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Preliminaries of Analytic Geometry and Linear Algebra 3D modelling

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from my lecture notes for the course Geo1004 (2015), 3D modelling of the built environment, at TU Delft, faculty of Architecture and the Built Environment

Published in: Engineering

Preliminaries of Analytic Geometry and Linear Algebra 3D modelling

  1. 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. 2. 2Challenge the future INVISIBLE DIRECTIONS Vector Mathematics in a Nutshell RenΓ© Descartes Image courtesy of David Rutten, from Rhinoscript 101
  3. 3. 3Challenge the future INVISIBLE DIRECTIONS Basic Operations 𝐴 = π‘Ž π‘₯ π’Š + π‘Ž 𝑦 𝒋 + π‘Ž 𝑧 π’Œ 𝐡 = 𝑏 π‘₯ π’Š + 𝑏 𝑦 𝒋 + 𝑏 𝑧 π’Œ 𝐴 + 𝐡 = (π‘Ž π‘₯ + 𝑏 π‘₯)π’Š + (π‘Ž 𝑦+𝑏 𝑦)𝒋 + (π‘Ž 𝑧+𝑏 𝑧)π’Œ Vector Addition Vector Length 𝐴 = π‘Ž π‘₯ 2 + π‘Ž 𝑦 2 + π‘Ž 𝑧 2
  4. 4. 4Challenge the future Dot Product: physical intuition… E.g. How to detect perpendicularity? β€’ Image courtesy of http://sdsu-physics.org
  5. 5. 5Challenge the future Dot Product: How is it calculated in analytic geometry? Image courtesy of http://sdsu- πœƒ B A π’Š. π’Š = 𝒋. 𝒋 = π’Œ. π’Œ = 1 π’Š. 𝒋 = 𝒋. π’Š = 0 𝒋. π’Œ = π’Œ. 𝒋 = 0 π’Œ. π’Š = π’Š. π’Œ = 0
  6. 6. 6Challenge the future Dot Product: How is it calculated in analytic geometry? 𝐴 = π‘Ž π‘₯ π’Š + π‘Ž 𝑦 𝒋 + π‘Ž 𝑧 π’Œ = π‘Ž π‘₯ π‘Ž 𝑦 π‘Ž 𝑧 π’Š 𝒋 π’Œ 𝐡 = 𝑏 π‘₯ π’Š + 𝑏 𝑦 𝒋 + 𝑏 𝑧 π’Œ = 𝑏 π‘₯ 𝑏 𝑦 𝑏 𝑧 π’Š 𝒋 π’Œ 𝐴. 𝐡 == 𝐴 . 𝐡 . πΆπ‘œπ‘ (πœƒ) πœƒ B A 𝐴. 𝐡 = π‘Ž π‘₯ π‘Ž 𝑦 π‘Ž 𝑧 𝑏 π‘₯ 𝑏 𝑦 𝑏 𝑧 = π‘Ž π‘₯ 𝑏 π‘₯ + π‘Ž 𝑦 𝑏 𝑦 + π‘Ž 𝑧 𝑏 𝑧
  7. 7. 7Challenge the future Cross Product: physical intuition… β€’ 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. 8. 8Challenge the future Cross Product: How is it calculated in analytic geometry? Images courtesy of Raja Issa, Essential Mathematics for Computational Design π’Š Γ— π’Š = 𝒋 Γ— 𝒋 = π’Œ Γ— π’Œ = 𝟎 π’Š Γ— 𝒋 = π’Œ 𝒋 Γ— π’Œ = π’Š π’Œ Γ— π’Š = 𝒋 𝒋 Γ— π’Š = βˆ’π’Œ π’Œ Γ— 𝒋 = βˆ’π’Š π’Š Γ— π’Œ = βˆ’π’‹
  9. 9. 9Challenge the future Cross Product: How is it calculated in analytic geometry? Images courtesy of Raja Issa, Essential Mathematics for Computational Design 𝐴 = π‘Ž π‘₯ π’Š + π‘Ž 𝑦 𝒋 + π‘Ž 𝑧 π’Œ = π‘Ž π‘₯ π‘Ž 𝑦 π‘Ž 𝑧 π’Š 𝒋 π’Œ 𝐡 = 𝑏 π‘₯ π’Š + 𝑏 𝑦 𝒋 + 𝑏 𝑧 π’Œ = 𝑏 π‘₯ 𝑏 𝑦 𝑏 𝑧 π’Š 𝒋 π’Œ 𝐴 Γ— 𝐡 = (π‘Ž π‘₯ π’Š + π‘Ž 𝑦 𝒋 + π‘Ž 𝑧 π’Œ) Γ— (𝑏 π‘₯ π’Š + 𝑏 𝑦 𝒋 + 𝑏 𝑧 π’Œ) = π’Š 𝒋 π’Œ π‘Ž π‘₯ π‘Ž 𝑦 π‘Ž 𝑧 𝑏 π‘₯ 𝑏 𝑦 𝑏 𝑧 𝐴 Γ— 𝐡 = 𝐴 . 𝐡 . 𝑆𝑖𝑛(πœƒ) 𝐴 Γ— 𝐡 = π‘Ž 𝑦 𝑏 𝑧 βˆ’ π‘Ž 𝑧 𝑏 𝑦 π’Š + π‘Ž 𝑧 𝑏 π‘₯ βˆ’ π‘Ž π‘₯ 𝑏 𝑧 𝒋 + π‘Ž π‘₯ 𝑏 𝑦 βˆ’ π‘Ž 𝑦 𝑏 π‘₯ π’Œ
  10. 10. 10Challenge the future INVISIBLE ORIENTATIONS Place things on planes! Planes in a Nutshell! Images courtesy of David Rutten, Rhino Script 101
  11. 11. 11Challenge the future Matrix Operations [Linear Algebra]: Look these up: β€’ Trivial Facts β€’ Identity Matrix β€’ Multiplication of Matrices 𝐴𝐡 β‰  𝐡𝐴 β€’ Transposed Matrix (𝐴 𝑇 ) 𝑇 = 𝐴 β€’ Systems of Linear Equations β€’ Determinant β€’ Inverse Matrix β€’ PCA: Eigenvalues & Eigenvectors Use MetaNumerics.DLL 𝐴𝐡𝑖,𝑗 𝑅×𝐢 = 𝐴 𝑖,π‘˜ Γ— 𝐡 π‘˜,𝑗 π‘š π‘˜=1 𝐴 𝑅×𝑀 βˆ— 𝐡 𝑀×𝐢 = 𝐴𝐡𝑖,𝑗 𝑅×𝐢
  12. 12. 12Challenge the future TRANSFORMATIONS β€’ Linear Transformations: Euclidean and Affine β€’ Homogenous Coordinate System β€’ Inverse Transforms? β€’ Non-Linear Transformations? Images courtesy of Raja Issa, Essential Mathematics for Computational Design πΏπ‘–π‘›π‘’π‘Žπ‘Ÿ π‘‡π‘Ÿπ‘Žπ‘›π‘ π‘“π‘œπ‘Ÿπ‘šπ‘Žπ‘‘π‘–π‘œπ‘›π‘  by Matrices
  13. 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!
  14. 14. 14Challenge the future Questions?

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