Projection Mapping Organized Collection of Points Credit: Alexandre Duret-Lutz \sourceLocated{Alexandre Duret-Lutz. Creative Commons Attribution Share-alike.} Produce a transformation of an object in another geometric context. "Here" and "there" pervade design. Eyes, ears, the sun, lights, structural elements, ducts and pipes all relate a "here" to some distant "there". Often the relationship between here and there is a geometric line or curve. Use this pattern to construct coherent, reproducible relationships between "here" and "there." Projection\index{transformation!projection|(} provides a conceptually simple, yet openended tool for producing new objects from old. Its origins lie in the Renaissance and before. For designers, projection is perhaps most associated with the field of descriptive geometry, an 18th Century invention \cite{MONGE1827} and one which, until recently, was a mandatory part of design curricula worldwide. Descriptive geometry codified procedures for deriving two dimensional representations of three dimension objects by projecting the three dimensional objects onto surfaces. Parametric modeling supports a much richer collection of projective ideas than was practical with older, manual techniques. With tongue somewhat in cheek, one could argue that parametric modeling is the 21st Century replacement for descriptive geometry. The basic idea of projection has three parts: (1) a source object to be projected, (2) a projector or projection method, and (3) a receiver, the object on which the projection appears. Its most simple form is orthogonal projection: points are projected onto a receiving plane such that the projection lines are perpendicular to the plane. The suite of projection and intersection tools available in parametric modeling enable a wide range of projective form-making ideas. The two main effects of projection are indirection and separation. With it, a model can be the indirect cause of a sculptural effect. With it, different aspects of an object can be separated into distinct views that may enable special views and inferences on the object. A very common example is a light (essentially a point source) that projects a pattern through a screen onto a surface. Every Projection has the three parts mentioned above: (1) the projected object, (2) the projection method and (3) the receiving object. The projected object is a point or any composite of points: a line, ray, line segment, curve, polygon, surface, or $3D$ object. The three most common projection methods are parallel projection in which all projecting ray are parallel; normal projection in which the projecting rays are normal to the receiving object; and perspective projection in which all projecting rays pass through a single point. There are a wide range of other methods. For instance, cartographic projections can be explained as the mapping of parametric coordinates from one surface to another. The most common receiving ojects are planes, polygons, surfaces, lines and curves, as well as composites of these. While possible, projections to points and $3D$ objects seem to be less common in practice. A wide variety of projections exist \cite{CSISS2009}. Computing projections involves either mathematical projection or intersection. Mathematical projection provides direct solutions to relatively simple situations such as projecting one vector $u$ onto another vector $v$ with result $w=((u.v)/|v|)v$. More complex situations involve intersecting objects. For example, projecting a point onto a surface amounts to computing the intersection between the surface and projecting ray. In many simple cases, a parametric modeler will provide direct tools for computing projections, for instance, projecting a line onto a plane. It is a fact of life though that designers will push these bounds. In these more complex situations, using the Projection pattern involves three steps: (1) sampling key object points, (2) projecting these points onto the receiver, and (3) reconstructing the object as projected on the receiver. Surface Sampler Project a collection of points onto a surface. The mathematical tools for describing surfaces tie the shape of the surface and its $uv$-parameterization to the control polygon. Often, only part of the surface is actually needed in a design. Projecting an Organized Collection of Points onto the surface\index{surface} provides a subset of the surface with its own independent parameterization. The source object is a point\index{point} collection. In this sample, the collection lies on a plane and is a simple array, but other geometric and data arrangements can be used. The projector is parallel projection, with projecting rays being parallel to a line from the centroid of the collection to a controlling point in space. Alternatively, the projector could be a perpendicular line\index{line} from the source plane and a control could allow the source to be moved within its plane. The receiver is the surface. Download ProjSurfaceSampler.gct \renewcommand{\sampleNewPage}[0]{false} Shadows Simulate a row of posts casting shadows on the ground as a light moves by. Start from a line (as the abstraction of a post) standing vertically on the $XY$-plane. Define a free point\index{point} as the moving light. The shadow point is the projection of the free point onto the $XY$-plane. The shadow is a line\index{line} between the base of the post and its shadow point. Replicating the startpoint of the posts gives a row of posts, each with its own shadow. In this case, the source is the free point, the projection is a perspective projection through the source and the receiver is the $xy$-plane. Download ProjPointPost.gct Skylight \sourceLocated{Pieter Morlion} Create a daylight "lens" that focuses on a circle. Two free-form surfaces\index{surface} represent a roof structure and ceiling. The $xy$-plane represents a floor. A circle\index{circle} on the floor can be daylit by projecting it through the ceiling and roof surfaces. The direction of daylight is nearly uniform, but the two separated holes will act as a very fuzzy lens to focus daylight on the circle. Similar to the Surface Sampler sample, the projection direction is controlled by a free point\index{point}. If the direction is chosen to lie within the sun's annual range, the sun will shine directly on the circle exactly twice a year. If the direction