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Content may be subject to copyright. # The construction of a pentagon can be performed using compass and a straightedge only. The construction algorithm is based on Euclid's axioms .

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This tutorial introduces the concepts and techniques of procedural & generative modeling. It starts with some introductory examples to motivate the main idea: to describe a shape using an algorithm. After the explanation of technical terms, the second Section focuses on technical details of algorithm descriptions, programming languages, grammars an...

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... ruler-and-compass constructions consist of repeated application of five basic constructions based on Euclid's axioms  using the points, lines and circles that have already been constructed. Based on these geometric primitives and a fixed set of operations, the ruler-and-compass constructions -such as illustrated in Figure 1 -are the first algorithmic descriptions of generative models. 1. Draw a circle in which to inscribe the pentagon and mark the center point O. ...
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... elementary data structures are commonly used in computer graphics (see Figure 10). By increasing the number of primitives a better approximation of a cylinder can be created. ...
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... quality of the approximation of an object's surface with polygons highly depends on the number of primitives and the modeling operations used. Once an approximation is found, all information about the approximated surface is often lost, see Figure 11. ...
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... Surfaces Subdivision surfaces are defined recursively starting with a given polygonal mesh (typically a quadrilateral, or a triangle mesh), as illustrated in Figure 10 (middle). A refinement scheme is applied to this mesh creating new vertices and faces converging to the limit subdivision surface (which is the surface produced by applying the refinement scheme infinitely many times). ...
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... to the fact that position and size of a voxel are pre- defined, voxels are good at representing regularly sampled spaces. The approximation of free-form shapes suffers from this inherent property, as can be seen in Figure 10 (right). Nevertheless, voxel representations do not suffer from numerical instabilities as they are typically defined on an integer grid. ...
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... this way, deformed splits can be carried out, the deforma- tion can be baked at any point to allow for straight splits in deformed geometry. An example is shown in Figure 13. ...
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... first step of the process is to register a generative model (including its free parameters) to a laser scan. Then, the difference between the generative model and the laser scan is stored in a texture, which can be applied to all instances of the same shape family as illustrated in Figure 14. ...
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... generative models represent an ideal object rather than a real one, the combination of noisy 3D data with an ideal description enhances the range of potential applications. This bridge between both Figure 13: Deformation aware shape grammars allow the integration of free-form deformation into a grammar- based system that is based on planar primitives and splits. Measurement about the available space for placing objects are taken in deformed space while splits are carried out in undeformed space. ...
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... modeling framework lets developers optimize the transcoding trade-off to create 3D interactive visualizations. The example augmented reality application shown in Figure 15 displays underground infrastructure created out of geographic information systems (GIS) data. ...
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... order to reduce the search space, an irregular grid is derived from the classifications, and the parsing algorithm is applied to yield the most probable application of rules that yields a classification label per grid cell. Such a parse tree can easily be converted into a procedural model, as can be seen in Figure 16. ...
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... system terminates, if every vertex has been assigned a state. This process is illustrated in Figure 17. ...
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... authors propose a similarity measure between the statistical model and a given input mesh that consists of three parts: shape distance, which measures the overall shape discrepancy, geometric distance, which reflects the statistics of geometry of its branches, and structural distance, which encodes the cost of transforming a graph representation of the statistical tree model into a graph representation of the input tree model. For some examples see Figure 18. The MCMC method has also been applied by other methods to find parameters of a statistical generative model: , , . ...
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... partial derivatives many numerical optimization routines cannot be used at all or in a limited way. The example data set shown in Figure 19 consists of laser-scanned cups and a generative cup descrip- tion. The algorithm is able to detect an instance of the generative cup. ...
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... can be combined by calling a sequence of different transfor- mations. Figure 21a shows an example of a profile polygon, copies of which are placed radially around the origin (Figure 21b). These polygons are first converted to double-sided faces. ...
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... can be combined by calling a sequence of different transfor- mations. Figure 21a shows an example of a profile polygon, copies of which are placed radially around the origin (Figure 21b). These polygons are first converted to double-sided faces. ...
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... polygons are first converted to double-sided faces. Then, corresponding back and front sides are connected (Figure 21c) and a subdivision surface is created (Figure 21d). The profile polygon, the rotation angle, and a set of custom parameters are the input to this transformation function (Figure 21f). ...
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... polygons are first converted to double-sided faces. Then, corresponding back and front sides are connected (Figure 21c) and a subdivision surface is created (Figure 21d). The profile polygon, the rotation angle, and a set of custom parameters are the input to this transformation function (Figure 21f). ...
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... corresponding back and front sides are connected (Figure 21c) and a subdivision surface is created (Figure 21d). The profile polygon, the rotation angle, and a set of custom parameters are the input to this transformation function (Figure 21f). In a post-processing step selected faces can be colored differently (Figure 21h). ...
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... profile polygon, the rotation angle, and a set of custom parameters are the input to this transformation function (Figure 21f). In a post-processing step selected faces can be colored differently (Figure 21h). The creation of the basic shape is separated from optional steps to create engravings, change materials, or add gems. ...
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... surfaces can be treated with various finishing techniques like polishing, brushing, or hammering. In order to account for these effects, a per-pixel shading model using an approximation of the Fresnel reflection term Figure 21: Parametric wedding ring construction: n copies of the the profile polygon (a) are radially placed around the center (b). Consecutive profiles are connected using a newly-designed operator to form a control mesh (c) that has a simple toroidal structure. ...