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.

Planes parallel to Coordinate Planes

A plane in space can be described by the equation A₁x + A₂y + A₃z = D.

If two of A_i = 0, we get a plane that is parallel to coordinate plane. What is coordinate plane? Coordinate plane is a plane containing two axes, for example xy-plane is a coordinate plane containing x- and y-axis. As such, there are 3 coordinate planes in space:

1. x = 0 : this is yz-plane (the blue plane).
2. y = 0 : this is xz-plane (the purple plane).
3. z = 0 : this is xy-plane (the brown plane).

Equations like x = 1, x = -2 are planes parallel to yz-plane. 

To sketch a plane that is parallel to coordinate plane, just use 2 pairs of lines that are parallel to the respective axes. For example, to sketch the plane x = 1 (parallel to yz-plane), draw two lines that are parallel y-axis and another 2 lines parallel to z-axis. Then, just label the x-intercept. You may refer to the video below to learn the drawing of plane.

You may also slide the slider at the left column to see the movement of the plane at the right. Take note that the x/y/z-intercept changes as the plane moves. You may also choose to play around with the planes here.

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