Blinn, How many ways can you draw a circle? Sabin, High accuracy geometric Hermite interpolation, Comput.
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Aided Geom. Design 4 , — Dokken, M. Lyche and K. Design 7 , 33— Floater, High order approximation of conic sections by quadratic splines, Comput.
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Design 12 , — Goldapp, Approximation of circular arcs by cubic polynomials, Comput. Design 8 , — Makarov, Splines with minimal defect and decomposition matrices, Communications in Mathematics and Applications 3 3 , — Rababah, High order approximation method for curves, Comput. Design 12 , 89— Rababah and S. Sculptured Surfaces It is well known that research studies of 3D geometric modeling started as part of CAM computer-aided manufacturing rather than CAD computer-aided design.
Inventing NC numerically-controlled machine tool at the beginning of the s at MIT Massachusetts Institute of Technology, US generated demand for digital mock-up of component parts, required to develop control programs for machine tools. Various research groups studied the principles of modeling 3D objects; their main customers were the largest companies in aerospace and car-manufacturing industries. He suggested a method for construction of smooth curves using a set of reference points that determine geometric properties.
How is it possible to constructively by geometric construction rather than abstract algebraic equations define a smooth surface of a required esthetic shape? The simplest method is to specify four points in a 3D space, which form a so-called bilinear patch : Fig. Bilinear patch A bilinear patch is a type of a ruled surface that fully consists of linear segments connecting corresponding points of two curves: Fig.
Ruled surface An MIT professor Steven Coons summarized such representation method for double curves surfaces that are named after him Coons patch : Fig.
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His techniques for boundary curves and blending functions formed the basis for further research in the field. Coons was the first to propose using rational polynoms to model conic sections. Hermite curve is a geometric representation of a cubic curve using end-points and tangent vectors. The shape of a Hermite curve can be controlled by varying vector directions and sizes: Fig. Any point on a polygonal curve always remains inside a convex closure of a set of reference points: Fig.
De Casteljau described them constructively, and the algorithm was named after him. The area for specifying surface parameters is rectangular. The second method uses natural generalization of Bernstein polynomials in the case of two variables.
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Another one is that their algebraic degree increases together with the number of reference points, which greatly complicates numerical calculations. Such curves are called splines ; the first reference to them was made by Isaac Schoenberg, an American mathematician of Romanian origin [Schoenberg ].
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The range of definition of a B-spline parameter is divided into knots, in accordance with the conjunction points of algebraic curves of the given degree. After a couple of errors were corrected, the code worked! It stimulated researchers to look for a way of representing all curves using a single shape. Another step towards discovery was introducing non-uniform B-splines, well-known from scientific literature, into industrial practice. Afterwards, a great deal of efforts was needed to persuade other TIGER developers to start employing the unified representation for all types of curves.
Initially it was intended that the company would provide consulting services in machine-building, but with time SDRC transformed in one of leading global CAD developers.
NLib algorithms are based on a classic monograph [Piegl ], and one of its authors - Wayne Tilleris is the founder and President of GeomWare. NLib is used by more than 85 companies involved in engineering software development. Subdivision Surfaces Subdivision surfaces can be considered as polygonal models that are iteratively constructed from a base mesh; that with each iteration becoming closer to the shape of a modelled surface. The two components of a subdivision surface are the base mesh and the algorithm for smoothing it. Historically, the theory of subdivision surfaces started with the work of an American designer Chaikin, who developed a method of iterative curve construction according to reference points [Chaikin ].
At the next stage, a new sequence of reference points is formed, which is built under special rules based on the first sequence.
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The process continues until the curve is not sufficiently smooth. Chaikin subdivision curve Soon it was proved that the curve generated by the Chaikin algorithm is nothing else but a quadric uniform B-spline. Chaikin method formed the basis for a family of algorithms developed by his followers. One of such algorithms was a method developed by Doo and Sabin for constructing quadric uniform B-spline surfaces using a base quadrangular mesh each facet in such a mesh is a convex quadrangle [Doo ].
Doo - Sabin method is that at each step each facet is replaced with a smaller facet with the same number of apexes.
Each apex of a smaller facet is the arithmetic average of the original apex, the centres of two adjacent edges and the facet centre. As a result, there is an unconnected mesh, where each new apex is then connected to all other apexes, obtained from the same old apex forming new facets. The resulting connected polyhedron is the basis for a new algorithm step. Doo-Sabin subdivision surface Post-graduate students at Utah Univestiy, Catmull and Clark, expanded the corner cutting method to construct uniform cubic B-splines [Catmull ]. Their method, like Doo - Sabin method, can work on base meshes of arbitrary topology locally the resulting surface is similar to a cubic B-spline.
The smoothing algorithm is based on iterative construction of a new mesh under somewhat different rules.