The ability to directly manipulate an embedded object in the free-form deformation (FFD) method improves controllability. However, the existing solution to this problem involves a pseudo-inverse matrix that requires complicated calculations. This paper solves the problem using a constrained optimization method. We derive the explicit solutions for deforming an object which is to pass through a given target point. For constraints with multiple target points, the proposed solution also involves simple calculations, only requiring solving a system of linear equations.We show that the direct manipulations exhibit the commutative group property, namely commutative, associative, and invertible properties, which further enhance the controllability of FFD. In addition, we show that multiple point constraints can be decomposed into separate manipulations of single point constraints, thus providing the user the freedom of specifying the constraints in any appropriate order.
In this paper, two explicit conversion formulae between triangular and rectangular Bézier patches are derived. Using the formulae, one triangular Bézier patch of degree n can be converted into one rectangular Bézier patch of degree $n \times n$. And one rectangular Bézier patch of degree m × n can be converted into two triangular Bézier patches of degree $m + n$ . Besides, two stable recursive algorithms corresponding to the two conversion formulae are given. Using the algorithms, when converting triangular Bézier patches to rectangular Bézier patches, we can computer the relations between the control points of the two types of patches for any $n\ge2$ based on the relationships for $n=1$. When converting rectangular Bézier patches to triangular Bézier patches, we can computer the relations between the control points of the two types of patches for any $m\ge 2, n \subset N^+ $ and $n \ge 2, m \subset N^+$ based on the relationships for $m=n=1$.
NURBS surfaces are among the most commonly used parametric surfaces in CAGD and Computer Graphics. This paper investigates shape modification of NURBS surfaces with geometric constraints, such as point, normal vector, and curve constraints. Two new methods are presented by constrained optimization and energy minimization. The former is based on minimizing changes in control net of surfaces, whereas the latter is based on strain energy minimization. By these two methods, we change control points and weights of an original surface, such that the modified surface satisfies the given constraints. Comparison results and practical examples are also given.
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Lee H, Han S. Reconstruction of 3D interacting solids of revolution from 2D orthographic views[J]. Computer-aided Design, 2005, 37(13): 1388-1398.
Ming Li · Frank C Langbein · Ralph R Martin. Detecting design intent in approximate CAD models using symmetry. 2010.
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Adem Cicek · Mahmut Gulesin. Reconstruction of 3D models from 2D orthographic views using solid extrusion and revolution. 2004.
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Zhang S, Shi Y, Fan H, et al. Serial 3D model reconstruction for machining evolution of rotational parts by merging semantic and graphic process planning information[J]. Computer-aided Design, 2010, 42(9): 781-794.
Zhe Wang · Mohammed Latif. Reconstruction of a 3D solid model from orthographic projections. 2003.
This paper presents a new approach for reconstructing solids with planar, quadric and toroidal surfaces from three-view engineering drawings. By applying geometric theory to 3-D reconstruction, our method is able to remove restrictions placed on the axes of curved surfaces by existing methods. The main feature of our algorithm is that it combines the geometric properties of conics with affine properties to recover a wider range of 3-D edges. First, the algorithm determines the type of each 3-D candidate conic edge based on its projections in three orthographic views, and then generates that candidate edge using the conjugate diameter method. This step produces a wire-frame model that contains all candidate vertices and candidate edges. Next, a maximum turning angle method is developed to find all the candidate faces in the wire-frame model. Finally, a general and efficient searching technique is proposed for finding valid solids from the candidate faces; the technique greatly reduces the searching space and the backtracking incidents. Several examples are given to demonstrate the efficiency and capability of the proposed algorithm.