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1 Challenge 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 Challenge the future
INVISIBLE DIRECTIONS
Vector Mathematics in a Nutshell
René Descartes
Image courtesy of David Rutten,
from Rhinoscript 101
3 Challenge the future
INVISIBLE DIRECTIONS
Basic Operations
𝐴 = 𝑎𝑥𝒊 + 𝑎𝑦𝒋 + 𝑎𝑧𝒌
𝐵 = 𝑏𝑥𝒊 + 𝑏𝑦𝒋 + 𝑏𝑧𝒌
𝐴 + 𝐵 = (𝑎𝑥 + 𝑏𝑥)𝒊 + (𝑎𝑦+𝑏𝑦)𝒋 + (𝑎𝑧+𝑏𝑧)𝒌
Vector Addition
Vector Length
𝐴 = 𝑎𝑥2 + 𝑎𝑦
2+ 𝑎𝑧
2
4 Challenge the future
Dot Product: physical intuition…
E.g. How to detect perpendicularity?
•
Image courtesy of http://sdsu-physics.org
5 Challenge the future
Dot Product: How is it calculated in analytic geometry?
Image courtesy of http://sdsu-
physics.org
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 Challenge the future
Dot Product: How is it calculated in analytic geometry?
𝐴 = 𝑎𝑥𝒊 + 𝑎𝑦𝒋 + 𝑎𝑧𝒌 = 𝑎𝑥 𝑎𝑦 𝑎𝑧𝒊𝒋𝒌
𝐵 = 𝑏𝑥𝒊 + 𝑏𝑦𝒋 + 𝑏𝑧𝒌 = 𝑏𝑥 𝑏𝑦 𝑏𝑧𝒊𝒋𝒌
𝐴 . 𝐵 == 𝐴 . 𝐵 . 𝐶𝑜𝑠(𝜃)
𝜃
B
A
𝐴 . 𝐵 = 𝑎𝑥 𝑎𝑦 𝑎𝑧
𝑏𝑥𝑏𝑦𝑏𝑧
= 𝑎𝑥𝑏𝑥 + 𝑎𝑦𝑏𝑦 + 𝑎𝑧𝑏𝑧
7 Challenge 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 Challenge 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 Challenge the future
Cross Product: How is it calculated in analytic geometry?
Images courtesy of Raja Issa, Essential Mathematics for Computational Design
𝐴 = 𝑎𝑥𝒊 + 𝑎𝑦𝒋 + 𝑎𝑧𝒌 = 𝑎𝑥 𝑎𝑦 𝑎𝑧𝒊𝒋𝒌
𝐵 = 𝑏𝑥𝒊 + 𝑏𝑦𝒋 + 𝑏𝑧𝒌 = 𝑏𝑥 𝑏𝑦 𝑏𝑧𝒊𝒋𝒌
𝐴 × 𝐵 = (𝑎𝑥𝒊 + 𝑎𝑦𝒋 + 𝑎𝑧𝒌) × (𝑏𝑥𝒊 + 𝑏𝑦𝒋 + 𝑏𝑧𝒌) =
𝒊 𝒋 𝒌𝑎𝑥 𝑎𝑦 𝑎𝑧𝑏𝑥 𝑏𝑦 𝑏𝑧
𝐴 × 𝐵 = 𝐴 . 𝐵 . 𝑆𝑖𝑛(𝜃)
𝐴 × 𝐵 = 𝑎𝑦𝑏𝑧 − 𝑎𝑧𝑏𝑦 𝒊 + 𝑎𝑧𝑏𝑥 − 𝑎𝑥𝑏𝑧 𝒋 + 𝑎𝑥𝑏𝑦 − 𝑎𝑦𝑏𝑥 𝒌
10 Challenge the future
INVISIBLE ORIENTATIONS
Place things on planes!
Planes in a Nutshell!
Images courtesy of David Rutten, Rhino Script 101
11 Challenge the future
Matrix Operations [Linear Algebra]:
Look these up:
• Trivial Facts
• Identity Matrix
• Multiplication of Matrices 𝐴𝐵 ≠ 𝐵𝐴
• Transposed Matrix (𝐴𝑇)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 Challenge 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 Challenge 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 Challenge the future
Questions?
