Perspective Transformations for Architectural Design Cornelie Leopold Abstract Geometric transformations enable to represent spatial objects in different ways dependent of the kind of transformation. The methods of representation had not been combined with the idea of transformation at the beginning of their development. An important step had been done in the intellectual history of perspective, by bringing together the mathematical concept of projections and, further on, transformations with the representational concept of perspective according to seeing and perceiving. The step to projective geometry gave the possibility to apply the perspective transfor- mation on objects in space and to receive again spatial objects by the transformation. The consequence had been a systematically theoretical work out of relief perspective as spatial transformation, which started as theatre stage design. This comprehension enables applications in architectural design processes. The fruitful interlacing of prac- tice in art and architecture with mathematical theory will be reflected and applied in architectural design as spatial transformation considering perception. Keywords Transformation · Architectural design · Relief perspective 1 From Perspective to Scenography The relationship between space and image, architectural spatial design and its percep- tion is the starting point of our research. Perspective had been developed in order to produce images of space according to our seeing [1]. The important historical step for drawing perspectives to represent according to our seeing had been done by Alberti. The further developed perspective machines, for example by Albrecht Dürer, helped to translate the concept of perspective in practical guiding tools for the production of the perspective images, therefore, as a tool in art. The mathematical concept of the perspective transformation as collineation later brought the possibility to apply the transformation to the spatial object, in order to receive a transformed spatial object. The perspectival transformed spaces or objects started in the sixteenth century with C. Leopold (B ) TU Kaiserslautern, Kaiserslautern, Germany e-mail: cornelie.leopold@architektur.uni-kl.de © Springer Nature Switzerland AG 2020 V. Viana et al. (eds.), Thinking, Drawing, Modelling, Springer Proceedings in Mathematics & Statistics 326, https://doi.org/10.1007/978-3-030-46804-0_6 77