曲面の1階微分量である法線ベクトルは 曲面間交線計算 NC工具の方向決定 組立て性の判定 オフセット曲面の計算等 様々な応用に直接結び付いている. 一方 自由曲線や自由曲面の表現としては ベジエ曲線およびベジエ曲面が その優れた性質から広く用いられている. このベジエ曲線ないし曲面の場合には 1階微分量である接ベクトルの軌跡 すなわちホドグラフが それぞれベジエ曲線ないしベジエ曲面として表現できることが知られている. つまり 接ベクトルに関しては ベジエ制御点網によって非常に厳しい存在領域が与えられる. しかし 曲面の法線ベクトルについては それほど厳しい存在域を示すことができず 従来は円錐 4角錐 6角錐等で包絡するにとどまっていた. 本研究では n×m次のテンソル積ベジエ曲面の非正規化法線ベクトルが(2n-1)×(2m-1)次のテンソル積ベジエ曲面 n次の三角形ベジエ曲面の非正規化法線ベクトルが(2n-2)次の三角形ベジエ曲面 としてそれぞれ表現できることを明らかにし これら法線ベクトルのベジエ制御点の計算式を示す. 最後に法線ベクトルのベジエ曲面をもとにして 法線ベクトルのより厳しい存在域計算方法を提案する.

Normal vectors of a surface are directly related to a lot of applications, such as surface-surface intersection, tool orientation for NC machining, assemblability of components, offset surface generation, and so on. Bezier curves and surfaces are commonly used by many applications because of their good properties. It is well known that hodographs of Bezier curves and surfaces, which are loci of tangent vectors, can be represented as Bezier curves and surfaces respectively. Therefore, tight bounds of tangent vectors of Bezier curves and surfaces can be obtained. However, bounds of normal vectors are usually approximated by cones, rectangular pyramids and hexagonal pyramids, because it is impossible to calculate such tight bounds of normal vectors of Bezier curves and surfaces as those of tangent vectors. This paper presents that unnormalized normal vectors of a tensor product Bezier surface of degree n×m can be represented as a tensor product Bezier surface of degree (2n-1)×(2m-1) and also that unnormalized normal vectors of a triangular Bezier surface of degree n can be represented as a triangular Bezier surface of degree 2n-2. The equations for calculating the control points of those Bezier normal vector surfaces are also presented. Furthermore, the method for determining tight bounds of normal vectors based on the Bezier normal vector surfaces is proposed.