Making a Goblet

Making a Goblet.

We can create a model for a goblet (or lots of other things) using a surface of revolution. First we need to decide on the shape of the goblet. One possibility would be to use the graph of a polynomial function revolved about an axis. If we know what we want the graph of the polynomial function to look like then we could choose some points that would be on the graph and use a regression analysis utility to create the polynomial function of a chosen degree that is the best fit to the data. One such utility would be a TI graphing calculator. Click here to see an example of finding the best fit fourth degree polynomial function for a given set of data points. Here is the polynomial regression utility I used to create the polynomial function used in creating the picture of a goblet linked to below. I plotted some points to get the shape I wanted. I then let the software determine the best fit fourth degree polynomial. The result was approximately the polynomial function below.

y = 0.0019x⁴ - 0.0566x³ + 0.5464x² - 1.6355x + 1.872

Here is a graph of the function over the interval used in modeling the goblet.

I then switched to

x = 0.0019z⁴ - 0.0566z³ + 0.5464z² - 1.6355z + 1.872 (1)

and formed the surface of revolution that would be my goblet by revolving the graph of (1) about the z-axis.

The equation of this surface would be

x² + y² = (0.0019z⁴ - 0.0566z³ + 0.5464z² - 1.6355z + 1.872)².

DPGraph Picture of the Goblet Blue DPGraph Picture of the Goblet

Here is a colorful goblet animation with a rounded top.

Here is a DPGraph Picture that starts out as the same goblet linked to above but in this image the DPGraph scrolbar can be used to change the coefficients of z and thus change the surface (dramatically).

Here is one more goblet created in a manner similar to that above. In this case I used a third degree polynomial function as my model, used an inequality in my DPGraph picture in order to get a different color for the interior and exterior of the goblet, and set the DPGraph transparency setting to 0.5 (you can also alter the transparency setting using the scrollbar). The equation of the second goblet is given below. The variable z ranges from 0 to 10.

x² + y² = (- 0.015588z³ + 0.2622756z² - 1.158147z + 1.679419)²

DPGraph Picture of the second goblet

Extra Credit: Make your own goblet or something else interesting and show it to me.

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Lane Vosbury, Mathematics, Seminole State College email: vosburyl@seminolestate.edu