Research On Geometric Modeling Of Weighted Lupa(?) Q-B(?)zier Curves

Parametric curve and surface modeling acts an important role in Computer Aided Geometric Design. This paper is based on the theory of curve modeling, including two parts below:Applying Mobius transformation to weighted Lupa(?) q-B(?)zier curve of degree n, those weights can be modified whereas the degree, these control points, the parametric domain and the shape of weighted Lupa(?) q-B(?)zier curve remain unchanged. Weighted Lupa(?) q-B(?)zier curve can be reparameterized by Mobius transformation such that changing the corresponding relationship between the point in the parameter domain and the point on the curve. What changes is the way of tracing the curve. Furthermore, this paper extends the shape invariant factor of quadratic weighted Lupa(?) q-B(?)zier curve to the weighted Lupa(?) q-B(?)zier curve of degree of n. Weight is the only shape invariant factor determined. It turns out the result that weighted Lupa(?) q-B(?)zier curve of degree n has n-1 shape invariant factors by Mobius transformation. Applying Mobius transformation yields tight the bounds on derivativess as well.This paper is focused on shape modification of weighted Lupa(?) q-B(?)zier curve. It presents two aspects:the one is quantitative correction of weighted Lupa(?) q-B(?)zier curve along the direction, which divide into quantitative correction along a fixed direction and quantitative correction along an arbitrary direcion. A point on a weighted Lupa(?) q-B(?)zier curve moves along a certain distance to another point, where the curve of new weight passed. Then it gets the formula of new weight. What quantitative correction along an arbitrary direcion in line with the actual is a general situation. Using twice the formula above. What the another is shape modification of weighted Lupa(?) q-B(?)zier curve based on single point constraint is a geometric constraint as for the one. It is a curve that passing through a known point. In the case of the minimun deformation of curve, which is passed through a target point. The known point and the target point correspond to the same value. Three ways address the problem above:(1)move control points; (2)change the weights; (3)move control points and change the weights. We suppose the perturbations of the control points (weights), find the relationship between the known point and target point, presents the lagrange multiplier method. At last, a new curve passed through the target point is generated. Practical examples are also given.