2.3: Parallel Curves

One of the most important topics in the theory of plane curves is the study of parallel curves. Imagine a curve is on the surface of a pool of water and that it has a built-in motor that impels it to vibrate steadily with a very small amplitude. Waves will recede from the curve at a constant rate. Since each point comprising a given wave travels at the same speed, at a particular time, each point on the wave will have been displaced the same distance from its starting position in the direction of the curve's normal vector at that point. This produces a sequence of parallel curves. If we parameterize the original curve as X(t), then the parallel curve at a distance d is defined as

Xd(t) = X(t) + d*U(t)

If the curve is a straight line, the locus of points at fixed distance from the line forms two parallel lines, and no matter how great the distance, these lines will never develop singularities. If the curve is a circle, the waves created at a given time form two concentric circles, one inside and one outside. However, when the distance equals the radius, one of the parallel curves degenerates to a point, which we consider a completely singular curve. For other curves, the singularity behavior of parallel curves can be quite a bit more complicated. Studying such singularities gives insight into the degree to which the curve differs from a straight line, that is, its curvature.

This demo shows a curve X(t) and its parallel curve at a distance r from it. The normals to X(t) connect the two curves. There is a tapedeck that controls t he value of r.



1. Describe the parallel curves of the parabola X(t) = (t, t2). For which values of r will the parallel curves have cusps?

2. Describe the parallel curves of the ellipse X(t) = (c*cos(t), sin(t)). For which values of r will the parallel curves have cusps?

3. Describe the parallel curves of the cardioid X(t) = (c + cos(t))*(cos(t), sin(t)).

4. Describe the parallel curves of the exponential spiral.
In the curvature section, we introduced the concept of parallel curves at a given distance d. At each stage we may scale the image so that it fits on the screen by multiplying the parallel curve Xd(t) = X(t) + d*U(t) by v = 1/(1+d). Thus we obtain

Xv(t) = (1 - v)X(t) + vU(t)

as v changes from 0 to 1. This means that the scaled parallel curve is just a linear interpolation between the curve and its normal image. Stated somewhat picturesquely, the normal circular image is the parallel curve at infinity. This becomes considerably more interesting in the case of hypersurfaces.


The Parallel at Distance v/(1-v) window shows the curve X(t) and its parallel at its actual distance. When the distance is large, you will have to rescale the display in order to see the parallel curve; as you do this, you can see that the original curve diminishes to a point in relative size. The Normalized Parallel Curve window shows the curve and the parallel curve scaled according to the function Xv(t) = (1-v)*X(t) + v*U(t). There is a tapedeck that controls the value of v.


What can be said about parallel curves and evolute curves of a piecewise circular curves, made up of a sequence of circular arcs with each one tangent to the next? For example, take a look at the portion of the golden spiral in the demonstration above. The golden spiral is made up of a sequence of circular arcs with larger and larger radii as we move outward. In the demo, the parallel curves are drawn at a distance v/(1-v) from the spiral. Use the tape deck in the control panel to run through different values of v.



1. The golden spiral is a piecewise circular curve in which the tangent vector is continuous at the connecting points. At what distances will the parallel curve have corners?

2. Now consider a piecewise circular curve that does not have a continuous tangent vector. Describe how the family of parallel curves behaves around a cusp or a corner.
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