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Ar1 twf030 lecture1.2
- 1. 11 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
- 2. 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
- 3. 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
- 4. 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
- 5. 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)
- 6. 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
- 7. 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 𝑡 𝑡 ∈ [0,2𝜋] 𝑡 = 2𝜋𝑖 𝑛 |𝑖 ∈[1,n]⊂ ℕ Plane Radius Circle
- 8. 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 𝑡 𝑡 ∈ [0,2𝜋] 𝑡 = 2𝜋𝑖 𝑛 |𝑖 ∈[1,n]⊂ ℕ Plane Radius Circle
- 9. 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
- 10. 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
- 11. 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
- 12. 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!
- 13. 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
- 14. 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 = 𝑎 𝑥Ԧ𝒊 + 𝑎 𝑦 Ԧ𝒋 + 𝑎 𝑧 𝒌 𝐵 = 𝑏 𝑥Ԧ𝒊 + 𝑏 𝑦 Ԧ𝒋 + 𝑏 𝑧 𝒌 Ԧ𝐴 + 𝐵 = (𝑎 𝑥 + 𝑏 𝑥)Ԧ𝒊 + (𝑎 𝑦+𝑏 𝑦)Ԧ𝒋 + (𝑎 𝑧+𝑏 𝑧)𝒌 Euclidean Vector Length Ԧ𝐴 = 𝑎 𝑥 2 + 𝑎 𝑦 2 + 𝑎 𝑧 2
- 15. 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
- 16. 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! ☺
- 17. 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.
- 18. 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 = 𝑎 𝑥 Ԧ𝒊 + 𝑎 𝑦 Ԧ𝒋 + 𝑎 𝑧 𝒌 = 𝑎 𝑥 𝑎 𝑦 𝑎 𝑧 𝒊 𝒋 𝒌 𝐵 = 𝑏 𝑥Ԧ𝒊 + 𝑏 𝑦 Ԧ𝒋 + 𝑏 𝑧 𝒌 = 𝑏 𝑥 𝑏 𝑦 𝑏 𝑧 𝒊 𝒋 𝒌 Ԧ𝐴. 𝐵 == Ԧ𝐴 . 𝐵 . 𝐶𝑜𝑠(𝜃) 𝜃 B A Ԧ𝐴. 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
- 19. 1919 • 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 Wiki Commons and Raja Issa, Essential Mathematics for Computational Design http://chortle.ccsu.edu/vectorlessons/vch12/vch12_4.html
- 20. 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! ☺
- 21. 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
- 22. 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 = 𝑎 𝑥 Ԧ𝒊 + 𝑎 𝑦 Ԧ𝒋 + 𝑎 𝑧 𝒌 = 𝑎 𝑥 𝑎 𝑦 𝑎 𝑧 𝒊 𝒋 𝒌 𝐵 = 𝑏 𝑥 Ԧ𝒊 + 𝑏 𝑦 Ԧ𝒋 + 𝑏 𝑧 𝒌 = 𝑏 𝑥 𝑏 𝑦 𝑏 𝑧 𝒊 𝒋 𝒌 Ԧ𝐴 × 𝐵 = (𝑎 𝑥 Ԧ𝒊 + 𝑎 𝑦 Ԧ𝒋 + 𝑎 𝑧 𝒌) × (𝑏 𝑥Ԧ𝒊 + 𝑏 𝑦 Ԧ𝒋 + 𝑏 𝑧 𝒌) = 𝒊 𝒋 𝒌 𝑎 𝑥 𝑎 𝑦 𝑎 𝑧 𝑏 𝑥 𝑏 𝑦 𝑏 𝑧 Ԧ𝐴 × 𝐵 = Ԧ𝐴 . 𝐵 . 𝑆𝑖𝑛(𝜃) Ԧ𝐴 × 𝐵 = 𝑎 𝑦 𝑏 𝑧 − 𝑎 𝑧 𝑏 𝑦 Ԧ𝒊 − 𝑎 𝑥 𝑏 𝑧 − 𝑎 𝑧 𝑏 𝑥 Ԧ𝒋 + 𝑎 𝑥 𝑏 𝑦 − 𝑎 𝑦 𝑏 𝑥 𝒌
- 23. 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
- 24. 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
- 25. 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
- 26. 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
- 27. 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
- 28. 2828 make a parametric staircase
- 29. 2929 make a parametric staircase https://collections.museumvictoria.com.au/articles/4624