Curves with One Fixed Variable

A curve in space can be expressed as < x, y, z > = < f(t), g(t), h(t) >. If one of the component is assigned to a constant, then we will get a curve on a plane. For example, if < x, y, z > = < 2cos(t), 2sin(t), 3 >, we get a circle of radius 2 units centered at (0, 0, 3) on the plane z = 3.

How do we know it's a circle? First, match the components and we get


When we find the sum of x² and y², we get , where x² + y² = 4 is an equation of a circle in xy-plane for a two dimensional system. Somehow, with the third component, we get another information, which is z = 3. That means, the circle is to be placed at the level of z = 3.

From the graph below, you can see that x² + y² = 4 is a circle in xy-plane. Somehow, in space, you need to define the value of z to keep it as a circle. Otherwise, the equation will give to a cylinder. You may slide the value of z to see the the position of the circle at different level. For better viewing, simply click here.

Betty, Created with GeoGebra, Screencast-O-Matic and Thomas Calculus, Pearson.