Ar1 twf030 lecture1.2
30
1 1 Preliminaries of Basics of Linear Algebra & Computer Geometry Dr.ir. Pirouz Nourian Assistant Professor of Design Informatics Department of Architectural Engineering & Technology Faculty of Architecture and Built Environment
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Transcript of Ar1 twf030 lecture1.2
11
Preliminaries of
Basics of Linear Algebra & Computer Geometry
Dr.ir. Pirouz Nourian
Assistant Professor of Design InformaticsDepartment of Architectural Engineering & TechnologyFaculty of Architecture and Built Environment
22
Try to guess how a line or a circle is represented in a computer
“If it looks like a duck, swims like a duck, and quacks like a duck, then it probably is a duck.”
Image: DUCK: GETTY Images; ILLUSTRATION: MARTIN O'NEILL, from http://www.nature.com/nature/journal/v484/n7395/full/484451a.html?message-global=remove
• Digital Geometry
• Data Models
• Euclidean World
• Cartesian World
• Vectors
o Sum
o Dot Product
o Cross Product
• Planes
o Locus
o Orientation
• Intersection
• Transformation
33
The image of a geometry is not the same as its representation
What you see is not what you get
Image: René Magritte, ceci n'est pas une pipe (this is not a pipe)
• Digital Geometry
• Data Models
• Euclidean World
• Cartesian World
• Vectors
o Sum
o Dot Product
o Cross Product
• Planes
o Locus
o Orientation
• Intersection
• Transformation
44
• Digital Geometry
• Data Models
• Euclidean World
• Cartesian World
• Vectors
o Sum
o Dot Product
o Cross Product
• Planes
o Locus
o Orientation
• Intersection
• Transformation
55
• Digital Geometry
• Data Models
• Euclidean World
• Cartesian World
• Vectors
o Sum
o Dot Product
o Cross Product
• Planes
o Locus
o Orientation
• Intersection
• Transformation
On terminology
• Geometry: Point (0D), Curve(1D), Surface(2D), Solid (3D) [free-form]
• Geometry: Point (0D), Line(1D), Polygon(2D), Polyhedron (3D) [piecewise linear]
• Topology: Vertex(0D), Edge(1D), Face(2D), Body(3D)
• Graph Theory: Object, Link, (and n-Cliques)
66
• Digital Geometry
• Data Models
• Euclidean World
• Cartesian World
• Vectors
o Sum
o Dot Product
o Cross Product
• Planes
o Locus
o Orientation
• Intersection
• Transformation
WYSIWYG versus WYSIWYM
𝑥2 + 𝑦2 = 𝑅2
The Product vs The Process
77
• Digital Geometry
• Data Models
• Euclidean World
• Cartesian World
• Vectors
o Sum
o Dot Product
o Cross Product
• Planes
o Locus
o Orientation
• Intersection
• Transformation
Parametric Modeling & Design• Thinking of parameters instead of numbers!• Same rationales, many alternatives!
▪ We could model an actual circle as a particular instance of a generic circle, which is the locus of points equidistant from a given point as C (center), at a given distance R (Radius), on a plane p.
▪ Parametric modeling is essential for formulating design problems▪ The same role algebra has had in the progress of mathematics, parametric modeling will have in systematic (research-oriented) design.
𝑥 = 𝑟𝑐𝑜𝑠(𝑡)𝑦 = 𝑟𝑠𝑖𝑛 𝑡𝑡 ∈ [0,2𝜋]
𝑡 =2𝜋𝑖
𝑛|𝑖 ∈[1,n]⊂ ℕ
Plane
RadiusCircle
88
• Digital Geometry
• Data Models
• Euclidean World
• Cartesian World
• Vectors
o Sum
o Dot Product
o Cross Product
• Planes
o Locus
o Orientation
• Intersection
• Transformation
Parametric Modeling & Design• Thinking of parameters instead of numbers!• Same rationales, many alternatives!
