A curve in the plane can be specified by a pair of parametric equations, (x(t),y(t)), for t in some interval. There are, of course, other ways of specifying curves, such as by geometric or mechanical constructions. A parabola can be defined geometrically as the set of points whose distance from a given point, called the focus, is the same as the distance to a given line, called the directrix. As an example of a mechanical construction, consider rolling a circle along a line. Define a curve by looking at the path followed by some selected point on the circle. The curve followed by that point is an example of a cycloid.
While all the Exhibits in the Plane Curve gallery are represented, ultimately, by parametric equations, many of them also have mechanical or geometric constructions that can be viewed in 3D-XplorMath-J as animations. In general, the animation will run when the Exhibit first appears. To see the animation again, select the "Create" command from the "Actions" menu.
There are three parameters that apply to every plane curve: Minimum Value of t, Maximum Value of t, and t Resolution. The minimum and maximum value of t determine the interval for which the parametric equations (x(t),y(t)) are displayed. The t-resolution is an integer that tells how many points on the curve are computed; the curve that is actually displayed is created by drawing straight lines from each computed point to the next. Individual exhibits can have additional parameters. Remember that all parameters can be changed using the "Set Parameters" command in the "Settings" menu.
Several commands in the "Actions" menu run animations that might help you to understand the curve. The "Show Tangents and Normals" command runs an animation in which the unit tangent and unit normal vectors to the curve are shown moving along the curve. The "Show Osculating Circles" command moves the "osculating circle" along the curve. The osculating circle at a point on a curve is the circle that best approximates the curve at that point. The radius of the osculating circle is a measure of curvature at that point: the greater the curvature, the smaller the radius. As the osculating circles move along the curve, their centers form a curve known as the evolute of the original curve. The evolute is drawn during the "Show Osculating Circles" animation. The "Display Evolute" checkbox in the Action Menu can be used to hide and show the evolute.
The line from a point on the curve to the center of the osculating circle at that point is normal to the curve. These lines are displayed by the command "Show Osculating Circles with Normals." By displacing points of the curve along these normal lines, one can produce curves that are parallel to the original curve. Use the "Show Parallel Curves" command to see these curves. It is interesting to observe that cusps on the parallel curves lie on the evolute of the original curve. Finally, the "Show Parallel Curves with Normals" command will draw the normals to the curve before showing the parallel curves.