Solving geometric constraints by homotopy

Geometric Constraint Solving Based on Connectivity of Graph

“…If the current node N is a leaf in the tree then it is a well-constrained problem that cannot be decomposed further. Solve N with numerical computation methods [12,21,27]. …”

“…There are four major approaches to geometric constraint solving: the numerical approach [12,21,27], the symbolic computation approach [7,19], the rule-based approach [2,6,20,29] and the graph-based approach [3,4,14,16,23,24,26]. This paper will focus on using graph algorithms to decompose a large constraint problem into smaller ones.…”

A robust and efficient method for solving point distance problems by homotopy

“…Subdivision methods [15,6] provide all the solutions but the number of boxes to be explored can be huge. In algebraic approaches, homotopy methods have been successfully studied in this area [3,11] but only for small size problems. Indeed, the number of homotopy paths to follow grows exponentially with the number of constraints.…”

“…Indeed, the number of homotopy paths to follow grows exponentially with the number of constraints. In [11] it is proposed to use the sketch to define a parameter-homotopy. A sole path is followed but a sole solution is obtained.…”

Morphogenesis-Based Deformable Models Application to 3D Medical Image Segmentation and Analysis

“…Application to the Elbow Joint Metaballs paradigm was chosen here to build our model by making a gaussian sum of implicit functions that represent our ossification centers like local morphogenetic fields controlled by a few parameters.The articulation to model is composed of three bones epiphyses : the distal region of humerus, the radius head, and the ulna. An homotopy process [4] for mesh constructing implies a progressive modelling from an initial state where each element is collapsed into a simple shape (sphere) with a known mesh for each of the three bone extremities. Then the parameters variations imply a simulated morphogenesis until an adult final stage.…”