Geometric-Modeling-in-Graphics-and-Visual-Computing-Updated Version.pptx

Geometric modelling

1. Business School University of Saint Thomas Mozambique . Barchelor’s Degree in Computer Science . GRAPHICAL USER INTERFACE Topic : Geometric Modeling in Graphics and Visual Computing . Authors:Erick .P.Wasonga,Daniel Darsamo,Scovia Kainza & Lucia Pedro. Lecturer: Marciano Ombe . 1

2. Geometric Modeling in Graphics and Visual Computing Modeling is fundamental to graphics and visual computing, enabling the representation and manipulation of visual content in 2D and 3D environments. This report focuses on geometric modeling, a critical component that involves processing geometric objects, organizing space, and implementing mathematical transformations. This study reviews theoretical concepts detailing various spaces and primitive objects. It explores the mathematical framework, highlighting points, scalars, and vectors. A case study illustrates the application of these theories, developing a modeling application using computer graphics techniques. Insights from this study provide a foundation for further research in graphics and visual computing. 2

3. Geometric Modeling: The Foundation of Computer Graphics Digital Sculpting Geometric modeling acts as the digital equivalent of sculpting tools, allowing the creation of virtual objects and environments. This process relies on mathematical principles to ensure accuracy and realism. Historical Evolution Geometric modeling evolved from simple 2D wireframes in the 1960s to today’s complex 3D renders, driven by advancements in hardware and algorithms. Understanding this field is essential for various applications. 3

4. 2D and 3D Primitives: Building Blocks of Models 2D Primitives In 2D modeling, the simplest elements are points, lines, and polygons. These primitives are the backbone of vector graphics, allowing images to scale infinitely without losing clarity. 3D Primitives 3D modeling introduces depth, adding a z-coordinate to create vertices. Advanced primitives like Bézier curves and NURBS enable complex shapes, vital in industries like automotive design. 4

5. Mathematical Spaces: Where Geometry Lives Euclidean Space The familiar world of classical geometry, used for rigid transformations, such as rotating a cube without distorting its shape. Affine Space Supports translations and scaling, preserving parallelism but not necessarily distances or angles. Crucial in CAD for scaling models. Vector Space Handles directional quantities, essential for tasks like calculating lighting in a 3D scene to determine shadows and highlights on objects. 5

6. Transformations: Moving and Reshaping Objects Matrices Matrices encode transformations in 3D space. A rotation matrix reorients an object around an axis, fundamental in animations. Scaling A scaling matrix enlarges or shrinks an object. Shearing slants objects, useful for simulating perspective or deformations. 6

7. Spatial Relationships: Position, Lengths, and Angles Position Defines where an object is located in space, tracked using (x,y,z) coordinates in a 3D game. Lengths Maintain proportions during transformations; scaling a building model requires halving window distances to preserve realism. Angles Ensure structural integrity; angles between robot arm segments determine its reach and flexibility. 7

8. Fundamental Data Types and Computational Power Data Types Points anchor objects in space. Scalars are single values. Vectors combine magnitude and direction. Computational Power Modern GPUs accelerate matrix operations, enabling real-time rendering. Parallel processing allows simultaneous transformations, lighting, and shading, creating seamless visuals. 8

9. Cutting-Edge Innovations: AI and Automation Machine Learning Automates tedious tasks; neural networks optimize 3D meshes by reducing polygon counts without losing detail. Generative Networks GANs create 3D models from 2D sketches, streamlining workflows for artists by generating models with proper proportions and textures. 9

10. Cutting-Edge Innovations: AI and Automation Video Gaming Models of characters, weapons, and environments rely on geometric primitives and transformations. MRI Scans MRI scans use 3D reconstructions of organs to aid diagnosis. 10 Figure 1. Super Mario Bros Figure 2. MRI Scan Figure 3. Architecture Using AUTOCAD CAD software models buildings to test structural integrity and energy efficiency . Architecture

11. .Application: Geometric Modeling with OpenGL This presentation explores the practical application of geometric modeling concepts through an interactive 3D cube transformation system. The C++/OpenGL application visualizes core geometric modeling principles, allowing users to dynamically manipulate transformations such as rotation, scaling, and shearing via keyboard input. Observe real-time changes in spatial properties. 11

12. 3D Primitive Modeling Vertices and Quads The cube is constructed using vertices (points) and quads (surfaces), representing the foundational primitives. This aligns with the concept of vertices and surfaces as basic 3D elements. The application demonstrates how these primitives are used to build more complex 3D shapes, providing a visual representation of the theoretical concepts. 12

13. Transformations: Rotation Rotation Matrix Applies a rotation matrix to alter the cube's orientation, preserving Euclidean distances and angles. This validates the use of matrix-based orientation changes. Scaling Matrix Uses a diagonal scaling matrix to resize the cube, maintaining shape proportions (affine transformation). This aligns with the concept of proportional resizing. Shearing Matrix Implements a custom shear matrix to skew the cube, demonstrating non-uniform deformation. This illustrates matrix-driven distortions. 13

14. Spatial Relationships Position Cube is centered at the origin (glTranslatef(0.0f, 0.0f, -5.0f)). Lengths Edges fixed at 2 units (from -1.0 to 1.0) but scale dynamically. Angles 90° angles preserved during rotation/scaling, distorted during shearing. The application reflects the impact of transformations on geometry, showcasing how spatial relationships are affected by different operations. 14

15. Matrix Operations Composite Transformations Order-sensitive matrix multiplication (rotation → scaling → shearing). 1 Homogeneous Coordinates OpenGL’s 4x4 matrices unify 3D transformations. 2 The application ties into the concept of matrix composition, demonstrating how multiple transformations can be combined into a single matrix operation. 15

16. Hardware Integration: GPU Acceleration 1 GPU Acceleration OpenGL offloads rendering to the GPU. 2 Real-Time Performance Achieved via parallel processing on modern GPUs. The application connects to the concept of GPU-accelerated transformations, showcasing how modern GPUs enable real-time performance through parallel processing. 16

17. User Interaction Key Action Theoretical Concept W/S Adjust scaling factor Affine transformations A/D Modify shear factor Non-uniform deformations ESC Exit program - The application demonstrates interactive parameter control, allowing users to explore the effects of different transformations in real-time. 17

18. Application Screenshots Default Cube Untransformed cube, no transformations Rotation Cube rotating around Y-axis. Shearing 18 Figure 2 Cube Rotating in the Y-axis Cube scaled : Enlarged Cube scaled : Shrunk

19. Conclusion: The Future of Geometric Modeling The integration of geometric objects, spaces, and transformation operations is crucial for creating realistic and interactive digital representations. Precise mathematical frameworks ensure accurate spatial relationships. The case study reinforces these insights by demonstrating a modeling application. Future work should explore machine learning to automate transformation pipelines and expand applications to support non-linear transformations and advanced rendering techniques. The fusion of robust theoretical models with innovative computational strategies will advance the field. 19