▪ We could model an actual circle as a particular instance of a generic circle, which is the locus of points equidistant from a given point as C (center), at a given distance R (Radius), on a plane p.
▪ Parametric modeling is essential for formulating design problems▪ The same role algebra has had in the progress of mathematics, parametric modeling will have in systematic (research-oriented) design.
𝑥 = 𝑟𝑐𝑜𝑠(𝑡)𝑦 = 𝑟𝑠𝑖𝑛 𝑡𝑡 ∈ [0,2𝜋]
𝑡 =2𝜋𝑖
𝑛|𝑖 ∈[1,n]⊂ ℕ
Plane
RadiusCircle
99
• Digital Geometry
• Data Models
• Euclidean World
• Cartesian World
• Vectors
o Sum
o Dot Product
o Cross Product
• Planes
o Locus
o Orientation
• Intersection
• Transformation
1010
• Digital Geometry
• Data Models
• Euclidean World
• Cartesian World
• Vectors
o Sum
o Dot Product
o Cross Product
• Planes
o Locus
o Orientation
• Intersection
• Transformation
1111
• Digital Geometry
• Data Models
• Euclidean World
• Cartesian World
• Vectors
o Sum
o Dot Product
o Cross Product
• Planes
o Locus
o Orientation
• Intersection
• Transformation
1212
• Digital Geometry
• Data Models
• Euclidean World
• Cartesian World
• Vectors
o Sum
o Dot Product
o Cross Product
• Planes
o Locus
o Orientation
• Intersection
• Transformation
Vectors in a Nutshell
Applications
• Any representation in Computer Graphics depends on vectors (points, lines, etc. are all eventually based on vectors)
• Any transformation (e.g. moving objects, rotating them, etc.)• It suffices to say there is no 3D geometry without vectors!
1313
• Digital Geometry
• Data Models
• Euclidean World
• Cartesian World
• Vectors
o Sum
o Dot Product
o Cross Product
• Planes
o Locus
o Orientation
• Intersection
• Transformation
Vectors in a Nutshell
René Descartes
Image courtesy of David Rutten,
from Rhinoscript 101
1414
• Digital Geometry
• Data Models
• Euclidean World
• Cartesian World
• Vectors
o Sum
o Dot Product
o Cross Product
• Planes
o Locus
o Orientation
• Intersection
• Transformation
Ԧ𝐴 = 𝑎𝑥Ԧ𝒊 + 𝑎𝑦 Ԧ𝒋 + 𝑎𝑧𝒌
𝐵 = 𝑏𝑥Ԧ𝒊 + 𝑏𝑦 Ԧ𝒋 + 𝑏𝑧𝒌
Ԧ𝐴 + 𝐵 = (𝑎𝑥 + 𝑏𝑥)Ԧ𝒊 + (𝑎𝑦+𝑏𝑦)Ԧ𝒋 + (𝑎𝑧+𝑏𝑧)𝒌
Euclidean Vector Length
Ԧ𝐴 = 𝑎𝑥2 + 𝑎𝑦
2+ 𝑎𝑧
2
1515
• Digital Geometry
• Data Models
• Euclidean World
• Cartesian World
• Vectors
o Sum
o Dot Product
o Cross Product
• Planes
o Locus
o Orientation
• Intersection
• Transformation
Exemplary application: detecting perpendicularity or similarity
𝑊 = 𝑭.𝑫 = 𝑭 . 𝑫 cos 𝜃
1616
• Digital Geometry
• Data Models
• Euclidean World
• Cartesian World
• Vectors
o Sum
o Dot Product
o Cross Product
• Planes
o Locus
o Orientation
• Intersection
• Transformation
Exemplary application: detecting perpendicularity or similarity
𝑊 = 𝑭.𝑫 = 𝑭 . 𝑫 cos 𝜃
Other applications:
• Computing ‘flux’ in a vector field (e.g. solar irradiation)
• Detecting perpendicularly
• Computing angles (with the help of an Arc Cosine function)
• A very long list of techniques and tricks in computational
geometry & computer graphics
• You cannot get by without knowing about dot products! ☺
1717
• Digital Geometry
• Data Models
• Euclidean World
• Cartesian World
• Vectors
o Sum
o Dot Product
o Cross Product
• Planes
o Locus
o Orientation
• Intersection
• Transformation
Dot Product: How is it calculated in analytic geometry?
