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insert a new control point into a curve without changing the shape of the curve. This is useful to allow a user to adjust this new control point, as opposed to only being able to adjust the existing control points. However, because the control grid of a B-Spline or NURBS surface has to be rectangular, it is only possible to insert an entire row or column of new control points.
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theory, do everything that NURBS can do. In practice, an enormous amount of programming was required to make NURBS work as well as they do, and creating the equivalent T-spline functionality would require similar effort. To smoothly join at points where more than three surface pieces meet, T-splines have been combined with
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B-Splines are a type of curve widely used in CAD modeling. They consist of a list of control points (a list of (X, Y) or (X, Y, Z) coordinates) and a knot vector (a list increasing numbers, usually between 0 and 1). In order to perfectly represent circles and other conic sections, a weight component
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To represent a three-dimensional solid object, or a patch of one, B-Spline or NURBS curves are extended to surfaces. These surfaces consist of a rectangular grid of control points, called a control grid or control net, and two knot vectors, commonly called U and V. During editing, it is possible to
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surfaces. They allow control points to be added to the control grid without inserting an entire new row or column. Instead, the new control points can terminate a row or column, which creates a "T" shape in the otherwise rectangular control grid. This is accomplished by assigning a knot vector to
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surfaces and make pieces easier to merge, but increases the book-keeping effort to keep track of the irregular connectivity. T-splines can be converted into NURBS surfaces, by knot insertion, and NURBS can be represented as T-splines without T's or by removing knots. T-splines can therefore, in
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M.A. Scott and R.N. Simpson and J.A. Evans and S. Lipton and S.P.A. Bordas and T.J.R. Hughes and T.W. Sederberg, Isogeometric boundary element analysis using unstructured T-splines, Computer
Methods in Applied Mechanics and Engineering, 2013 254. p
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surfaces of any connectivity and topology, such as holes, branches, and handles. However, none of T-splines, subdivision surfaces, or NURBS surfaces can always accurately represent the (exact, algebraic)
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Thomas W. Sederberg, Jianmin Zheng, Tom Lyche, David Cardon, G. Thomas
Finnigan, Nicholas North: T-Splines Simplification and Local Refinment, from ACM Trans. Graph. (SIGGraph 2004)
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are alternative technologies. Subdivision surfaces, as well as T-spline and NURBS surfaces with the addition of geometrically continuous constructions, can represent
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granted patent number 7,274,364 for technologies related to T-Splines. T-Splines, Inc. was founded in 2004 to commercialize the technologies and acquired by
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where a row of control points is allowed to terminate without traversing the entire surface. The control net at a terminated row resembles the letter "T".
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of two surfaces within the same surface representation. Polygon meshes can represent exact intersections but lack the shape quality required in
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G. Westgaard, H Nowacki, Construction of fair surfaces over irregular meshes, Symposium on Solid
Modeling and Applications 2001: 88-98
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Thomas W. Sederberg, Jianmin Zheng, Almaz
Bakenov, Ahmad Nasri: T-Splines and T-NURCCS, from ACM Trans. Graph. (SIGGRAPH 2003)
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J. Fan, J Peters, On Smooth
Bicubic Surfaces from Quad Meshes, ISVC 2008, see also: Computer Aided Design 2011, 43(2): 180-187
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Thomas W. Sederberg, Jianmin Zheng, Almaz
Bakenov, Ahmad Nasri: T-Splines and T-NURCCS, from ACM Trans. Graph. (SIGGRAPH 2003)
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164:'s variant of the subdivision surfaces has the advantage of edge weights. T-splines do not yet have edge weights.
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J Peters, Biquartic C^1 spline surfaces over irregular meshes, Computer Aided Design 1995 27 (12) p 895--903
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each individual control point, and creating some rules around how control points are added or removed.
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constructions of degree 3 by 3 (bi-cubic) and, more recently, of degree 4 by 4 (bi-quartic).
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Modeling surfaces with T-splines can reduce the number of control points in comparison to
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is often added, which extends B-Splines to rational B-Splines, commonly called
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113:. A NURBS curve represents a 1D perfectly smooth curve in 2D or 3D space.
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Reconsideration of T-spline data models and their exchanges using STEP
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The T-spline patent, US patent 7,274,364, expired in 2024.
298:"Autodesk Acquires T-Splines Modeling Technology Assets"
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Method of defining complex surfaces in computer graphics
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Transitioning from NURBS to T-splines (67-minute video)
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T-splines were initially defined in 2003. In 2007 the
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may be too technical for most readers to understand
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69:Learn how and when to remove this message
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51:make it understandable to non-experts
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188:Technical articles about T-splines
97:. A T-spline surface is a type of
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198:NURBS and CAD: 30 Years Together
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