Det Teknisk-Naturvidenskabelige Fakultet Første Studieår AALBORG UNIVERSITET Arkitektur Og Design MATEMATIK OG FORM

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1 Det Teknisk-Naturvidenskabelige Fakultet Første Studieår AALBORG UNIVERSITET Arkitektur Og Design MATEMATIK OG FORM 11 June Lecture in English FO, Grasshopper: Tegning og analyse af kurver Group 1 8:15-9:15 Lecture at Auditorium 3 Lecturer: Dario Parigi 9:15-11:00 Task Check at group room Group 2 9:15-10:15 Lecture at Auditorium 3 Lecturer: Dario Parigi 10:15-12:15 Task Check at group room Contents: Parametric curves in grasshopper Cubic polynomial curves in Grasshopper: Bezier curves, NURBS curves Evaluating parametric curves Tasks group room task: see document Literature R. Issa, Essential Mathematics for computational design, 2nd ed. (pages 22-36)

2 ZENTRUM PAUL KLEE Renzo Piano Building Workshop

3 The project structure is formed by a series of undulated beams concentric to a circle. The beams height further away from the circle center decreases gradually. The resulting geometry is particularly complex because each beam is different from the other, either in plan and in elevation. Without a parametric definition of the project it would be impossible to define its geometry, and to explore different design proposals. Above we can see the elevation of a single undulated beam. For its definition 14 points are needed. We first want to define the position of the points in plan, then we will fix their position in elevation. The parametric process outlined 1. 2D parametric grid. The design concept is translated into a two dimensional parametric grid, which determines the positions of the ridges and valleys of the roof 2. 3D parametric waves: the three-dimensional geometries of the waves are built on the grid. From the result, any kind of section and view can be derived to check the geometry during the design phase 3. Deduction of steel geometries. The geometry of the curved I-beams is mapped to two-dimensional plans for the contractor, including webs and flanges. Also the geometry for the sheet metal cladding of the I-beams and the roof is unfolded We will implement steps 1 and 2

4 STEP1 2D layout in plan The plan is defined by 30 beams, each one describing an arc of circumference, for each beam we need 14 equally spaced points, so the base grid will have 14*30 =420 points. 1- Draw a circumference with an arbitrary radius (use a slider). 2- On this circumference draw 14 points enclosed in an arch of 1/10 of the circumference. HINT: use the eval component to generate the points on the circumference. If selecting "reparametrized" in the "eval" component the entire circumference can be described with a parameter going from 0 to 1, Find out which parameter you should input in order to define the points in the 1/10 long arch.. Use the component "range " to generate the list of numbers (the parameters). Starting from these points we draw 14 segments radiating outwards the circle, defining the depth of the building in plan. each segment is direction point to the center of the circle. HINT: use the components "vect2pt" and "line SDL" to draw these segments.

5 3- Now we can define 30 points in each of the radiating segments, one for each of the 30 beams crossing the segments. If we merely create 30 points using "eval" and "range" component from the radiating segments, we will get a list of 420 points that we won't be able to use for the geometry of the beams. Instead we need ordered set of points. HINT: you should "graft " the 14 segment before the division into 30 points, in order to obtain 14 separate sets of 30 points each. Then with the component "flip" you can turn the data structure into 30 sets, each one containing the 14 base points for the undulated beam. set set 14 sets with 30 points NO 30 sets with 14 points after "flipping" YES

6 Review - This is how your drawing should look like at this point The dotted line in the grasshopper component signals that there is a tree data structure (the 30 sets of 14 points...).

7 3D parametric waves Now we define the elevation of the points of the undulated beams. We first build 14 points in the "xz" plane, equally spaced, two by two at two different heights, and reference them into grasshopper into a "point" component. The idea is to use the z coordinate of these points as the z coordinate of the points of the undulated beams. HINT: Use "point decompose "component and then the "point xyz" component. If you managed to complete this operation, the z coordinate of the 30 sets of 14 points, base geometry for the undulated beams, is now inherited by the set of 14 points just defined. With the component "Curve" you will generate the geometry of the arches with a NURBS curve. You can control the degree of the curve with the input "D" The idea is however that you can modify the z coordinate of the 14 points before using it as the z coordinate for the undulated beams.use first a component "multiplication" to increase or decrease the height with a "slider"

8 In the project the height of the undulated beams is decreasing gradually. Also, the height is not decreasing linearly, but following a different gradient. We chose to define its gradient with a "Bezier" curve, taken from the component, "graph mapper ", then right click> graph type> Bezier. The Bezier curve is not generating a real geometry in this case, but it is used only for generating coefficients that are used to set the gradient of the decreasing arches height. HINT: use the output of the "graph mapper" as multiplier of the z coordinates of the 14 points before using them as the "z" coordinates of the undulated beams. You will need to use "graft" in order to provide each arch (or branch) with different coefficients, decreasing gradually.

9 Use "loft" component to generate the surface. The curves should be "flattened" before.

10 Use "extrude" to turn the curves of the beams into solid geometry. You will need to use the component two times consecutively, one to define the beams width, the other to define the beams height.