The design and manipulation of geometric spaces are fundamental concepts in mathematics, engineering, and design, with wide applications in building and urban planning, mapping, civil engineering, interior design, and industrial design. With the rapid development of digital technologies, working with geometric spaces is no longer limited to manual calculations and procedures. Instead, digital systems and applications can be developed to perform a range of defined operations on spatial forms, including measurement, subdivision, integration, comparison, and the identification of boundaries, empty areas, and overlapping regions. This approach aims to transform geometric forms from visual representations into digital data that can be accurately and efficiently analyzed and processed by computers. Defined operations refer to a set of procedures predetermined to be performed on a geometric space according to clear mathematical rules. When dealing with a rectangle, for example, a system can calculate its area and perimeter, divide it into equal parts, identify a specific section, or compare it with another geometric area. These operations become particularly important when dealing with complex spaces consisting of multiple geometric forms. In architectural design, for instance, they can be used to analyze the areas of rooms, corridors, and service spaces and examine the functional relationships between them in order to achieve a more efficient and organized spatial distribution. The design process begins by identifying the type of geometric space to be processed, whether it is a square, rectangle, triangle, circle, or irregular shape. The required data are then identified, including dimensions, angles, coordinates, and boundaries. The next stage involves determining the operations to be performed by the system, such as calculating area and perimeter, measuring distances, subdividing spaces, combining shapes, identifying intersection points, and detecting empty or overlapping areas. A fundamental stage is the development of the mathematical rules and algorithms governing these operations. The area of a rectangle, for example, is calculated by multiplying its length by its width, while the calculation of a circle depends on its radius and that of a triangle depends on its base and height. Irregular shapes can be represented using a set of points and coordinates and then processed through appropriate geometric algorithms to calculate their areas and analyze their properties. For a computer to process geometric spaces, they must first be converted into a digital representation based on a coordinate system, in which the positions of points and components forming the shape are represented by numerical values. Once this digital representation is established, the program can automatically perform the required operations and recalculate the results whenever the dimensions or boundaries of the shape are modified. For example, when the length of one side of a given space changes, the system can immediately update its area and perimeter without requiring all calculations to be performed manually again. Colors, symbols, and visual elements can also be employed to distinguish different areas, such as built-up spaces, corridors, and service areas, thereby improving data interpretation and spatial understanding. The practical applications of such systems extend across various fields. In architecture, they can be used to analyze interior spaces and organize functional areas and design elements. In urban planning, they can support the analysis of land parcels, roads, and green spaces. In agriculture, they can be applied to calculate land areas and identify irrigation zones, while in civil engineering they can assist in estimating material quantities associated with dimensions and areas. This approach also provides an important foundation for Geographic Information Systems (GIS), which rely on the analysis of locations, distances, boundaries, and spatial relationships between different elements. The success of designing defined operations on geometric spaces largely depends on the accuracy of the data, mathematical rules, and algorithms employed. Therefore, dimensions and coordinates must be carefully verified, while potential errors resulting from measurement or data entry should be addressed. In addition, the system interface should be designed to be clear and user-friendly, allowing users to enter data and obtain results without requiring advanced knowledge of programming or mathematics. These systems can also be further enhanced by integrating artificial intelligence and digital design technologies, enabling more advanced spatial analysis and the generation of alternative design solutions based on functional, aesthetic, and engineering criteria. In conclusion, designing defined operations on geometric spaces provides an effective means of transferring traditional geometric processes into a digital environment characterized by accuracy, speed, and flexibility. Successful system development depends on identifying the type of geometric space, digitally representing its data, defining the required operations, and applying appropriate mathematical rules and algorithms. This approach contributes to reducing errors, saving time, improving data analysis, and supporting design decision-making. It also opens promising opportunities for developing intelligent systems capable of processing geometric spaces and proposing design solutions with greater efficiency.