This paper presents a scale method for developing high dimensional scale functions to blend implicitly defined objects. Scale functions are differentiable on the entire domain except the origin, provide blending range control, and behave like Min/Max operators everywhere, so even a successive composition of blending operations containing overlapped blending regions can be generated smoothly. Because the scale method is a generalized method, implicit or parametric curves, such as cubic Bezier curves, rational conic curves, and implicit conics and hyper‐ellipsoids, can be used to develop scale functions. As a result, it can enhance the flexibility of generating the implicitly blending surfaces in Ricci's constructive geometry, soft objects modeling, and implicit sweep objects. ACM CSS: I.3.5 Computer Graphics—Computational Geometry and Object Modeling ‐ Curve, surface, solid and object representations

No abstract

Abstract-This paper proposes spherical product functions to define implicit spherical product surfaces. A spherical product function is composed of a contour function and a profile function and its iso-surface's shape is generated by modulating and translating the iso-curve of the contour function through the points on the iso-curve of the profile function. This paper also shows if contour and profile functions are ray-linear, then an implicit spherical product surface can be parameterized and hence have both the advantages of implicit and parametric surfaces. Moreover, this paper proposes ray-linear two-branch and one-branch linear and super-hyperbolic functions that can be used to construct new contour and profile functions with asymmetric or symmetric iso-curves. Based on them, an implicit spherical product surface can has asymmetric or symmetric contour and profile and also has a parametric representation.Index Terms-Implicit surface, blending operations, parametric surface.

In soft object modeling, in order to make the shapes of primitive soft objects more diverse, this paper proposes ray-scaling operations to deform a primitive soft object. The proposed ray-scaling operations perform scaling via a given scaling object. But unlike scaling of affine transform which gives the same scaling factor to all the points, ray-scaling operations give each point in different directions an individual and a different scaling factor obtained from the ray-scaling object. As a result, by choosing suitable scaling objects, ray-scaling operations are able to erode or dilate a soft object locally with many special deformation effects. This paper also applies super-ellipsoids, super-quadrics and generalized distance functions to define a scaling object, and then successfully deforms a sphere into a six-balled, eight-balled or ten-balled object by using scaling objects of polyhedron, each of whose facets can deform to be a ball.