![11Challenge the future
Matrix Operations [Linear Algebra]:
Look these up:
• Trivial Facts
• Identity Matrix
• Multiplication of Matrices 𝐴𝐵 ≠ 𝐵𝐴
• Transposed Matrix (𝐴 𝑇
)
𝑇
= 𝐴
• Systems of Linear Equations
• Determinant
• Inverse Matrix
• PCA: Eigenvalues & Eigenvectors
Use MetaNumerics.DLL
𝐴𝐵𝑖,𝑗 𝑅×𝐶
= 𝐴 𝑖,𝑘 × 𝐵 𝑘,𝑗
𝑚
𝑘=1
𝐴 𝑅×𝑀 ∗ 𝐵 𝑀×𝐶 = 𝐴𝐵𝑖,𝑗 𝑅×𝐶](https://image.slidesharecdn.com/preliminaries-160130163247/85/Preliminaries-of-Analytic-Geometry-and-Linear-Algebra-3D-modelling-11-320.jpg)
Preliminaries of Analytic Geometry and Linear Algebra 3D modelling
- 1. 1Challenge the future Preliminaries Basic Vector Mathematics for 3D Modeling Ir. Pirouz Nourian PhD candidate & Instructor, chair of Design Informatics, since 2010 MSc in Architecture 2009 BSc in Control Engineering 2005 MSc Geomatics, GEO1004, Directed by Dr. Sisi Zlatanova
- 2. 2Challenge the future INVISIBLE DIRECTIONS Vector Mathematics in a Nutshell René Descartes Image courtesy of David Rutten, from Rhinoscript 101
- 3. 3Challenge the future INVISIBLE DIRECTIONS Basic Operations 𝐴 = 𝑎 𝑥 𝒊 + 𝑎 𝑦 𝒋 + 𝑎 𝑧 𝒌 𝐵 = 𝑏 𝑥 𝒊 + 𝑏 𝑦 𝒋 + 𝑏 𝑧 𝒌 𝐴 + 𝐵 = (𝑎 𝑥 + 𝑏 𝑥)𝒊 + (𝑎 𝑦+𝑏 𝑦)𝒋 + (𝑎 𝑧+𝑏 𝑧)𝒌 Vector Addition Vector Length 𝐴 = 𝑎 𝑥 2 + 𝑎 𝑦 2 + 𝑎 𝑧 2
- 4. 4Challenge the future Dot Product: physical intuition… E.g. How to detect perpendicularity? • Image courtesy of http://sdsu-physics.org
- 5. 5Challenge the future Dot Product: How is it calculated in analytic geometry? Image courtesy of http://sdsu- 𝜃 B A 𝒊. 𝒊 = 𝒋. 𝒋 = 𝒌. 𝒌 = 1 𝒊. 𝒋 = 𝒋. 𝒊 = 0 𝒋. 𝒌 = 𝒌. 𝒋 = 0 𝒌. 𝒊 = 𝒊. 𝒌 = 0
- 6. 6Challenge the future Dot Product: How is it calculated in analytic geometry? 𝐴 = 𝑎 𝑥 𝒊 + 𝑎 𝑦 𝒋 + 𝑎 𝑧 𝒌 = 𝑎 𝑥 𝑎 𝑦 𝑎 𝑧 𝒊 𝒋 𝒌 𝐵 = 𝑏 𝑥 𝒊 + 𝑏 𝑦 𝒋 + 𝑏 𝑧 𝒌 = 𝑏 𝑥 𝑏 𝑦 𝑏 𝑧 𝒊 𝒋 𝒌 𝐴. 𝐵 == 𝐴 . 𝐵 . 𝐶𝑜𝑠(𝜃) 𝜃 B A 𝐴. 𝐵 = 𝑎 𝑥 𝑎 𝑦 𝑎 𝑧 𝑏 𝑥 𝑏 𝑦 𝑏 𝑧 = 𝑎 𝑥 𝑏 𝑥 + 𝑎 𝑦 𝑏 𝑦 + 𝑎 𝑧 𝑏 𝑧
- 7. 7Challenge the future Cross Product: physical intuition… • Image courtesy of http://hyperphysics.phy-astr.gsu.edu Images courtesy of Raja Issa, Essential Mathematics for Computational Design E.g. How to detect parallelism?
