Geometric Transformations: Projective Geometry Fundamentals

This lecture introduces the fundamental concepts of projective geometry, focusing on the transformations that preserve certain properties. The instructor discusses the historical context of projective geometry, referencing key figures such as Georg Mohr and Lorenzo Mascheroni, who demonstrated that constructions traditionally done with a ruler and compass can also be achieved using only a compass. The lecture emphasizes the importance of projections, defining them as transformations involving three elements: the observer, the object, and the projection surface. Various types of projections are explored, including modern perspective and stereographic projection, highlighting their applications in architecture. The instructor also addresses the significance of invariants in projective geometry, such as alignments and intersections, and how these properties are maintained despite transformations. The lecture concludes with a discussion on the practical implications of projective geometry in architectural design, emphasizing its relevance in creating accurate representations of three-dimensional objects on two-dimensional surfaces.