𝜃
B
A
Ԧ𝒊. Ԧ𝒊 = Ԧ𝒋. Ԧ𝒋 = 𝒌. 𝒌 = 1
Ԧ𝒊. Ԧ𝒋 = Ԧ𝒋. Ԧ𝒊 = 0
Ԧ𝒋. 𝒌 = 𝒌. Ԧ𝒋 = 0
𝒌. Ԧ𝒊 = Ԧ𝒊. 𝒌 = 0
So we do not have to do it by ‘drawing’ vectors and finding the angle between them with an angle ruler and a calculator! We do it algebraically instead.
1818
• Digital Geometry
• Data Models
• Euclidean World
• Cartesian World
• Vectors
o Sum
o Dot Product
o Cross Product
• Planes
o Locus
o Orientation
• Intersection
• Transformation
Dot Product: How is it calculated in analytic geometry?
Ԧ𝐴 = 𝑎𝑥 Ԧ𝒊 + 𝑎𝑦 Ԧ𝒋 + 𝑎𝑧𝒌 = 𝑎𝑥 𝑎𝑦 𝑎𝑧𝒊𝒋𝒌
𝐵 = 𝑏𝑥Ԧ𝒊 + 𝑏𝑦 Ԧ𝒋 + 𝑏𝑧𝒌 = 𝑏𝑥 𝑏𝑦 𝑏𝑧
𝒊𝒋𝒌
Ԧ𝐴. 𝐵 == Ԧ𝐴 . 𝐵 . 𝐶𝑜𝑠(𝜃)
𝜃
B
A
Ԧ𝐴. 𝐵 = 𝑎𝑥 𝑎𝑦 𝑎𝑧
𝑏𝑥𝑏𝑦𝑏𝑧
= 𝑎𝑥𝑏𝑥 + 𝑎𝑦𝑏𝑦 + 𝑎𝑧𝑏𝑧
1919
• Digital Geometry
• Data Models
• Euclidean World
• Cartesian World
• Vectors
o Sum
o Dot Product
o Cross Product
• Planes
o Locus
o Orientation
• Intersection
• TransformationImages courtesy of Wiki Commons and
Raja Issa, Essential Mathematics for Computational Design
http://chortle.ccsu.edu/vectorlessons/vch12/vch12_4.html
2020
• Digital Geometry
• Data Models
• Euclidean World
• Cartesian World
• Vectors
o Sum
o Dot Product
o Cross Product
• Planes
o Locus
o Orientation
• Intersection
• Transformation
Exemplary application: setting up a local coordinate system
• computing torque, electromotive force, etc in physics
• detecting parallelism
• a long list of techniques and tricks in computer graphics and computational
geometry
• computing volumes of polyhedrons
• Conclusion: you cannot get by without knowing about cross products either! ☺
2121
• Digital Geometry
• Data Models
• Euclidean World
• Cartesian World
• Vectors
o Sum
o Dot Product
o Cross Product
• Planes
o Locus
o Orientation
• Intersection
• Transformation
Cross Product: How is it calculated in analytic geometry?