- 8. 8Challenge the future Cross Product: How is it calculated in analytic geometry? Images courtesy of Raja Issa, Essential Mathematics for Computational Design 𝒊 × 𝒊 = 𝒋 × 𝒋 = 𝒌 × 𝒌 = 𝟎 𝒊 × 𝒋 = 𝒌 𝒋 × 𝒌 = 𝒊 𝒌 × 𝒊 = 𝒋 𝒋 × 𝒊 = −𝒌 𝒌 × 𝒋 = −𝒊 𝒊 × 𝒌 = −𝒋
- 9. 9Challenge the future Cross Product: How is it calculated in analytic geometry? Images courtesy of Raja Issa, Essential Mathematics for Computational Design 𝐴 = 𝑎 𝑥 𝒊 + 𝑎 𝑦 𝒋 + 𝑎 𝑧 𝒌 = 𝑎 𝑥 𝑎 𝑦 𝑎 𝑧 𝒊 𝒋 𝒌 𝐵 = 𝑏 𝑥 𝒊 + 𝑏 𝑦 𝒋 + 𝑏 𝑧 𝒌 = 𝑏 𝑥 𝑏 𝑦 𝑏 𝑧 𝒊 𝒋 𝒌 𝐴 × 𝐵 = (𝑎 𝑥 𝒊 + 𝑎 𝑦 𝒋 + 𝑎 𝑧 𝒌) × (𝑏 𝑥 𝒊 + 𝑏 𝑦 𝒋 + 𝑏 𝑧 𝒌) = 𝒊 𝒋 𝒌 𝑎 𝑥 𝑎 𝑦 𝑎 𝑧 𝑏 𝑥 𝑏 𝑦 𝑏 𝑧 𝐴 × 𝐵 = 𝐴 . 𝐵 . 𝑆𝑖𝑛(𝜃) 𝐴 × 𝐵 = 𝑎 𝑦 𝑏 𝑧 − 𝑎 𝑧 𝑏 𝑦 𝒊 + 𝑎 𝑧 𝑏 𝑥 − 𝑎 𝑥 𝑏 𝑧 𝒋 + 𝑎 𝑥 𝑏 𝑦 − 𝑎 𝑦 𝑏 𝑥 𝒌
- 10. 10Challenge the future INVISIBLE ORIENTATIONS Place things on planes! Planes in a Nutshell! Images courtesy of David Rutten, Rhino Script 101
- 11. 11Challenge the future Matrix Operations [Linear Algebra]: Look these up: • Trivial Facts • Identity Matrix • Multiplication of Matrices 𝐴𝐵 ≠ 𝐵𝐴 • Transposed Matrix (𝐴 𝑇 ) 𝑇 = 𝐴 • Systems of Linear Equations • Determinant • Inverse Matrix • PCA: Eigenvalues & Eigenvectors Use MetaNumerics.DLL 𝐴𝐵𝑖,𝑗 𝑅×𝐶 = 𝐴 𝑖,𝑘 × 𝐵 𝑘,𝑗 𝑚 𝑘=1 𝐴 𝑅×𝑀 ∗ 𝐵 𝑀×𝐶 = 𝐴𝐵𝑖,𝑗 𝑅×𝐶
- 12. 12Challenge the future TRANSFORMATIONS • Linear Transformations: Euclidean and Affine • Homogenous Coordinate System • Inverse Transforms? • Non-Linear Transformations? Images courtesy of Raja Issa, Essential Mathematics for Computational Design 𝐿𝑖𝑛𝑒𝑎𝑟 𝑇𝑟𝑎𝑛𝑠𝑓𝑜𝑟𝑚𝑎𝑡𝑖𝑜𝑛𝑠 by Matrices
- 13. 13Challenge the future TOPOLOGY in GH: Use matrices to represent graphs Connectivity, Adjacency and Graphs in GH We will see more about topology in solids and meshes!