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
Preliminaries of Analytic Geometry and Linear Algebra 3D modelling
Jan 23, 2018
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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
RenΓ© Descartes
Image courtesy of David Rutten,
from Rhinoscript 101
3 Challenge the future
INVISIBLE DIRECTIONS
Basic Operations
π΄ = ππ₯π + ππ¦π + ππ§π
π΅ = ππ₯π + ππ¦π + ππ§π
π΄ + π΅ = (ππ₯ + ππ₯)π + (ππ¦+ππ¦)π + (ππ§+ππ§)π
Vector Addition
Vector Length
π΄ = ππ₯2 + ππ¦
2+ ππ§
2
4 Challenge the future
Dot Product: physical intuitionβ¦
E.g. How to detect perpendicularity?
β’
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
π
B
A
π . π = π . π = π. π = 1
π . π = π . π = 0
π . π = π. π = 0
π. π = π . π = 0
6 Challenge the future
Dot Product: How is it calculated in analytic geometry?
π΄ = ππ₯π + ππ¦π + ππ§π = ππ₯ ππ¦ ππ§πππ
π΅ = ππ₯π + ππ¦π + ππ§π = ππ₯ ππ¦ ππ§πππ
π΄ . π΅ == π΄ . π΅ . πΆππ (π)
π
B
A
π΄ . π΅ = ππ₯ ππ¦ ππ§
ππ₯ππ¦ππ§
= ππ₯ππ₯ + ππ¦ππ¦ + ππ§ππ§
7 Challenge the future
Cross Product: physical intuitionβ¦
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 Challenge the future
Cross Product: How is it calculated in analytic geometry?
Images courtesy of
Raja Issa, Essential Mathematics for Computational Design
π Γ π = π Γ π = π Γ π = π
π Γ π = π
π Γ π = π
π Γ π = π
π Γ π = βπ
π Γ π = βπ
π Γ π = βπ
9 Challenge the future
Cross Product: How is it calculated in analytic geometry?
Images courtesy of Raja Issa, Essential Mathematics for Computational Design
π΄ = ππ₯π + ππ¦π + ππ§π = ππ₯ ππ¦ ππ§πππ
π΅ = ππ₯π + ππ¦π + ππ§π = ππ₯ ππ¦ ππ§πππ
π΄ Γ π΅ = (ππ₯π + ππ¦π + ππ§π) Γ (ππ₯π + ππ¦π + ππ§π) =
π π πππ₯ ππ¦ ππ§ππ₯ ππ¦ ππ§
π΄ Γ π΅ = π΄ . π΅ . πππ(π)
π΄ Γ π΅ = ππ¦ππ§ β ππ§ππ¦ π + ππ§ππ₯ β ππ₯ππ§ π + ππ₯ππ¦ β ππ¦ππ₯ π
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:
β’ Trivial Facts
β’ Identity Matrix
β’ Multiplication of Matrices π΄π΅ β π΅π΄
β’ Transposed Matrix (π΄π)π= π΄
β’ Systems of Linear Equations
β’ Determinant
β’ Inverse Matrix
β’ PCA: Eigenvalues & Eigenvectors
Use MetaNumerics.DLL
π΄π΅π,π π ΓπΆ = π΄ π,π Γ π΅ π,π
π
π=1
π΄ π Γπ β π΅ πΓπΆ = π΄π΅π,π π ΓπΆ
12 Challenge 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
πΏπππππ πππππ ππππππ‘ππππ 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?
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