004 LectureDM2_2011

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TRANSFORMATION DM2 | Lecture 4 IMAGE BY http://ds13.uforg.net/page/2/

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Lecture on transformation

Transcript of 004 LectureDM2_2011

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TRANSFORMATIONDM2 | Lecture 4

IMAGE BY http://ds13.uforg.net/page/2/

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This series of lectures will focus on the many types of transformations that are used in constructing the proper geometry for specific architectural applications.

+ Planar Transformations will be explored.

+ Tilings - Regular and Semi Regular tessellations will be examined and how they transition into building components.

+ Motion, Sweeping, and Shape Evolution

+ Skinning

TRANSFORMATIONS

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Basic Transformations are congruence transformations (translation, rotation, reflection) which preserve all lengths and angles occurring on an object.

Slightly more general are similarity transformations, which still preserve angles but multiply all distances by the same factor.

Shear transformations preserve the area of the transformed objects.

Scaling transformations provide even more freedom for shape modification (but it’s still linear)

PLANAR TRANSFORMATIONS

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Escher illustrates a perfect way vari-ous types of transformations are cre-ated. We study his work to learn about tiling’s which can be used to create so-phisticated facades or surface tilings.

He used his knowledge of the properties of planar transformations to generate nontrivial tessellations.

M.C.ESCHER

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Located on the BART transit line.

Ceramic hexagon tiled wall.

Form features a hexagon shape blended into a spherical form.

Completed in 1973

SAN FRANCISCO RAILWAY STATION

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FOA - FOREIGN OFFICE ARCHITECTS.

2005 World Exposition Aichi, Japan

FOA created this seemingly irregular hexagonal tiling by simply shifting some of the internal vertices of groups of 8 tiles. By coloring the tiles in various colors, this further breaks up the under laying regular pattern.

Constructed of slip cast glazed ceramic.

SPANISH PAVILION BY FOA

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Only three cladding materials; sand-stone, zinc (perforated and solid) and glass have been used within a modular basis established by the triangular pin-wheel grid

This fractal incremental system uses a single triangles, whose proportion is maintained across the single tile shape, the panel composed of five tiles and the construction module of the mega-panel composed of five panels.

The unique quality of the pinwheel grid lies in the possibility of surface figu-ration and framing shapes to be indepen-dent from the grid’s smallest component unit, the triangle.

This grid allows the building facades to be treated in a continuous changing and visually dynamic way, instead of being traditionally composed as a regularly repeating flat surface.

FEDERATION SQUARE - LAB ARCHITECTS

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A transformation is a general term for four specific ways to manipulate the shape of a point, a line, or shape. The original shape of the object is called the pre-image and the final shape and position of the object is the image under the transformation.

Types of transformations in math • Translation• Reflection• Rotation

Translation – a translation is defined by a translation vector t, which specifies the direc-tion and the magnitude of the translation.

TRANSLATION, ROTATION AND REFLECTION OF A PLANE

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A simple example that is not as trivial as the square and the hexagon is the chair fam-ily of polynomial. Chairs are not symmetric as the square and the regular hexagon are, and still they manage to tile the plane in the same simple way, just by translating them in two different directions.

TRANSLATION, ROTATION AND REFLECTION OF A PLANE

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Rotation – We define a rotation by a fixed point c, the center of rotation, and the rotational angle p

The order in which you perform transforma-tions can affect the final result. Consider, for example, translating and rotating an image. If you perform the transformations in this order, you end up with a rotated model translated, for example, down the X-axis, as shown

TRANSLATION, ROTATION AND REFLECTION OF A PLANE

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We have now looked at how planar congru-ence’s transformations as effective tools for positioning objects in the plane.

They are very useful in creating regular tessellations and tiling’s.

The art of designing tiling’s and patterns has a long history and is therefore well developed

TILINGS

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A tessellation is a way of filling the en-tire plane with congruent shapes without overlaps or gaps.

The term tiling is sometimes used to de-scribe a special tessellation of the plane using planar polygons.

There are 14 types classes of convex pen-tagonal tilings known and three classes of tilings with irregular hexagonal tiles. There exist only three regular tessella-tions due to the fact that the vertex angle of the tiles must be a divisor of 360 de-grees, Therefore, we only have regular tes-sellations with regular triangles, squares and hexagons.

REGULAR AND SEMI-REGULARTESSELLATIONS

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Quadrangular Tessellation

Triangular tessellation

Irregular Tilings

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Tessellations that use two or more differ-ent regular polygons, we add the rule that every vertex must have exactly the same con-figuration. This means that every vertex there has to be the same number and the same sequence of congruent regular polygons.

SEMI REGULAR TESSELATIONS

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a built project which demonstrates a system of flat panel tessellation derived from com-plex surfaces to enable ease in constructa-bility. Each panel’s uniqueness is afforded by the efficiency of digital fabrication while coded parametric relationships allow an emergent structural efficiency. Recently the development of planar quadri-lateral meshes has become a strong interest in the architectural community due to their potential ease for constructing complex sur-faces. The project responds to this problem and proposes a method for flat panelization of free form surfaces which provides large scale, efficient and economic construction from flat sheet material.

Tesselion was constructed using:-189 panels (7 rows of 27 panels)

EXAMPLES IN ARCHITECTURAL TILINGSSKYLER TIBITS - SJET

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It was proposed as a dynamically reconfigu-rable surface capable of real-time respon-siveness to events in the theatre, such that movement or sound can create actual defor-mation of the architectural surface. Ef-fectively Aegis is a dynamically reconfigu-rable screen where the calculating speed of the computer is deployed to a matrix of ac-tuators ( 896 pneumatic pistons ) that drive a ‘deep’ elastic surface. The implicit sug-gestion is one of a physically responsive architecture where the building develops an electronic central nervous system, the sur-faces responding instinctively to any digi-tal input (sound, movement, Internet, etc).-CNC routed 1/16” white aluminum-1 week of assembly

EXAMPLES IN ARCHITECTURAL TILINGSDECOI - HYPOSURFACE

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