Parallel curve

Parallel curves of the graph of

y=1{,}5\sin(x)

for distances

d=0{,}25,\dots ,1{,}5

Two definitions of a parallel curve: 1) envelope of a family of congruent circles, 2) by a fixed normal distance

The parallel curves of a circle (red) are circles, too

A parallel of a curve is the

envelope of a family of congruent circles centered on the curve.

It generalises the concept of parallel lines. It can also be defined as a

curve whose points are at a fixed normal distance from a given curve.^[1]

These two definitions are not entirely equivalent as the latter assumes smoothness, whereas the former does not.^[2]

In Computer-aided design the preferred term for a parallel curve is offset curve^[2]^[3]^[4] (In other geometric contexts, the term offset can also refer to translation.^[5]) 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.^[6]

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.^[7]

Except in the case of a line or circle, the parallel curves have a more complicated mathematical structure than the progenitor curve.^[1] For example, even if the progenitor curve is smooth, its offsets may not be so; this property is illustrated in the adjacent figure using a parabola as progenitor curve.^[2] 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.^[3]

The notion also generalizes to 3D surfaces, where it is called offset surface.^[8] Increasing a solid volume by a (constant) distance offset is sometimes called dilation.^[9] The opposite operation is sometimes called shelling.^[8]

Parallel curve of a parametricaly given curve

If there is a regular parametric representation ${\vec {x}}=(x(t),y(t))$ 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 $|d|$ :

${\vec {x}}_{d}(t)={\vec {x}}(t)+d{\vec {n}}(t)$ with the unit normal ${\vec {n}}(t)$ .

In cartesian coordinates:

x_{d}(t)=x(t)+{\frac {d\;y'(t)}{\sqrt {x'(t)^{2}+y'(t)^{2}}}}

y_{d}(t)=y(t)-{\frac {d\;x'(t)}{\sqrt {x'(t)^{2}+y'(t)^{2}}}}\ .

Distance parameter $d$ may be negative, too. 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 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:^[10]

${\vec {x}}'_{d}(t)\parallel {\vec {x}}'(t)\quad$ , that means: the tangent vectors for a fixed parameter are parallel.
$\kappa _{d}(t)={\frac {\kappa (t)}{1-d\kappa (t)}}\quad$ , with $\kappa (t)$ the curvature of the given curve and $\kappa _{d}(t)$ the curvature of the parallel curve for parameter $t$ .

As for parallel lines, a normal line to a curve is also normal to its parallels.
When parallel curves are constructed they will have cusps 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.

If the given curve is polynomial (that means $x(t)$ und $y(t)$ are polynomials) the parallel curves are in general no more polynomial. In CAD area this is a draw back, because CAD systems use polynomials or rational curves. In order to get at least rational curves, the sqare 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.^[11]

Parallel curves of an implicit curve

Parallel curves of the implicit curve (red) with equation

x^{4}+y^{4}-1=0

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

\;f(x,y)=x+y-1=0\;

→ distance function:

\;h(x,y)={\frac {x+y-1}{\sqrt {2}}}=d\;

(Hesse normalform)

Circle

\;f(x,y)=x^{2}+y^{2}-1=0\;

→ distancefunction:

\;h(x,y)={\sqrt {x^{2}+y^{2}}}-1=d\;.

In general, presuming certain conditions, one can prove the existence of an oriented distance function $h(x,y)$ . In practice one has to treat it numerically ^[12] Considering parallel curves the following is true:

The parallel curve for distance d is the level set $h(x,y)=d$ of the corresponding oriented distance function $h$ .

Properties of the distance function:^[10] ^[13]

$|\operatorname {grad} h({\vec {x}})|=1\;,$
$h({\vec {x}}+d\operatorname {grad} h({\vec {x}}))=h({\vec {x}})+d\;,$
$\operatorname {grad} h({\vec {x}}+d\operatorname {grad} h({\vec {x}}))=\operatorname {grad} h({\vec {x}})\;.$

Example:
The diagram shows parallel curves of the implicit curve with equation $\;f(x,y)=x^{4}+y^{4}-1=0\;.$
Remark: The curves $\;f(x,y)=x^{4}+y^{4}-1=d\;$ are not parallel curves, because $\;|\operatorname {grad} f(x,y)|=1\;$ is not true in the area of interest.

Further examples

Involutes of a circle

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:^[14]

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 $n$ has as (two-sided) offsets algebraic curves of degree $4 n - 2$ . In particular, a cubic Bezier curve has as (two-sided) offsets algebraic curves of degree 10.

