C~1 Blending Interpolation And Blending Surfaces For Quadratic Function

In the theory of surface modeling, blending surfaces is a fundamental problem, that is to find a C¹ blending of the surfaces with lower degree for some given surfaces. In fact this problem is spline with boundary by viewpoint of spline function,it isn't convenient that we construct piecewise surfaces by spline function.We try to find blending surfaces directly by the theory of smooth cofactor.This is motive of the paper.We only study the problem of blending surfaces quadratic function in practice.This dissertation is organized as follows:Chapter 1:We introduce the past and the present in the problem of blending surfaces.We introduce the basis theory of multi-polynomial interpolation and multi-polynomial spline.Chapter 2:We study C¹ blending surfaces on the corner and triangle.At first,given a corner and two related quadratic functions (located at different sides of it) that are C¹ continous on the corner, we present the existence of C¹ blending of the two surfaces with another two quadratic functions and the necessaryand sufficient condition of the existence of C¹ blending of the two surfaces with another quadratic function with the help of the theory of smooth cofactor. We gain the following theorem:Theorem 1 Suppose two lines l_i=α_i+ y+γ_i,i=1,2, intersect at P = (x_*,y_*),andα₁≠α₂.given two related quadratic Junctions g₁,g₂ located at two sides of∠P, their values and first derivatives are equivalent at point P,that is,g₁,g₂ are C¹ continuous.then exist quadratic function f is C¹ continuous with g₁,g₂ along l₁,l₂,respectively if and only if g₁,g₂,l₁,l₂ satisfy(α₁+α₂)α₁₁¹²=α₂₀¹²+α₁α₂α₀₂¹²whereα₂₀¹²,α₁₁¹²,α₀₂¹² are the coefficients of term x²,xy,y² of g₁-g₂,respectively.Theorem 2 Suppose two lines l_i=α_i+y+γ_i,i=1,2, intersect at P=(x_*,y_*),andα₁≠α₂.given two related quadratic functions g₁,g₂ located at two sides of Z.P, their values and first derivatives are equivalent at point P,that is,g₁,g₂ are C¹ continuous.for arbitrary line l₃ cross point P,l₃=α₃x+y+γ₃,α₃≠α₁,α₃≠α₂,then exist quadratic function f₁,f₂ is C¹ continuouswith g₁,g₂ along l₁,l₂,respectively, f₁,f₂ is C¹ continuous along l₃.For open side and the related quadratic functions that are C¹ continuous, we can construct piecewise quadratic functions that blend C¹ smoothly with them with the above method, and the number of the functions are the smallest.Moreover, with multivariate interpolation theory, we proved that for 3 given quadratic functions that are C¹ continuous on consecutive vertexes, then there exists a quadratic functions that blends smoothly with the given quadratic functionsrespectively. Therefore, we have the following theorem:Theorem 3 Givenα△P₁₂P₂₃P₃₁ with three sides l_i,i=1,2,3.If we define 3 quadratic functions g_i,i=1,2,3 along l_i,i=1,2,3 outside the triangle respectively, and we assume that g_i,g_j are C¹ continuous at P_ij, then there existsαquadratic function f that blends smoothly with g_i,i=1,2,3 along l_i,i=1,2,3 respectively.Chapter 3: The blending of quadratic functions on a polygonal domain.Upon the results of Chapter 2, for given quadratic functions defining out of a polygonal domain satisfying that they are continuous at consecutive vertexes, we will propose the method for constructing piecewise quadratic functions that blend smoothly with the given surfaces. Furthermore, the number of the functions is small to the full.First, we will discuss the blending of quadratic