preliminaries of analytic geometry and linear algebra 3d modelling
TRANSCRIPT
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?