Parallel Curve

parallel curves
Two Artistic Rendering by Parallels of a Deltoid

Mathematica Notebook for This Page parallel_gallery.nb.zip

History

Leibnitz considered parallel curves in 1692-4, following Huygens' (1673) study of involutes.

Description

Parallel are not only defined for lines but for curves as well. Parallel of a curve is the envelope of circles whose centers are on the curve. Alternatively, parallels can be defined as the locus of points Q1, Q2 on the normals of the curve, where Q1 and Q2 are d distant from the curve.

Step by step description:

  1. Draw a normal line of length 2 d centerd on a point P on the curve.
  2. Repeat this for other points p on the curve.
  3. The locus of end points of normal lines are the parallel curves to the given curve with offset d.

Parallel curves may not be similar to the orginal curve if d is large. Below are three illustrations for the parallels of sine curve. Notice the wedge shaped loop that is not in the original curve.

parallel curves parallel curves
Parallels of sine curve.
parallel curves parallelSin
Generating Parallels.

Formula

Given a parametric curve {xf[t], yf[t]}, the parametric formulas for its parallel (two branches) with a positive offset d is

{ xf[t] + d yf'[t]/Sqrt[xf'[t]^2 + yf'[t]^2],
  yf[t] - d xf'[t]/Sqrt[xf'[t]^2 + yf'[t]^2] }

This formula is easily derived by considering the unit normal of curve at t, scale it by d, then move it to the point on the curve. If the curve is written in complex notation z[t], then the formula for parallel is simply d*N[t]+z[t], where N[t] is the unit normal at t.

Properties

Parallels and Evolute

Theorem: The locus of Cusps of a curve C's parallel curves is the Evolute of C. This is a alternative definition of evolute.

parallel curves parallel curves parallel curves
Sine curve and its parallels, its evolute, and the two superimposed. The evolute is drawn from light blue to dark blue. Note the asymptotes in the evolute. The asymptotes happens at inflection points. Graphics code: sine_parallel_evolute.nb.zip.

Graphics Gallery

parallel curves
Parallels of the Lemniscate of Bernoulli.

Animation of parallels of a 3-pedaled Rose with varing distance: trifolium_parallel.mov

parallel curves
Parallels of a hypotrochoid. parallel trochoid template.

Related Web Sites

See: Websites on Plane Curves, Printed References On Plane Curves.

Robert Yates: Curves and Their Properties.

Parallel curve.

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