• Honam Mathematical Journal
• /
• v.45 no.2
• /
• pp.269-284
• /
• 2023

In this paper we study generalizations of the concept of pure-direct-projectivity from module categories to Grothendieck categories. We examine for which categories or under what conditions pure-direct-projective objects are direct-projective, quasi-projective, pure-projective, projective and flat. We investigate classes all of whose objects are pure-direct-projective. We give applications of some of the results to comodule categories.

• Journal of the HCI Society of Korea
• /
• v.1 no.1
• /
• pp.37-44
• /
• 2006

Specular reflections provide the visual feedback that describes the material type of an object, its local shape, and lighting environment. In photorealistic rendering, there have been a number of research available to render specular reflections effectively based on a local reflection model. In traditional cel animations and cartoons, specular reflections plays important role in representing artistic intentions for an object and its related environment reflections, so the shapes of highlights are quite stylistic. In this paper, we present a method to render and control stylized specular reflections using projective textures based on principal curvature analysis. Specifying a texture as a pattern of a highlight and projecting the texture on the specular region of a given 3D model, we can obtain a stylized representation of specular reflections. For a given polygonal model, a view point, and a light source, we first find the maximum specular intensity point, and then locate the texture projector along the line parallel to the normal vector and passing through the point. The orientation of the projector is determined by the principal directions at the point. Finally, the size of the projection frustum is determined by the principal curvatures corresponding to the principal directions. The proposed method can control the position, orientation, and size of the specular reflection efficiently by translating the projector along the principal directions, rotating the projector about the normal vector, and scaling the principal curvatures, respectively. The method is be applicable to real-time applications such as cartoon style 3D games. We implement the method by Microsoft DirectX 9.0c SDK and programmable vertex/pixel shaders on Nvidia GeForce FX 7800 graphics subsystems. According to our experimental results, we can render and control the stylized specular reflections for a 3D model of several ten thousands of triangles in real-time.