A centroid of an area correlates to the center of mass of an object. It is equal to the center of gravity if the area's density is consistent. Draw a coordinate system on foam board. Using that system, accurately draw areas of space on the board. Use an integration formula to determine the centroid of those areas, then cut them out to test your calculations. If the area lays flat when suspended in the air, you have calculated correctly. If it leans, your calculations are off.
Soap bubbles have what are known as "minimal surfaces," meaning their mean curvature is zero. Finding a minimal surface of a boundary with specified constraints is a calculus problem called "Plateau's problem." Dipping a wire loop into dish soap creates these bubbles which are some of the few physical examples of minimal surfaces. After blowing your bubbles, observe the differences between their shapes and consistencies before plugging their diameters, shapes and curvatures into your equation.
Lissajous curves are the family of curves described by a set of equations expressing explicit functions of a number of independent variables. They are determined by rectangular boundaries as opposed to circular ones. A Harmonograph measures these curves and can be built using two pendulums to dictate the movement of a pen. Simple Harmonographs can create figure eights, ellipses and spirals by varying the frequency and force applied to the pendulums in relation to each other within the apparatus.
Accurate mapping has always been difficult to portray on paper, but with the advent of Global Positioning Systems, maps are becoming much more precise. If you examine levels of rainfall, temperature variance or topography, for instance, you can calculate gradients, draw flow lines and draw directional derivatives based on their average value and functions, which you can see on GPS maps. The next step is to make a map yourself. Using Mercantor projection, draw out a three-dimensional sphere (usually the globe) on a two-dimensional surface. The coordinates on a sphere are close to longitude and latitude but do not match exactly.