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Three dimensional mosaics with variable-sized tiles. Visual Comput 2008 報告者 : 丁琨桓. Introduction. Three dimensional mosaics, or surface mosaics, are a beautiful art form where a sculpture is made from putting together tiles on a given shape. Previous work.

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## Three dimensional mosaics with variable-sized tiles

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**Three dimensional mosaics with variable-sized tiles**Visual Comput 2008 報告者:丁琨桓**Introduction**• Three dimensional mosaics, or surface mosaics, are a beautiful art form where a sculpture is made from putting together tiles on a given shape.**Previous work**• In computer graphics 2D mosaics have been fully explored. • centroidal Voronoi diagram**Previous work**• 3D mosaics are much harder since the tiles have to be positioned on the surface of a non-planar object being decorated. • If the shape is complex, adequate tile positioning is a real challenge.**Previous work**• [Surface mosaics,2006] addressed the problem of mosaics with tiles of the same size. • Using the same tile size for the whole surface is not the best choice, since this size could be too big for some locations with high curvature.**Algorithm**• Step1: Tiles are initially distributed randomly over the surface • Higher curvature places with higher density of bigger tiles • Smaller curvature places with fewer bigger tiles • Step2: relaxation procedure • move tiles away from one another, leaving some gap for grout and avoiding collisions among tiles. • Step3: rendering specific effects • achieve a more realistic result**Evaluating curvatures using model vertex data**• [ Re-tiling polygonal surfaces, SIGGRAPH 1992 ] • The method gives a good approximation of the exact curvature, using only the model’s polygonal data. • For each vertex the method finds an associated curvature of this vertex with respect to all edges connected to it.**Evaluating curvatures using model vertex data**The radius of curvature r : r = tan(θ)|P-A|/2 Point C bisects the Angle APB In 3D the normal vector at P approximates the line segment PC. The term θis estimated with the dot product between a normalized vector A – P and the normal vector at P. Approximation of the curvature in 2D**Evaluating curvatures using model vertex data**Red is mapped to vertices of higher curvature whereas blue is mapped to relatively flat regions Radius of curvature (Rc) in the plane**Mapping curvatures into tile size**A : the total area of the object’s surface 2h: the average tile size and h is half this size N : user-specified number of tiles h : half the tile size r : the radius of the circle**Mapping curvatures into tile size**Function for mapping curvatures into tile sizes**Distributing random points on the surface of a polyhedral**model • distributed randomly over the surface • polygon capacity Ai : the area of polygon i rci : the polygon radius of curvature f : the mapping function**Distributing random points on the surface of a polyhedral**model • Polygons with higher curvature, i.e., smaller radius of curvatures, will receive more tiles. distributed randomly distributed with capacity function**Relaxation of points on the surface of the model**• move the tiles away from each other, to avoid intersections using a repulsive force • repulsive force is proportional to tile size, such that small tiles will concentrate in strongly curved places, and big tiles will push smaller ones to curved regions • f = Kf * ( 1 – d/(r1 + r2))**Relaxation of points on the surface of the model**• f = Kf * ( 1 – d/(r1 + r2)) • d is the distance between the particles • r1 and r2 are the radii of the ideal circles around the tile. r : the radius of the circle**Relaxation of points on the surface of the model**r1 r2 r1 r2 d d f > 0 f < 0 f = Kf * ( 1 – d/(r1 + r2))**Adjusting the orientation of the tiles**• [ Texture Synthesis on Surfaces, SIGGRAPH 2001 ] • Vector field**Rendering**• To make the results more visually appealing to the user, the final shape of the tiles may be controlled by four parameters Square tiles, turned into general quadrilateral tiles**Rendering**Comparison of tiles with and without random variation in the shape. Random variables U1, U2, V1, and V2 with values between 85% and 115% of h**Result**# of tiles :7000 Tsmin : 0.4h Tsmax : 3.15h Rcmin : 0 Rcmax : 25h**Result**# of tiles :7000 Tsmin : 0.1h Tsmax : 2.3h Rcmin : 0.5 Rcmax : 20h**Result**Effect of varying the size of tiles ( number of tiles : 4000)**Comparison**surface mosaic mosaics with variable-sized tiles Comparison with previous result from [surface mosaic]**Conclusion**• This paper presented a solution efficiently computes the distribution, placement and rendering of tiles • Author plan to extend this work by allowing tiles of variable shapes, not only squares.

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