Literally Changing the Point of View

For many years computer graphics has been used in calculus-level classrooms to visualize diverse mathematical objects. In the two dimensional case, this usually encompasses reproducing images that are familiar from centuries of masterful art practiced in books and on blackboards. Unfortunately, in the three dimensional case, many uses are of a disappointingly similar character, instead of pouncing on the opportunity to get new and deeper insights that previously were beyond reach. We demonstrate how three dimensional visualization can link together formerly disparately treated topics, cultivate the central mathematical ideas of lifting and projecting, and foster the habit of exploring new points of view. This article gives three innovative examples that do the latter. It challenges the reader to reconsider all situations that involve more than two variables: maybe there is a better three-dimensional view that is not part of the tradition just because one could not do it at the time. The examples discussed address constrained optimization problems, compositions of functions, and the product rule of differentiation.