuous wave and also a smooth periodic function. It occurs often i ...
as progenitor curve.
In general, even if a curve is
rational
Rationality is the quality of being guided by or based on reasons. In this regard, a person acts rationally if they have a good reason for what they do or a belief is rational if it is based on strong evidence. This quality can apply to an abili ...
, 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 or parallel surface.
Increasing a solid volume by a (constant) distance offset is sometimes called ''dilation''.
[http://www.cc.gatech.edu/~jarek/papers/localVolume.pdf, p. 3] 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 (s. above) leads to the following parametric representation of the parallel curve with distance
:
:
with the unit normal
.
In cartesian coordinates:
:
:
The distance parameter
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:[E. Hartmann]
''Geometry and Algorithms for COMPUTER AIDED DESIGN.''
S. 30.
*
that means: the tangent vectors for a fixed parameter are parallel.
*
with
the
curvature
In mathematics, curvature is any of several strongly related concepts in geometry. Intuitively, the curvature is the amount by which a curve deviates from being a straight line, or a surface deviates from being a plane.
For curves, the canon ...
of the given curve and
the curvature of the parallel curve for parameter
.
*
with
the
radius of curvature
In differential geometry, the radius of curvature, , is the reciprocal of the curvature. For a curve, it equals the radius of the circular arc which best approximates the curve at that point. For surfaces, the radius of curvature is the radius o ...
of the given curve and
the radius of curvature of the parallel curve for parameter
.
* 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
cusp
A cusp is the most pointed end of a curve. It often refers to cusp (anatomy), a pointed structure on a tooth.
Cusp or CUSP may also refer to:
Mathematics
* Cusp (singularity), a singular point of a curve
* Cusp catastrophe, a branch of bifurc ...
s when the distance from the curve matches the radius of
curvature
In mathematics, curvature is any of several strongly related concepts in geometry. Intuitively, the curvature is the amount by which a curve deviates from being a straight line, or a surface deviates from being a plane.
For curves, the canon ...
. 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.
If the given curve is polynomial (meaning that
and
are polynomials), 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:
: ''Line''
→ distance function:
(Hesse normalform)
: ''Circle''
→ distance function:
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 of the corresponding oriented distance function
.
Properties of the distance function:
*
*
*
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
*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:
[http://faculty.engineering.ucdavis.edu/farouki/wp-content/uploads/sites/41/2013/02/Introduction-to-PH-curves.pdf, p. 16 "taxonomy of offset curves"]
* 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
In mathematics, an affine algebraic plane curve is the zero set of a polynomial in two variables. A projective algebraic plane curve is the zero set in a projective plane of a homogeneous polynomial in three variables. An affine algebraic plane ...
of degree 8.
* A
Bézier curve
A Bézier curve ( ) is a parametric curve used in computer graphics and related fields. A set of discrete "control points" defines a smooth, continuous curve by means of a formula. Usually the curve is intended to approximate a real-world shape ...
of degree has as (two-sided) offsets
algebraic curve
In mathematics, an affine algebraic plane curve is the zero set of a polynomial in two variables. A projective algebraic plane curve is the zero set in a projective plane of a homogeneous polynomial in three variables. An affine algebraic plane ...
s of degree . In particular, a cubic Bezier 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, you 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
cusp
A cusp is the most pointed end of a curve. It often refers to cusp (anatomy), a pointed structure on a tooth.
Cusp or CUSP may also refer to:
Mathematics
* Cusp (singularity), a singular point of a curve
* Cusp catastrophe, a branch of bifurc ...
s when the distance from the curve matches the radius of
curvature
In mathematics, curvature is any of several strongly related concepts in geometry. Intuitively, the curvature is the amount by which a curve deviates from being a straight line, or a surface deviates from being a plane.
For curves, the canon ...
at the sharp corner.
Normal fans
As described
above, the parametric representation of a parallel curve,
, to a given curver,
, with distance
is:
:
with the unit normal
.
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:
:
where
.
