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

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!

14. 14Challenge the future
Questions?