One common advanced geometry subject is 3-D geometry. This does not just mean cubes and cylinders but animals, humans, art, buildings and other complex objects. Describing the dimensions of these items is much more difficult then for the traditional shapes. Engineers and mathematicians must work together to graph the variation, texture and dynamic movement to create the geometry of 3-D objects through computer programs.
Differential Geometry studies problems in geometry using the calculus functions of integration and derivatives. Curved objects in 3-D require these types of calculations. For example, the geometry of a saddle (which has the mathematical name hyperbolic paraboloid) is actually quite complex. Differential geometry experts further complicate their analysis by studying shapes printed on top of curved shapes.
Topology is an important area of advanced mathematics that combines geometry and set theory. It describes continuous objects such as a rubber band that can be stretched, pulled and tangled. Topology describes the abstract forms of objects while ignoring the details. For example, a topologist would see a dough nut and coffee cup as mathematically the same form, according to a mathematical joke.
Algebraic Geometry is another advanced subject that combines advanced algebra such as commutative algebra with geometric functions. It combines topology with number theory and geometry. In addition, it uses polynomial equations to solve geometric problems. Experts in algebraic geometry tend to focus on certain shapes such as planes, curves, ovals and parabolas which can easily be described with their algebraic tools.