Geometric Properties of Ellipses and Their Applications

This lecture covers the geometric properties of ellipses, focusing on their parametric equations and the relationship between ellipses and epicyclic motion. The instructor begins by discussing the intrinsic axial triangle and its significance in defining ellipses. The construction of ellipses using a compass is demonstrated, highlighting the identification of foci. The lecture then transitions to the parametric equations for ellipses, explaining how they relate to circular bases and projections. The instructor emphasizes the importance of understanding the elliptical structure in architectural contexts, particularly in stereotomy. The discussion includes the challenges of constructing elliptical arches and the methods used to achieve accurate representations. The lecture concludes with a review of the Cartesian equation of an ellipse and its derivation from the parametric form, reinforcing the connection between geometric principles and practical applications in design and architecture.