The resulting definition of the parallel curve
provides the desired behavior:
:
Algorithms
In general, the parallel curve of a
Bézier curve
A Bézier curve ( ) is a parametric curve used in computer graphics and related fields. A set of discrete "control points" defines a smooth, continuous curve by means of a formula. Usually the curve is intended to approximate a real-world shape ...
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.
If there is a regular parametric representation
of the given surface available, the second definition of a parallel curve (see above) generalizes to the following parametric representation of the parallel surface with distance
:
:
with the unit normal
.
Distance parameter
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:
*
that means: the tangent vectors for fixed parameters are parallel.
*
that means: the normal vectors for fixed parameters match direction.
*
where
and
are the
shape operator
In mathematics, the differential geometry of surfaces deals with the differential geometry of smooth surfaces with various additional structures, most often, a Riemannian metric.
Surfaces have been extensively studied from various perspectives ...
s for
and
, respectively.
:The principal curvatures are the
eigenvalues of the
shape operator
In mathematics, the differential geometry of surfaces deals with the differential geometry of smooth surfaces with various additional structures, most often, a Riemannian metric.
Surfaces have been extensively studied from various perspectives ...
, the principal curvature directions are its
eigenvectors, the
Gaussian curvature is its
determinant, and the mean curvature is half its
trace.
*
where
and
are the inverses of the
shape operator
In mathematics, the differential geometry of surfaces deals with the differential geometry of smooth surfaces with various additional structures, most often, a Riemannian metric.
Surfaces have been extensively studied from various perspectives ...
s for
and
, respectively.
:The principal radii of curvature are the
eigenvalues of the inverse of the
shape operator
In mathematics, the differential geometry of surfaces deals with the differential geometry of smooth surfaces with various additional structures, most often, a Riemannian metric.
Surfaces have been extensively studied from various perspectives ...
, 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
A parallel of a curve is the envelope of a family of congruent 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'' ...
.
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
Adobe Illustrator is a vector graphics editor and design program developed and marketed by Adobe Inc. Originally designed for the Apple Inc., Apple Macintosh, development of Adobe Illustrator began in 1985. Along with Creative Cloud (Adobe's shi ...
has added somewhat similar facility in version
CS5
Adobe Creative Suite (CS) is a discontinued software suite of graphic design, video editing, and web development application software, applications developed by Adobe Systems.
The last of the Creative Suite versions, Adobe Creative Suite 6 (CS6) ...
, although the control points for the variable width are visually specified.
[http://design.tutsplus.com/tutorials/illustrator-cs5-variable-width-stroke-tool-perfect-for-making-tribal-designs--vector-4346 application of the generalized version in Adobe Illustrator CS5 (als]
video
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
(this parameterization by normal exists for curves whose curvature is strictly positive or negative, and thus convex, smooth, and not straight). The parametric representation of the general offset curve of
offset by
is:
:
where
is the unit normal of
.
Note that the trival offset,
, gives you ordinary parallel (aka, offset) curves.
Geometric properties:
*
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.
*
with
the
curvature
In mathematics, curvature is any of several strongly related concepts in geometry. Intuitively, the curvature is the amount by which a curve deviates from being a straight line, or a surface deviates from being a plane.
For curves, the canon ...
of the general offset curve,
the curvature of
, and
the curvature of
for parameter
.
*
with
the
radius of curvature
In differential geometry, the radius of curvature, , is the reciprocal of the curvature. For a curve, it equals the radius of the circular arc which best approximates the curve at that point. For surfaces, the radius of curvature is the radius o ...
of the general offset curve,
the radius of curvature of
, and
the radius of curvature of
for parameter
.
*When general offset curves are constructed they will have
cusp
A cusp is the most pointed end of a curve. It often refers to cusp (anatomy), a pointed structure on a tooth.
Cusp or CUSP may also refer to:
Mathematics
* Cusp (singularity), a singular point of a curve
* Cusp catastrophe, a branch of bifurc ...
s when the
curvature
In mathematics, curvature is any of several strongly related concepts in geometry. Intuitively, the curvature is the amount by which a curve deviates from being a straight line, or a surface deviates from being a plane.
For curves, the canon ...
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.