Algorithms

An efficient algorithm for offsetting is the level approach described by Kimel and Bruckstein (1993). ^[15]

There are numerous approximation algorithms for this problem. For a 1997 survey, see Elber, Lee and Kim's "Comparing Offset Curve Approximation Methods".^[16]

Generalizations

Offset surface of a complex irregular shape

The problem generalizes fairly obviously to higher dimensions e.g. to offset surfaces, and slightly less trivially to pipe surfaces.^[17] 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.^[18] For curves embedded in 3D surfaces the offset may be taken along a geodesic.^[19]

Another way to generalize it is (even in 2D) to consider a variable distance, e.g. parametrized by another curve.^[20]^[21] One can for example stroke (envelope) with an ellipse instead of circle^[20] as it is possible for example in METAFONT.^[22] More recently Adobe Illustrator has added somewhat similar facility in version CS5, although the control points for the variable width are visually specified.^[23] In contexts where it's important to distinguish between constant and variable distance offsetting the acronyms CDO and VDO are sometimes used.^[9]

References

1 2 Willson, Frederick Newton (1898). Theoretical and Practical Graphics. Macmillan. p. 66. ISBN 1-113-74312-3.
1 2 3 Devadoss, Satyan L.; O'Rourke, Joseph (2011). Discrete and Computational Geometry. Princeton University Press. pp. 128–129. ISBN 1-4008-3898-3.
1 2 Sendra, J. Rafael; Winkler, Franz; Pérez Díaz, Sonia (2007). Rational Algebraic Curves: A Computer Algebra Approach. Springer Science & Business Media. p. 10. ISBN 978-3-540-73724-7.
↑ Agoston, Max K. (2005). Computer Graphics and Geometric Modelling: Mathematics. Springer Science & Business Media. p. 586. ISBN 978-1-85233-817-6.
↑ Vince, John (2006). Geometry for Computer Graphics: Formulae, Examples and Proofs. Springer Science & Business Media. p. 293. ISBN 978-1-84628-116-7.
↑ Marsh, Duncan (2006). Applied Geometry for Computer Graphics and CAD (2nd ed.). Springer Science & Business Media. p. 107. ISBN 978-1-84628-109-9.
↑ http://www.slideshare.net/Mark_Kilgard/22pathrender, p. 28
1 2 Agoston, Max K. (2005). Computer Graphics and Geometric Modelling. Springer Science & Business Media. pp. 638–645. ISBN 978-1-85233-818-3.
1 2 http://www.cc.gatech.edu/~jarek/papers/localVolume.pdf, p. 3
1 2 E. Hartmann: Geometry and Algorithms for COMPUTER AIDED DESIGN. S. 30.
↑ Rida T. Farouki: Pythagorean-Hodograph Curves: Algebra and Geometry Inseparable (Geometry and Computing). Springer, 2008, ISBN 978-3-540-73397-3.
↑ E. Hartmann: Geometry and Algorithms for COMPUTER AIDED DESIGN. S. 81, S. 30, 41, 44.
↑ J.A. Thorpe: Elementary topics in Differential Geometry, Springer-Verlag, 1979, ISBN 0-387-90357-7.
↑ http://faculty.engineering.ucdavis.edu/farouki/wp-content/uploads/sites/41/2013/02/Introduction-to-PH-curves.pdf, p. 16 "taxonomy of offset curves"
↑ Kimmel and Bruckstein (1993) Shape offsets via level sets CAD (Computer Aided Design) 25(3):154–162.
↑ http://www.computer.org/csdl/mags/cg/1997/03/mcg1997030062.pdf
↑ Pottmann, Helmut; Wallner, Johannes (2001). Computational Line Geometry. Springer Science & Business Media. pp. 303–304. ISBN 978-3-540-42058-3.
↑ Chirikjian, Gregory S. (2009). Stochastic Models, Information Theory, and Lie Groups, Volume 1: Classical Results and Geometric Methods. Springer Science & Business Media. pp. 171–175. ISBN 978-0-8176-4803-9.
↑ Sarfraz, Muhammad, ed. (2003). Advances in geometric modeling. Wiley. p. 72. ISBN 978-0-470-85937-7.
1 2 Brechner, Eric L. (1992). "5. General Offset Curves and Surfaces". In Barnhill, Robert E. Geometry Processing for Design and Manufacturing. SIAM. pp. 101–. ISBN 978-0-89871-280-3.
↑ "Properties of Generalized Offset Curves and Surfaces". Hindawi.com. Retrieved 2014-08-15.
↑ https://www.tug.org/TUGboat/tb16-3/tb48kinc.pdf
↑ 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 (also video)

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.

