Rotating about Y-AXIS: Creating Surfaces of Revolution in GGB AR
- Tim Brzezinski
Consider the graph of an equation graphed in the coordinate plane. In calculus, we often end up studying the solid of revolution formed by rotating the graph of such an equation about the Y-AXIS. In GeoGebra's 3D Graphing Calculator, this is actually quite easy to do. The silent screencast below illustrates how easy this actually is. Here, we illustrate how to EASILY rotate two functions ( and about the Y-AXIS.
Try it yourself before moving forward! Note: You don't have to use the function(s) illustrated above. You can use ANY FUNCTION!
However, GeoGebra's Augmented Reality app currently only allows users to plot to surfaces of the form . That is, z need to be written as a function of x and y. Yet in order for us to be able to rotate graphs of equations about the Y-AXIS in GeoGebra Augmented Reality, we are restricted, at the moment, to use equations that can be rewritten so that x is written explicitly as one (or more) function(s) of y. To see this in action, move the LARGE YELLOW POINT, in the applet below, up the y-axis. Note how these cross sections parallel to the xz-plane are always circles with RADIUS = x.
To carry this out in GeoGebra Augmented Reality, we first need to write x explicitly in terms of y That is, we need to write x as a function of y. Thus, . Give this, the equation of ANY circular cross section above is Upon solving the equation above for z, we obtain and . Thus, given this, any surface of revolution formed by rotating the graph of about the Y-AXIS can be considered to be these 2 SURFACES PUT TOGETHER: z = a surface with POSITIVE OUPUTS (top half) z = a surface with NEGATIVE OUTPUTS (bottom half). Since our original equations (above) were , this is equivalent to . Thus, , and we obtain = blue surface shown above. = pink surface shown above.