DA Architectural Geometry with Grasshopper

Transcription

1 DA Architectural Geometry with Grasshopper Toni Kotnik Professor of Design of Structures Lecture 1 Curves 1

2 DA Architectural Geometry with Grasshopper Toni Kotnik Professor of Design of Structures Prologue 1 Associative Geometry 2

3 UN Studio: Mercedes Benz Museum, Stuttgart, Germany,

4 building design production geometry UN Studio: Mercedes Benz Museum, Stuttgart, Germany,

5 UN Studio: Mercedes Benz Museum, Stuttgart, Germany,

6 Louis Kahn: Margaret Esherick House, Philadelphia, USA,

7 Louis Kahn: Margaret Esherick House, Philadelphia, USA,

8 Francesco Borromini: San Carlo alla Quattro Fontane, Rome, Italy, &

9 Francesco Borromini: San Carlo alla Quattro Fontane, Rome, Italy, &

10 DA Architectural Geometry with Grasshopper Toni Kotnik Professor of Design of Structures Prologue 2 Dynamic Simplicity 10

11 ijp: Henderson Waves Bridge, Singapore,

12 12

13 modern manufacturing requires control over movement in space of robotic arm 13

14 Linkage Francis van Schooten ( ): mechanism to draw an ellipsoid Franz Reuleaux ( ): Peaucellier Linkage,

15 Linkage analytic approach a cycloid is traced by a point P on a circle of radius a when a concentric circle of radius b rolls without slipping along a straight line ( x y ) = (a sin(t) + t b a cos(t) + b) geometric approach understand a curve as the result of simultaneous actions of several forces respectively simple movements: circular movement + linear movement curve viewed not as analytic problem but as geometric process, as coupling of simple operations and movements out of which more complex pattern emerge as a result 15

16 Königs Architekten / Peter Kulka / Cecil Balmond: Stadium, Chemnitz, Germany,

17 DA Architectural Geometry with Grasshopper Toni Kotnik Professor of Design of Structures From Line to NURBS 17

18 Point & Vector y a vector is a mathematical object that has a direction and an intensity the position vector v can be identified with its endpoint P v P 1 P 2 v (c d) P 3 (a b) P v (c d) v but the position vector v is not equal to its endpoint P v v v v (c d) position vector x v 18

19 Point & Vector 19

20 Line y l a line l is defined by a point P and a direction vector v this results in a natural parametrization of the line with respect to the scaling factor t of v l (t) = P + t v P v for the distance d between two points on the line this implies v d( l (t 1 ),l (t 2 ) ) = t 1 -t 2 v x attention: in Rhino the direction vector is always a unit vector, i.e. d( l (t 1 ),l (t 2 ) ) = t 1 -t 2 20

21 Bézier-Curve P 2 using the parametrization of two intersecting lines simultaneously t [0,1] P 12 t v 1 t P(t) v 12 P 23 t v 2 P 12 = P 1 + t v 1 P 23 = P 2 + t v 2 a unique point along the connecting line can be defined P(t) = P 12 + t v 12 P 1 P 3 The resulting curve is called a Bézier-curve of degree 3 Exercise: construct the resulting curve P 21

22 Bézier-Curve reach of influence of control point P 1 control polygon Bézier curve of degree 5 control point P 0 B P 4 P 2 P 3 problem 1: no localized control of form problem 2: averaging out of curvature Bézier curve of degree 7 P 2 P 4 P 0 B P 6 P 1 P 3 P 5 22

23 B-Spline reach of influence of control point limited P 1 P 3 Bézier curve of degree 3 B B 2 B 1 Bézier curve of degree 3 knot B-spline of degree 3 P 0 P 2 with N i,k B-spline basic functions as weights 23

24 B-spline of degree 7 Bézier-curve of degree 7 B-spline of degree 5 B-spline of degree 3 B-spline of degree 2 24

25 NURBS Non-Uniform Rational B-Splines problem with B-spline: a B-spline is a polynomial curve, that means important curves like circles can only be approximated but not represented in a precise manner! offset-curve curves surface-surface intersection NURBS B-splines Bézier-curves 25

26 DA Architectural Geometry with Grasshopper Toni Kotnik Professor of Design of Structures Curvature 26

27 Francesco Borromini: San Carlo alla Quattro Fontane, Rome, Italy, &

28 Francesco Borromini: San Carlo alla Quattro Fontane, Rome, Italy, &

29 Herzog & de Meuron: University Library, Cottbus, Germany,

30 Herzog & de Meuron: University Library, Cottbus, Germany,

31 circles as well-known and easy-to-construct curvy curves with radius as measurement for curvature! 31

32 Curvature for a circle the curvature к is defined as the invers of the radius r к = 1/r r P c the curvature к is a measure for the roundness of the circle; by means of the limit circle r(t) the local behaviour of a curve at point c(t) can be approximated c к(t) = 1/r(t) P 32

33 the limit circle is unique in the case of smooth curves like NURBS and enables the definition of a curvature graph for a curve c 33

34 Joinig of Curves different degrees of smoothness of joining two curves are possible dependent on the continuity of the curvature graph c 4 same tangent direction same curvature value same tangent direction different curvature direction c 5 c 3 c 1 different tangent direction different curvature direction same tangent direction different curvature value c 2 34

35 Frenet Frame based on the limit circle a point P = c(t) a local coordinate-system at P can be defined bi-normal direction tangent plane tangent direction normal direction orientation by right-hand rule 35

36 Exercise: construct a paperstrip model 36

37 Exercise: construct a pipe with varying diameter defined by the curvature of the guiding curve 37