External links

Parallel curves on MathWorld
Visual Dictionary of Plane Curves 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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[Willson-1] 1 2 Willson, Frederick Newton (1898). Theoretical and Practical Graphics. Macmillan. p. 66. ISBN 1-113-74312-3.

[DevadossO'Rourke2011-2] 1 2 3 Devadoss, Satyan L.; O'Rourke, Joseph (2011). Discrete and Computational Geometry. Princeton University Press. pp. 128–129. ISBN 1-4008-3898-3.

[SendraWinkler2007-3] 1 2 Sendra, J. Rafael; Winkler, Franz; Pérez Díaz, Sonia (2007). Rational Algebraic Curves: A Computer Algebra Approach. Springer Science & Business Media. p. 10. ISBN 978-3-540-73724-7.

[Agoston2005m-4] Agoston, Max K. (2005). Computer Graphics and Geometric Modelling: Mathematics. Springer Science & Business Media. p. 586. ISBN 978-1-85233-817-6.

[Vince2006-5] Vince, John (2006). Geometry for Computer Graphics: Formulae, Examples and Proofs. Springer Science & Business Media. p. 293. ISBN 978-1-84628-116-7.

[Marsh2006-6] Marsh, Duncan (2006). Applied Geometry for Computer Graphics and CAD (2nd ed.). Springer Science & Business Media. p. 107. ISBN 978-1-84628-109-9.

[Kilgard-7] ttp://www.slideshare.net/Mark_Kilgard/22pathrender, p. 28

[Agoston2005-8] 1 2 Agoston, Max K. (2005). Computer Graphics and Geometric Modelling. Springer Science & Business Media. pp. 638–645. ISBN 978-1-85233-818-3.

[jarek-9] 1 2 http://www.cc.gatech.edu/~jarek/papers/localVolume.pdf, p. 3

[hart30-10] 1 2 E. Hartmann: Geometry and Algorithms for COMPUTER AIDED DESIGN. S. 30.

[11] Rida T. Farouki: Pythagorean-Hodograph Curves: Algebra and Geometry Inseparable (Geometry and Computing). Springer, 2008, ISBN 978-3-540-73397-3.

[12] E. Hartmann: Geometry and Algorithms for COMPUTER AIDED DESIGN. S. 81, S. 30, 41, 44.

[13] J.A. Thorpe: Elementary topics in Differential Geometry, Springer-Verlag, 1979, ISBN 0-387-90357-7.

[farouki-slides-14] ttp://faculty.engineering.ucdavis.edu/farouki/wp-content/uploads/sites/41/2013/02/Introduction-to-PH-curves.pdf, p. 16 "taxonomy of offset curves"

[15] Kimmel and Bruckstein (1993) Shape offsets via level sets CAD (Computer Aided Design) 25(3):154–162.

[16] ttp://www.computer.org/csdl/mags/cg/1997/03/mcg1997030062.pdf

[PottmannWallner2001-17] Pottmann, Helmut; Wallner, Johannes (2001). Computational Line Geometry. Springer Science & Business Media. pp. 303–304. ISBN 978-3-540-42058-3.

[Chirikjian2009-18] Chirikjian, Gregory S. (2009). Stochastic Models, Information Theory, and Lie Groups, Volume 1: Classical Results and Geometric Methods. Springer Science & Business Media. pp. 171–175. ISBN 978-0-8176-4803-9.

[Sarfraz2003-19] Sarfraz, Muhammad, ed. (2003). Advances in geometric modeling. Wiley. p. 72. ISBN 978-0-470-85937-7.

[barn-20] 1 2 Brechner, Eric L. (1992). "5. General Offset Curves and Surfaces". In Barnhill, Robert E. Geometry Processing for Design and Manufacturing. SIAM. pp. 101–. ISBN 978-0-89871-280-3.

[21] "Properties of Generalized Offset Curves and Surfaces". Hindawi.com. Retrieved 2014-08-15.

[22] ttps://www.tug.org/TUGboat/tb16-3/tb48kinc.pdf

[23] ttp://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 (also video)

Differential transforms of plane curves
Unary operations	Evolute Involute Dual curve Inverse curve Parallel curve Isoptic
Unary operations defined by a point	Pedal & Contrapedal curves Negative pedal curve Pursuit curve Caustic
Unary operations defined by two points	Strophoid
Binary operations defined by a point	Roulette Cissoid
Operations on a family of curves	Envelope