![]() ![]() The theory and numerical scheme presented here are sufficiently general to handle nonconvex patchy colloids with arbitrary surface patterns characterized by a wetting angle, e.g., amphiphilicity. The merits of the tessellation method prove to outweigh those of traditionally used semianalytic approaches, especially when it comes to generality and applicability.ĪB - We present a numerical technique, namely, triangular tessellation, to calculate the free energy associated with the adsorption of a colloidal particle at a flat interface. It is verified that the numerical method presented is accurate and sufficiently stable to be applied to more general situations than presented in this paper. We ignore interfacial deformation due to capillary, electrostatic, or gravitational forces, but the method can be extended to take such effects into account. N2 - We present a numerical technique, namely, triangular tessellation, to calculate the free energy associated with the adsorption of a colloidal particle at a flat interface. T2 - Towards nonconvex patterned colloids The barycentric coordinates are given by the built-in variable gl_TessCoord.T1 - Triangular tessellation scheme for the adsorption free energy at the liquid-liquid interface ![]() \[\begin$) of the PN Triangle, while the tessellation evaluation shader will compute the positions ($b(u,v)$) and normals ($n(u,v)$) of the final tessellated geometry. The displacement field $b(u,v)$ of a PN Triangle is defined as follows: This is the stage where the new vertices can be assigned their own attributes, using information provided by the tessellator and/or varyings from previous stages.ĭisplacement field Let a triangle be defined by 3 vertices $V_i = (P_i, N_i)$, where $P_i$ is the position and $N_i$ the normalized normal of the ith vertex. The tessellation evaluation stage begins once the hardware tessellator has finished subdividing a patch, and has to be programmed using GLSL's tessellation evaluation shading language. This means that per patch varyings are evaluated several times, so heavy computations should be avoided or limited if possible (see the code of the shaders I present for an example). It is important to note that when using the programmable approach, the shader is executed for each vertex of the input patch (clearly for performance reasons, as this model allows to evaluate the data in parallel). Its main goal is to provide subdivision levels to a tessellator (implemented in specific hardware in OpenGL4 GPUs) which performs the actual subdivision later in the pipeline. The tessellation control stage can be programmed using GLSL's tessellation control shading language, or replaced by calls to the glPatchParameter function. It has two additional programmable processing stages: the tessellation control stage and the tessellation evaluation stage. OpenGL4 adds a new pipeline for geometry processing which is specifically intended for patch primitives. That's something I'll probably detail in another article :-).īrief overview of tessellation in OpenGL4 This is why minification scenarios should be avoided when rendering with GPU rasterizers : they generate aliased images at sub optimal speed. ( *) GPUs are also very inefficient when it comes to synthesizing polygons which have a size lower than a few pixels, also called micropolygons. Both of these algorithms are easy to implement and can be used transparently with "conventionnal" triangle meshes. Today I'll introduce PN triangles and Phong tessellation, algorithms which try to solve this matter, and can be implemented in hardware using GLSL's tessellation control and evaluation shaders available with OpenGL4. On modern hardware, subdivision surfaces - called patches in OpenGL4 - can limit the magnification problem to a certain extent. Although triangle meshes can reproduce any kind of surface, GPUs can only render them efficiently for a single scale/viewing distance: we get lack of detail above this scale, and aliasing below( *). This is still mostly the case today, especially for video games. For a long time, GPUs could only synthesize triangles, and since there was no other way of generating images in real-time, performance critical applications used triangle meshes to describe their objects and render them. ![]() Z-Buffering is the only rendering algorithm which is hardware accelerated by a widely available and affordable component: the GPU. ![]()
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