Tiling the hyperbolic plane by iterated reflection in the sides of a kaleidoscopic polygon can be employed to make many beautiful and artistic patterns in the plane. These constructions may be greatly assisted by computer methods, provided we can construct the sides and vertices of these polygons, and hence the reflections in the sides of these polygons. We show how simple analytic geometry may be used to construct kaleidoscopic triangles and quadrilaterals. This work was motivated by work with undergraduates Dawn Haney, Lori McKeough, and Brandi Smith in the Rose-Hulman NSF-REU Tilings project. site. Indeed, the methods of the paper have been used for visualization in a classification project of all divisible tilings of the hyperbolic plane. A description of the project is available at the same site. More importantly, figures depicting the divisible tilings in the classification -- using the methods of this paper.