# Parametric Equations, Integration, and Gothic Architecture

We investigated the shapes in gothic window architecture and created a mock-up of a gothic style window from the equations we found using Mathematica. Specifically, we used parametric equations to model the clover-shaped windows that are common motif often found in gothic architecture. We also modeled the parabolic-like curves of a pointed arch, two rectangular windows below it, and the linear base of the window. After finding the parametric equations that correspond to each shape, we integrated these regions to find the surface area of the glass needed for each window.

After having all the littles shapes, we wanted to combine them altogether to create the completed look of a window. Since each piece has different domains, when we used the command of Show[p1,p2,p3,p4,p4], they did not really go along with each other into the way we would like them to be, so we had to shift our parabolas up to make them cover the clover piece. We also changed the size of rectangles to make them fit into the window shape. We played with the coefficients of the two parabola functions to make them more pointed. Lastly, we added a horizontal line at the bottom of the window by the equation f(x)=-25.