Place Holder Organized Collection of Points Use proxy objects to organize complex inputs when making collections. \label{sec:pattern:PlaceHolder} Designs have parts. Parametric modeling enables a single model to represent variations of a part, for example different window designs. Copying the model, one copy for each part, and adjusting the inputs to the copies is an effective strategy. Typically a part has multiple inputs---customizing each one is a lot of work. Use this pattern when you are able to describe the multiple inputs to an model through a smaller number (preferably one) of abstract proxy objects. A very common situation is to array a complex module across a surface or along a set of curves. If this module requires multiple point-like inputs themselves defined on the target, organizing these inputs is sure to be complex and error prone. If you can define the inputs to the complex module through a simple construct such as a polygon\index{polygon}, it is often much easier to place the module. An arrangement of polygons on the goal surface creates proxies on which the module can be later (and easily) placed. Place Holders have two parts. The first is the proxy object itself: a simple object that carries the inputs for the module. For example, a rectangular module may require four input points, one for each corner. A four-sided polygon can act as a proxy for these points: each of the vertices of the polygon provides one of the points. The proxy simplifies the arguments needed for the module: instead of four points, use only one polygon. The second part relates the proxy object to the model. For example, a polygon proxy can be placed using a rectangular array of points by relating the $ijth$ polygon's vertices to the points $\afPt[p]_{i,j}, \afPt[p]_{i+1,j}, \afPt[p]_{i+1,j+1} \afTMath{and} \afPt[p]_{i,j+1}$Pij, Pi+1j, Pi+1j+1 and Pij+1.. The code placing a generic object such as a polygon is more simple and reusable than the code for a specific module. \renewcommand{\sampleNewPage}[0]{false} Hedgehog Use a collection of points as Place Holders to locate components (spines) that are perpendicular to a surface. Every point\index{point} on a surface\index{surface} defines a single frame\index{frame} comprising the surface\index{surface!normal} normal and the two vectors\index{vector} of principal curvature\index{surface!curvature}. This \index{surface!principal directions} is sufficient information to place and size spine-like objects on the surface. The point provides location, the surface normal provides the direction for the top of the spine and the vectors of principal curvature provide information for further adapting the spine to context. Start with an Organized Collection of Points defined by $u$ and $v$ parameters on the surface. Instead of points, use coordinate systems---remember they have points inside them! Each of the coordinate system\index{frame} points will serve as the base of a spine. Define two graph variables $count$ and $height$. Use $count$ to control the Organized Collection of Points to produce $count$ coordinate systems in each parametric direction. At each of the coordinate systems, use the $z$-direction of the coordinate system and the parameter $height$ to define a cone. \begin{bodyNote} \pbOneCol[r] {\pbIncludeGraphics[Samples/PlHoHedgehog/Gruz_Anita_Martinz.jpg]{width=\currentWidth} \source{Anita Martinz}} \end{bodyNote} Download PlHoHedgehog.gct Truss Use lines as Place Holders to locate the members of a truss. Each member of a truss might carry information such as the section of the member its moment of inertia, material and modulus of elasticity. In addition, the parametric model for truss members may be able to shape its ends depending on the context in which it is placed. Placing a truss member though requires only the baseline along which the member lies. First, develop a feature representing a truss member and requiring only a line\index{line} as a geometric input. Second, (and in a new model!) create an abstract truss comprising line segments to represent the truss members. By applying the truss member feature to these baseline Place Holders, the detailed truss members are put in place. This is, of course, a simplification of a real truss member Place Holder in which the truss member parametric model would need sufficient information to shape its sectional properties end details. Taking this next step would require that the Place Holders become spatially more sophisticated and that the truss member feature use that new information to specify its details. \begin{bodyNote} \pbOneCol[r] {\pbIncludeGraphics[Samples/PlHoTruss/firthofForth_Kenneth_Barker.jpg]{width=\currentWidth} \source{Kenneth Barker}} \end{bodyNote} Download PlHoTrussBeam.gct and PlHoTrussSystem.gct Paper Folding Use quadrilaterals as Place Holders to simulate the paper folding effects in the following