Space Curves Gallery

A curve in three-dimensional space can be specified, for example, by a triple of parametric equations, (x(t),y(t),z(t)), for t in some interval. All the curves in the "Space Curves" gallery are ultimately represented in this way, although a few, such as the "Curves of Constant Curvature," are actually defined by other properties. There are three parameters that apply to every space 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),z(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.

When a space curve is drawn in a standard Monocular view, it is projected onto a plane before it is drawn, and several points on the curve can project onto the same point in the plane. When this happens, the back curve is drawn with a gap to make it clear which piece of the curve is on top. The "Thick Curve" option in the "Action" menu can be turned on to give an even clearer visual presentation of the "over-and-under" properties of the drawing. The "Thick Curve" representation gives a surprisingly three-dimensional appearance to the drawing, especially when the curve is rotating.

The "Show Repere Mobile" command in the "Action" menu shows a triple of three orthogonal unit vectors moving along the curve. The three vectors are the unit tangent vector, the standard unit normal, and the bi-normal vector that is obtained as the cross product of the unit tangent and unit normal vectors. (The standard unit normal is the one the points in the direction of the center of the osculating circle.)

The "Show Osculating Circles" command shows an animation in which the osculating circle rolls along the curve. The osculating circle at a point on the curve is the circle that best approximates the curve at that point. The radius of the osculating circle is a measure of the curvature of the curve at that point. As the osculating circles moves along the curve, their centers form another curve, which is called the evolute of the original curve. The evolute is drawn by the "Show Osculating Circles" animation, and it remains after the animation ends. The "Display the Evolute" option in the "Action" menu can be used to hide and show the evolute.

Sometimes it is useful to view a space curve as a "tube," a surface that is drawn around the one-dimensional curve. The "Action" menu contains a choice two views of the space curve: "View as Curve" shows the usual curve, and "View as Tube" shows a surface surrounding the curve. The space curve has parameters "Tube Size" and "Tube Sides" that determine, respectively, the radius of the tube and the number of sides in the polygonal cross-section of the tube. The values of these parameters can, of course, be set using the "Set Parameters" command in the "Settings" menu.