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  1. 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. 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. 3. 33 The image of a geometry is not the same as its representation What you see is not what you get Image: Ren 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. 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. 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. 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 2 + 2 = 2 The Product vs The Process
  7. 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. = () = [0,2] = 2 | [1,n] Plane Radius Circle
  8. 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. = () = [0,2] = 2 | [1,n] Plane Radius Circle
  9. 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. 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. 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. 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. 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 Ren Descartes Image courtesy of David Rutten, from Rhinoscript 101
  14. 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 = + + = + + + = ( + ) + ( + ) + ( + ) Euclidean Vector Length = 2 + 2 + 2
  15. 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 = . = . cos
  16. 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 = . = . cos 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. 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? B A . = . = . = 1 . = . = 0 . = . = 0 . = . = 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. 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? = + + = = + + = . == . . () B A . = = + +
  19. 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. 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. 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 = = = = = = = = =
  22. 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 = + + = = + + = = ( + + ) ( + + ) = = . . () = +
  23. 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. 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. 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. 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. 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. 28. 2828 make a parametric staircase
  29. 29. 2929 make a parametric staircase https://collections.museumvictoria.com.au/articles/4624