image. \begin{bodyNote} \pbOneCol[r] {\pbIncludeGraphics[Samples/PlHoPaperFolding/PlH0PaperFoldingImage.jpg]{width=\currentWidth} \source{Zhenyu Qian}} \end{bodyNote} Parametrically modeling folded paper is hard. The problem is physics -- paper has actual dimensions and folds in it constrain the spatial configurations it can achieve. These real-world constraints inevitably imply that any model will require the solution of simultaneous equations, which propagation-based systems cannot do. (The Goal Seeker pattern demonstrates a partial solution to this problem.) That said, approximations can be useful in design and this sample shows a way to simulate a folded paper system, ceding from reality some dimensional variation in the individual panels. In a folded structure, the pattern of folds can be thought of as separate from the size and location of the folded panels. Further, the folding pattern will belong to one of the $17$ possible symmetry\index{symmetry} groups on the plane (each group represents one of the fundamentally different ways of arranging a collection of like motifs on the plane (\citet[pp. 37-45]{GRUNBAUM1987}; \citet{Weisstein2009WallPaper})). In each such group, there is a repeating module that imposes geometric conditions on where the paper edge must be to connect to the next module. The modeling task splits into three parts: the paper folds, ensuring geometric connection at the joints and arranging the resulting module across a surface. The choice of module is key to clarity and simplicity. This sample comprises a collection of identical parallelograms (for symmetry aficiandos, arranged in symmetry group $pmg$ in crystallographic notation). It is much simpler though to combine parts of six parallelograms\index{parallelogram} to form a module that has simple translational symmetry only (symmetry group $p1$). Using two whole and four half parallelgrams defines a module in which it is easy (or at least, easier) to relate the geometric boundary conditions to the proxy Place Holder. \begin{bodyNote} \pbOneCol{\input{\GCFigsPath/Patterns/PlaceHolderPaperFoldingCell0.tex}} \end{bodyNote} To connect adjacent modules requires coincidence along each edge and at each vertex at which modules join. Four edge points connect two modules each and four vertex points connect four modules each. The edge points are easy: they lie at edge midpoints on the Place Holder, so are guaranteed to coincide. Vertex point coincidence requires that, at each vertex, adjacent modules share a common vector from the vertex to the module point. Here, this vector is computed as a global property of the surface. With slightly more work, the Place Holder object could hold individual direction vectors\index{vector!direction} at each of its vertices. An Organized Collection of Points using a rectangular grid of the $u$ and $v$ parameters on a parametric surface locates a collection of quadrilateral\index{quadrilateral} polygons\index{polygon}. Using these quadrilaterals as Place Holders, the surface gets covered by the folded modules and looks like a folded paper sculpture. It isn't one of course: on a general surface, edge lengths will differ from the initial paper and polygons will not be planar. In more constrained domains it is feasible to model folded paper. For instance, the Goal Seeker pattern can be used to find the feasible configurations for folding models of Persian Rasmi domes\index{Rasmi dome} from single sheets of paper \cite{Maleki2008A}. \begin{bodyNote} \pbTwoCol {\pbIncludeGraphics[\extImagesPath/Rasmi/MalekiPaperDome.png]{width=\currentWidth/2}} {\currentWidth/2} {\pbIncludeGraphics[\extImagesPath/Rasmi/MalekiDome.png]{width=\currentWidth/2}} {\currentWidth/2} \figcaption{A dome structure folded from a planar assembly of triangles. \source{Maryam Maleki}} \end{bodyNote} Download PlHoPaperFolding01.gct and PlHoPaperFolding02.gct Mattress Use octagons\index{octagon} and quadrilaterals\index{quadrilateral} as Place Holders to simulate the shape of an air mattress. There are two special features in this sample. 1) There are two types of Place Holders: the quadrilateral to place the rectangular pillow and the octagon to place the octagonal pillow. 2) The combined Place Holder is hierarchical: one simple Place Holder defines the actual Place Holders containing both quadrilaterals and octagons. The first Place Holder is the familiar one from previous samples: the quadrilaterals created by $uv$ points on the surface. This acts as a Place Holder for placing octagon Place Holders on face centres and quadrilateral Place Holders on vertices. Once the Place Holder structure is complete, the two different modules are located on each of the quadrilaterals and octagons respectively. \begin{bodyNote} \pbOneCol {\tikzPlaceHolder[Explanatory figure here]{\currentWidth}{\currentWidth*0.75}} \end{bodyNote} Download PlHoSquarePillow.gct, PlHoPolygonPillow.gct and PlHoMattress.gct