Parallel curve: Difference between revisions

Revision as of 02:47, 1 April 2020

Revision as of 15:53, 6 January 2021

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:<math> y_d(t)= y(t)-\frac{d\; x'(t)}{\sqrt {x'(t)^2+y'(t)^2}} \ .</math>

Distance parameter <math>d</math> 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:<ref name="hart30">E. Hartmann: [http://www.mathematik.tu-darmstadt.de/~ehartmann/cdgen0104.pdf ''Geometry and Algorithms for COMPUTER AIDED DESIGN.''] S. 30.</ref>===

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==Parallel curve to a curve with a corner==

[[File:Parallel curves to a curve with a discontinuous normal.png|thumb|Parallel curves to a curve with a discontinuous normal around 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 discontinousdiscontinuous 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 [[Cusp (singularity)|cusp]]s when the distance from the curve matches the radius of [[curvature]] at the sharp corner.

===Normal fans===

As described [[#Parallel curve of a parametrically given curve|above]], the parametric representation of a parallel curve, <math>\vec x_d(t)</math>, to a given curver, <math>\vec x(t)</math>, with distance <math>|d|</math> is: