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Parallel CurvesOne 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. Imagine it has a built-in (zero-dimensional) 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 is how we define a parallel curve , Xd(t) =X(t) +dU(t) , at distance d. 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.
Demonstration 1: Normal Vectors and Parallel Curves You can choose the range over which normals should be drawn and the number of normals to display by modifying the s domain. You can choose the length of the normals by modifying the beginning and end of the len interval. You can choose the minimum and maximum distance from the curve at which parallel curves should be drawn and the number of parallel curves to draw by modifying the d domain. For many curves, as the vectors get longer they will intersect one another. Observe the patterns these intersections make.
Demonstration 1a: Moving Parallel Curves This demo is the same except that d, the length of the normals and the distance of the parallel curves, is a variable over which the parallel curves can be animated. Only two parallel curves are drawn; however, you can still choose the number of normals to be drawn by modifying s. This demo displays normal vectors and parallel curves of a plane curve. Look particularly at the places where parallel curves have cusps.
Describe the parallel curves of the parabola
X(t)
=(t,t2)
. A good way to look at it is to input it as t->[t, c t
t] and to slowly increase c from zero. For which values of
d
will the parallel curves have cusps?
Describe the parallel curves of the ellipse
X(t)
=c(cos(t),sin(
t))
. For which values of
d
will the parallel curves have cusps?
Describe the parallel curves of the
cardioid
. To observe this curve, you can erase all of the
curve
type-in, and then type Cardioid.
Optional There exist many similarities between the theory of parallel curves in two dimensions and the theory for three dimensions, as you may observe by turning to Lab3, Parallel Curves . |