In this thesis, we continue the research of investigating the central fan question through numerical computation of minimal surfaces and the author develops a new conformal map in hopes of producing a more accurate and simplistic algorithm. The conformal mapping module adds greater power and accuracy to existing toolboxes" designed for mathematical and technical problems that require those special transformations. In addition, the investigator numerically solves the Riemann-Hilbert problem and explores whether enough data for conjectures exist for central fans. With the use of MATLAB for numerical computations, we develop tools to analyze different capillary surfaces at reentrant corners to determine where a central fan does and does not exist. The author computes minimal surfaces from their Enneper-Weierstrass representation based on holomorphic functions, (f, g) , determined by their boundary conditions. Here f is determined by solving the Riemann-Hilbert problem defined by the geometry of the boundary, while we choose g to be the identity map applied to the conformal image of the boundary onto the unit circle. The understanding of capillary surfaces has applications outside the field of mathematics, while we choose g to be the identity map applied to the conformal image of the boundary onto the unit circle. The understanding of capillary surfaces has applications outside the field of mathematics, especially in engineering. Computing the surfaces will allow us to consider the flow of fluids in zero gravity and other areas where surface tension plays an important role such as in instruments and components built in spacecraft. This study will enable engineers to test their designs by computing the capillary surfaces in many engineering applications such as DNA microarray processors, microthermal technologies, and fluid dynamics in zero gravity.