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
(this parameterization by normal exists for surfaces whose
Gaussian curvature is strictly positive, and thus convex, smooth, and not flat). The parametric representation of the general offset surface of
offset by
is:
:
where
is the unit normal of
.
Note that the trival offset,
, gives you ordinary parallel (aka, offset) surfaces.
Geometric properties:
*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.
*
where
and
are the
shape operator
In mathematics, the differential geometry of surfaces deals with the differential geometry of smooth surfaces with various additional structures, most often, a Riemannian metric.
Surfaces have been extensively studied from various perspectives ...
s for
and
, respectively.
:The principal curvatures are the
eigenvalues of the
shape operator
In mathematics, the differential geometry of surfaces deals with the differential geometry of smooth surfaces with various additional structures, most often, a Riemannian metric.
Surfaces have been extensively studied from various perspectives ...
, the principal curvature directions are its
eigenvectors, the
Gaussian curvature is its
determinant, and the mean curvature is half its
trace.
*
where
and
are the inverses of the
shape operator
In mathematics, the differential geometry of surfaces deals with the differential geometry of smooth surfaces with various additional structures, most often, a Riemannian metric.
Surfaces have been extensively studied from various perspectives ...
s for
and
, respectively.
:The principal radii of curvature are the
eigenvalues of the inverse of the
shape operator
In mathematics, the differential geometry of surfaces deals with the differential geometry of smooth surfaces with various additional structures, most often, a Riemannian metric.
Surfaces have been extensively studied from various perspectives ...
, 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
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
(this parameterization by normal exists for surfaces whose
Gaussian curvature is strictly positive, and thus convex, smooth, and not flat). The parametric representation of the general offset surface of
offset by
is:
:
where
is the unit normal of
. (The trival offset,
, gives you ordinary parallel surfaces.)
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 The Weingarten equations give the expansion of the derivative of the unit normal vector to a surface in terms of the first derivatives of the position vector of a point on the surface. These formulas were established in 1861 by the German mathematic ...
for the
shape operator
In mathematics, the differential geometry of surfaces deals with the differential geometry of smooth surfaces with various additional structures, most often, a Riemannian metric.
Surfaces have been extensively studied from various perspectives ...
, 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:
:
where
is the shape operator for
.
Next, we use the
Weingarten equations The Weingarten equations give the expansion of the derivative of the unit normal vector to a surface in terms of the first derivatives of the position vector of a point on the surface. These formulas were established in 1861 by the German mathematic ...
again to replace
:
:
where
is the shape operator for
.
Then, we solve for
and multiple both sides by
to get back to the
Weingarten equations The Weingarten equations give the expansion of the derivative of the unit normal vector to a surface in terms of the first derivatives of the position vector of a point on the surface. These formulas were established in 1861 by the German mathematic ...
, this time for
:
:
:
Thus,
, and inverting both sides gives us,
.
See also
*
Bump mapping
Bump mapping is a texture mapping technique in computer graphics for simulating bumps and wrinkles on the surface of an object. This is achieved by perturbing the surface normals of the object and using the perturbed normal during lighting calcul ...
*
Distance function and
signed distance function
In mathematics and its applications, the signed distance function (or oriented distance function) is the orthogonal distance of a given point ''x'' to the boundary of a set Ω in a metric space, with the sign determined by whether or not ''x' ...
*
Distance field
*
Offset printing
Offset printing is a common printing technique in which the inked image is transferred (or "offset") from a plate to a rubber blanket and then to the printing surface. When used in combination with the lithographic process, which is based on t ...
*
Tubular neighborhood
In mathematics, a tubular neighborhood of a submanifold of a smooth manifold is an open set around it resembling the normal bundle.
The idea behind a tubular neighborhood can be explained in a simple example. Consider a smooth curve in the ...
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
*
*
*
*
Free online version
*
* Pages listed are the general and introductory material.
*
External links
Xah Lee
* http://library.imageworks.com/pdfs/imageworks-library-offset-curve-deformation-from-Skeletal-Anima.pdf application to animation; patented as http://www.google.com/patents/US8400455
* http://www2.uah.es/fsegundo/Otros/Offset/16-SanSegundoSendraSendra-1532.pdf
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