functions on a quadrangulardomain. Suppose that the equations for the four sides of the quadrangleP ₁₂P₂₃P₃₄P₄₁ arel_i=α_ix+y+γ_i,i=1,2,3,4.Obviously,α_i-α_j≠0,(i,j)=(1,2), (2,3),(3,4),(4,1).We always assume that the quadrangle P₁₂P₂₃P₃₄P₄₁ is convex. Given quadratic functions g_i,i=1,2,3,4, along l_i out of P₁₂P₂₃P₃₄P₄₁, and they are C¹ continuous at the consecutive vertexes, that isHence,g_i(x,y)-g_j(x,y) can be written asg_i(x,y)-g_j(x,y)=α₂₀^ij(x-x_ij)²+2α₁₁^ij(x-x_ij)(y-y_ij+α₀₂^ij(y-y_ij)²,(i,j) = (1,2),(2,3),(3,4),(4,1).The assumption above will be called Assumption 1, Theorem 4 to 6 are all discussedunder it.Theorem 4 Assumption 1 is satisfied by g_i,l_i who also satisfy(α₁+α₂)α₁₁¹²=α₂₀¹²+α₁α₂α₀₂¹². To do the subdivision: connect P₂₃P₄₁ leads to line l₅, then there exist quadratic functions f₁,f₂ such that f₁ blends C¹ smoothly with g₁,g₂ along l₁,_l2 respectively,f₂ blends C¹ smoothly with g₃,g₄ along l₃,l₄ respectively, and f₁,f₂ blends smoothly along l₅.Theorem 5 Assumption 1 is satisfied by g_i,l_i who also satisfyThen there exist a quadratic function f that blends C¹ smoothly with g₁,g₂,g₃,g₄ along l₁,l₂,l₃,l₄ respectively.Theorem 6 Assumption 1 is satisfied by g_i,l_i. To do the subdivision: connectP₁₂P₃₄ and P₂₃P₄₁ lead to lines l₅,l₆ and intersection point O. Then there exist quadratic functions fi,i = 1,2,3,4 such that f_i blends C¹ smoothly with g_i along l_i respectively, and f₁ blends smoothly with f₂,f₄ along l₅,l₆ and f₃ blends smoothly with f₂,f₄ along l₆,l₅.Next, we will discuss the blending of quadratic functions on a pentagonal domain. Let the equations of the sides of pentagon P₁₂P₂₃P₃₄P₄₅P₅₁ arel_i=α_ix+y+γ_i,i=1,2,3,4,5 Obviously,α_i-α_j≠0,(i,j) = (1,2),(2,3),(3,4),(4,5),(5,1).We always assume that P₁₂P₂₃P₃₄P₄₅P₅₁ are convex. Given quadratic functions g_i,i=1,2,3,4,5 along l_i out of P₁₂P₂₃P₃₄P₄₅P₅₁, and suppose that their values and the values of their derivatives of order 1 at consecutive vertexes are equal to each other, that isHence,g_i(x, y)-g_j(x, y) can be written asg_i(x, y)-g_j(x, y) =α₂₀^ij(x-x_ij)²+2α₁₁^ij(x-x_ij)(y-y_ij)+α₀₂^ij(y-y_ij)²,(i,j) = (1,2),(2,3),(3,4),(4,5),(5,1).We call this assumption Assumption 2, Theorem 7,8,9, are all discussed under it.Theorem 7 Assumption 2 is satisfied by g_i,l_i who also satisfyTo do the subdivision: connect P₂₃P₅₁ and P₂₃P₄₅ leads to lines l₆,l₇, then there exist quadratic functions f₁,f₂,f₃ such that f₁ blends C¹ smoothly with g₁,g₂ along l₁,l₂ respectively, f₃ blends C¹ smoothly with g₃,g₄ along l₃,l₄ respectively, f₂ blends C¹ smoothly with g₅ along l₅, and f₁,f₂ blends C¹ smoothly along l₆ and f₂,f₃ blends C¹ smoothly along l₇.Theorem 8 Assumption 2 is satisfied by g_i,l_i who also satisfy To do the subdivision: connect P₂₃P₅₁ leads to line l₆, then there exist quadratic functions f₁,f₂ such that f₁ blends C¹ smoothly with g₁,g₂ along l₁,l₂ respectively, f₂ blends C¹ smoothly with g₃,g₄,g₅ along l₃,l₄,l₅ respectively, and f₁,f₂ blends C¹ smoothly along l₆.Theorem 9 Assumption 2 is satisfied by g_i,l_i who also satisfyTo do the subdivision: connect P₂₃P₅₁,P₂₃P₄₅ and P₃₄P₅₁ leads to lines l₆,l₇,l₈, then there exist quadratic functions f₁,f₂,f₃,f₄,f₅ such that f₁ blends C¹ smoothly with g₁,g₂ along l₁,l₂ respectively, f₃,f₄,f₅ blends C¹ smoothly with g₃,g₄,g₅ along l₃,l₄,l₅ respectively, f₁,f₂ blends C¹ smoothly along l₆,f₂,f₃ blends C¹ smoothly along l₇,f₃,f₄ blends C¹ smoothly along l₈, f₄,f₅ blends C¹ smoothly along l₇, f₂,f₅ blends C¹ smoothly along l₈.