Images courtesy of
Raja Issa, Essential Mathematics for Computational Design
Ԧ𝒊 × Ԧ𝒊 = Ԧ𝒋 × Ԧ𝒋 = 𝒌 × 𝒌 = 𝟎
Ԧ𝒊 × Ԧ𝒋 = 𝒌
Ԧ𝒋 × 𝒌 = Ԧ𝒊
𝒌 × Ԧ𝒊 = Ԧ𝒋
Ԧ𝒋 × Ԧ𝒊 = −𝒌
𝒌 × Ԧ𝒋 = −Ԧ𝒊
Ԧ𝒊 × 𝒌 = −Ԧ𝒋
2222
• Digital Geometry
• Data Models
• Euclidean World
• Cartesian World
• Vectors
o Sum
o Dot Product
o Cross Product
• Planes
o Locus
o Orientation
• Intersection
• Transformation
Cross Product: How is it calculated in analytic geometry?
Images courtesy of Raja Issa, Essential Mathematics for Computational Design
Ԧ𝐴 = 𝑎𝑥 Ԧ𝒊 + 𝑎𝑦 Ԧ𝒋 + 𝑎𝑧𝒌 = 𝑎𝑥 𝑎𝑦 𝑎𝑧𝒊𝒋𝒌
𝐵 = 𝑏𝑥 Ԧ𝒊 + 𝑏𝑦 Ԧ𝒋 + 𝑏𝑧𝒌 = 𝑏𝑥 𝑏𝑦 𝑏𝑧
𝒊𝒋𝒌
Ԧ𝐴 × 𝐵 = (𝑎𝑥 Ԧ𝒊 + 𝑎𝑦 Ԧ𝒋 + 𝑎𝑧𝒌) × (𝑏𝑥Ԧ𝒊 + 𝑏𝑦 Ԧ𝒋 + 𝑏𝑧𝒌) =
𝒊 𝒋 𝒌𝑎𝑥 𝑎𝑦 𝑎𝑧𝑏𝑥 𝑏𝑦 𝑏𝑧
Ԧ𝐴 × 𝐵 = Ԧ𝐴 . 𝐵 . 𝑆𝑖𝑛(𝜃)
Ԧ𝐴 × 𝐵 = 𝑎𝑦𝑏𝑧 − 𝑎𝑧𝑏𝑦 Ԧ𝒊 − 𝑎𝑥𝑏𝑧 − 𝑎𝑧𝑏𝑥 Ԧ𝒋 + 𝑎𝑥𝑏𝑦 − 𝑎𝑦𝑏𝑥 𝒌
2323
• Digital Geometry
• Data Models
• Euclidean World
• Cartesian World
• Vectors
o Sum
o Dot Product
o Cross Product
• Planes
o Locus
o Orientation
• Intersection
• Transformation
2424
• Digital Geometry
• Data Models
• Euclidean World
• Cartesian World
• Vectors
o Sum
o Dot Product
o Cross Product
• Planes
o Locus
o Orientation
• Intersection
• Transformation
Images courtesy of David Rutten, Rhino Script 101
2525
• Digital Geometry
• Data Models
• Euclidean World
• Cartesian World
• Vectors
o Sum
o Dot Product
o Cross Product
• Planes
o Locus
o Orientation
• Intersection
• Transformation
2626
• Digital Geometry
• Data Models
• Euclidean World
• Cartesian World
• Vectors
o Sum
o Dot Product
o Cross Product
• Planes
o Locus
o Orientation
• Intersection
• Transformation
http://geomalgorithms.com/a05-_intersect-1.html
2727
• Digital Geometry
• Data Models
• Euclidean World
• Cartesian World
• Vectors
o Sum
o Dot Product
o Cross Product
• Planes
o Locus
o Orientation
• Intersection
• Transformation
• Linear Transformations: Euclidean and Affine (Translation [movement], Rotation, Scaling,etc.)
• Homogenous Coordinate System
• Inverse Transforms?
• Non-Linear Transformations?
Images courtesy of Raja Issa, Essential Mathematics for Computational Design
2828
make a parametric staircase
2929
make a parametric staircase
https://collections.museumvictoria.com.au/articles/4624