is chosen to lie on either of the two solstice paths, this reduces to exactly once per year. Fixed architecture can have difficulty in responding to moving phenomena. The projection of a circle onto a surface, or even an angled plane, is no longer a circle. While some parametric modelers may provide curve onto surface projection tools, a good approximation can be had by projecting sampled points on the circle and reconstructing the curve\index{curve} from the projected points. The resulting curve will not exactly coincide with the surface in which it should lie. Alternatively, if the modeller has surface trimmming tools, trimming the surface with a sweep of the circle along the projection line will yield a new surface with a hole. When rotating the model, you can see that three circles perfectly coincide at a specific viewing angle (in a parallel viewing projection). A famous example of projection used in form-making is Le Corbusier's 1953 Monastery of Sainte-Marie de La Tourette in France (shown in the image above). Download ProjSkylight.gct Spotlight Model a metaphorical spotlight projecting a circle shape onto several surfaces. This sample is very similar to the previous one. The main difference is that all the projecting rays intersect at the light point. To calculate the projection, each projecting ray starts from the light point and goes through the sampling points of the base circle. While rotating the model in a parallel projection view, you can see the projected circles do not coincide at any angle. If you use a perspective view the projected circles\index{circle} coincide when the camera and light source coincide. \begin{bodyNote} \pbOneCol {\input{\GCFigsPath/Patterns/ProjectionSpotlightExplain0.tex}} \end{bodyNote} Download ProjSpotlight.gct Solar Polygon Shadow The shadow of a polygon cast by the sun on a curved surface. \index{point|(} \index{line|(} \index{polygon|(} \index{curve|(} \index{surface|(} The source is a polygon, the receiver a free-form surface. The projector is the sun, therefore the projecting rays are parallel. Straight lines project as curves onto a surface. For specific surface types, such as conic sections\index{conic section}, there are closed-form equations for these curves. For free-form surfaces, approximations must suffice. Even though a curve projection tool may be available in your chosen parametric modeler, techniques for making such an approximation are an important part of a modeler's toolkit. The key is sampling. Sample each line of the source with a sequence of points. The choice of how many points depends on the complexity of the receiving surface: high curvature and rapidly changing surface normals will require more samples. Project the sampled points onto the surface and reconstruct a curve "on" the surface from the sampled points. The word "on" is advisory---the curve will not lie exactly on the surface. Much representation is approximation. If the curve must exactly coincide with the surface, either sample very densely or find a modeler that supports exact curve-to-surface projection. It is clear that the shadow is no longer bounded by four straight lines, but by four curves. Note too a further simplifying assumption. Non-planar polygons can thought of as defining a minimal surface. If the source polygon is nonplanar then its orientation must be such that no part of this minimal surface projects outside of the projection of the polygon's edges. Else, the shadow will not model reality. This may be good enough---again, much representation in design is approximate. Download ProjParaShadow.gct Spotlight Polygon Shadow The shadow of a polygon cast by the spotlight on a curved surface. This sample is very similar to the previous one. The main difference is that all the projecting paths intersect at the spotlight point. To calculate the projection, each projecting path starts from the spotlight. When the spotlight moves closer to the polygon, the size of shadow increases quickly. \index{point|)} \index{line|)} \index{polygon|)} \index{curve|)} \index{surface|)} Download ProjPointShadow.gct Pinhole Camera Model a pinhole camera. A pinhole camera is a camera without a conventional glass lens. An extremely small hole in very thin material can focus light by confining all rays from a scene through a single point. In order to produce a reasonably clear image, the aperture diameter has to be less than about 1/100 of the distance between the pinhole and the screen. The principle of a pinhole camera is that light rays from an object pass through a small hole to form an image on the screen (shown in image above). To model this effect, simply place a point\index{point} (modeling the pinhole) between the source and receiver and project from the source through the pinhole to the receiver. Use the sampling technique from the Solar Polygon Shadow sample above to reconstruct the source on the receiver. Note that the image is reflected both top-to-bottom and left-to-right. This is equivalent to a $180\; degree$ rotation about the axis normal to the receiving plane and through the pinhole (providing the receiver is a plane).\index{transformation!projection|)} Download ProjCamera.gct Kaleidoscope Model a kaleidoscope. The kaleidoscope is a tubelike instrument containing loose bits and pieces that are reflected by mirrors so that various symmetrical patterns appear as the instrument is rotated. Many people have played with this toy. This sample does not seriously follow the principles of mirroring in the tube. Instead, it uses Projection and replication to simulate the effect of a kaleidoscope. The source curve lies on a plane (modeling the loose bits and pieces). You create a free point and vector to model the central axis of the tube. Build three symmetric surfaces surrounding the axis to model the internal mirrors. The first step projects the curve from the target coordinate system to the surrounding mirrors. Then the second Projection is from the mirrors to the bottom plan. As you can see, the result of these two steps is totally different from a direct Projection. Since the three mirrors in a real kaleidoscope reflect images among each other, you can replicate the resulting image and get a kaleidoscope effect. When you move either the target coordinate system or the central axis, the bottom image shifts and varies. Download ProjParaKaleidoscope.gct