Transformer A related pattern Change an object's form by translating and/or rotating its rigid components. In physics, a rigid body is an idealization of a solid body of finite size in which deformation is neglected. Although most architecture structures are, in essence, rigid bodies as a whole, most have mechanisms within them (think of a door hinge). Interest in larger scale mechanisms in increasing, though the most obvious examples are still to be found in stadium roofs \cite{aGoodRetractableStadiumRoof}. Use this pattern when you are making quick approximate models of the motion of an assembly of rigid bodies. Apart from such systems as tensioned membranes, bending members or inflatable structures, mechanisms are needed for movement in design. In such a structure, every components is a rigid body and links between components restrict feasible relative motion. Components can rotate and slide with respect to each other. Some parametric modeling systems provide kinematic analysis packages, which compute allowable states of mechanisms. Certainly use these when they are available, but understanding how to model simple mechanisms directly remains an important technique. The Transformer pattern has two basic features: rigid bodies maintain the same shapes and sizes during a transformation, and these bodies slide and/or rotate along a path or according to a rule. Understanding and codifying the rule is critical to modeling a mechanism. First the components must be rigid bodies. Before creating the model, you should distinguish the rigid bodies from other parts in the models. While the body can be complex, it will typically be based on a Jig: a relatively small collection of simple geometric elements. Except for position, these must be indepdent of any parameter that drives the mechanism. In a model of a rigid body, make its Jig as simple as possible and then define all the detail solely in terms of the Jig. Secondly, the transformation is constrained by the components. After the first step, the rigid body has one or several geometries as the anchor. Making this anchor transformable is the key. Some properties of this anchor could be variables such as the parameter along the curve, the number of the panels or the angle of rotation. Some properties could also be driven by independent Controllers. Scissor Arm Model a scissor mechanism, that is, an arm whose horizontal extension is controlled by its handle's vertical movement along a line. In this model, lines model the scissor arm rigid bodies. All lines are of the same length. The anchor comprises a fixed line in space and one of its endpoints. The next step is to define the anchor. There are several parts available to be used as the anchor - the start point or the endpoint of frames. In this sample file, you take the two frames crossing point as the anchor. This series of points slide on both X and Y directions, anchoring a middle line path between the two controlling points and a changeable distance between each other. Since all the frames are of the same length, after calculating the first anchor point, it is easy to locate all the rest anchors. Finally, you apply crossing frames on each anchor points and get the extendible frames. Download Tran_Arm.gct Square Use polygons with right angles as a shutter. Four polygons, each having one right angle, can be arranged in spiral pattern around a square. Changing the size of the square moves the polygons such that they model a shutter. Polygon sides appear to slide along each other. A parametric model for this is almost trivial. Download Tran_Square_Simple.gct Rotary action of the shutter, and perhaps more geometric insight, can be had by realizing the that shutter corners move along arcs of circles with centres at $90\; degree$ intervals on the circumference of a circle of the same size. The lines defining the right-angle sides start at the shutter corners and pass through the guide circle intersections. Download Tran_Square.gct Shades Model vertical and horizontal shades covering a window. Sliding blinds are a common window covering. Overlapping horizontal and vertical blinds open a view slowly at first, then rapidly as they approach their full travel. Of course, they require double the material of a single blind system. Modeling such blinds is straigtforward, except for one trick: coordinating smooth movement across all of the blinds. The trick is that the points controlling the location of the blinds are parametric points on the window diagonal and are themselves controlled by the $slidePoint$, a single parametric point on any curve, anywhere in the model. Here is how it works. Download Tran_HVShades_Feature.gct Download Tran_HVShades.gct Differential A mechanical differential mechanism links rotation through three axes. Using coordinate systems multiple axes can be linked. Of course, this is a representational device only---actually constructing such a system could be a challenge. In this sample, a vertical axes is linked to multiple axes at $90\; degrees$ to the vertical axis. The horizontal rotations are a function of the vertical rotation. The modeling process is straightforward. To clarify the geometric structure, each successive coordinate system is displaced by a small amount. Reducing this amount to zero brings all objects to a central point. The $Z$-axis of the $differentialCS$ coordinate system defines the vertical axis. The origin of the $inputCS$ coordinate system coincides with $differentialCS$ and is rotated by $input\; degrees$ about the vertical axis. Place $count$ coordinate systems (call these $outputBase$) on the $XY$-plane of the $inputCS$, each rotated about the vertical axis by $360/count\; degrees$ from each other. Place an $outputCS$ coordinate system, coincident with each $outputBase$, with $Z$-axes aligned with the corresponding $outputBase$ coordinate systems. Bind the rotation around the $Z$-axis of each outputCS to a function (the simplest is $f(x)=x$) of $input\; degrees$. Attach an object to each $outputCS$. The objects will rotate as the input rotation changes. In this sample the rigid bodies are circle segments, governing by parameters for their subtending angle and radius. When the subtending angle is $180\; degrees$, and $input$ is $0\; degrees$, the overall form approximates a sphere. When $input$ is $90\; degrees$, all objects lie in the $XY$-plane. Download Tran_Differential.gct Shutters Simulate the shutters' movement along a free curve. \NtS{The function in this script doesn't work and is very long. Debug and simplify.} In the image, Mercedes Benz ML320 uses shutters to cover its curved sunroof. It is easy to draw the shutters of a planar door, but tricky to deal with a curved surface. No need for reticence, the shutters are rigid bodies. Each piece of shutters not only slide along the curve, but also cover on the previous piece so that there is no leaking. In this model, the most interesting part is not the anchor of shutter, but the rest part of the shutter. One side of the shutter slides along the curve. Each shutter's anchor is located and indentified through its index number. For the rest part of the shutter, to cover the previous piece property, there are two conditions: covering on the previous piece (A) or snaping to previous piece's starting points and sticking out (B). How to distinguish these two conditions? You can draw two directions (direction 1 from the current shutter's start point to previous shutter's start point, and direction 2 from the current start point to the previous's endpoint) and get their cross product direction. Wether or not this direction is positive determine which condition it belongs to. Through using "if...else" statement in the script, you can make your shutters responsive to the different conditions along the free curve. Download Tran_Shutters.gct