boundary representation

Boundary Representation Models: Validity and Rectification

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• Nicholas M. Patrikalakis 3 ,
• Takis Sakkalis 3 &
• Guoling Shen 3  

Model validity, especially of manifold boundary representation (B-rep) models, has long been recognized as an important problem. This paper reviews issues on model validity of existing models. In particular, we present a set of sufficient conditions for representational validity of a typical B-rep data structure; we propose a rectify-by-reconstruction approach to the B-rep model rectification problem, and present results on the inherent complexity of the corresponding boundary reconstruction problem. Further, we develop the concept of an interval solid model, associated with a solid, for achieving numerical robustness. Finally, we present a set of sufficient conditions so that an interval solid model is “approximately equal” to its associated solid.

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Nicholas M. Patrikalakis, Takis Sakkalis & Guoling Shen

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Department of Engineering, University of Cambridge, Cambridge, CB2 1PZ, UK

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Ralph Martin MA, PhD, FIMA, CMath, MBCS, CEng

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Patrikalakis, N.M., Sakkalis, T., Shen, G. (2000). Boundary Representation Models: Validity and Rectification. In: Cipolla, R., Martin, R. (eds) The Mathematics of Surfaces IX. Springer, London. https://doi.org/10.1007/978-1-4471-0495-7_23

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What is Boundary Representation (B-Rep)

• Post author: Sandeep Verma
• Reading time: 2 mins read

B-rep or boundary representation is a rendering technique for solid modeling. It is a popular approach to create a solid model of a physical object. Brep is that a three-dimensional object model is enclosed by surfaces or faces and has its own interior and exterior. It describes the shape as a collection of surfaces which separate its interior from the external environment. It is suitable for complex designs, Polygon facets are one of the examples of boundary representation.

These are the following primitives of B-rep:- 1) Vertices: It is a point where two or more edges meet with another. 2) Edges:  It is a line or curve enclosed between two vertices. 3) Faces: It is a surface or plane of the solid. 4) Loop: It is a hole in a face. 5) Genus: it is through a hole in a solid.

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Meshes vs B-Reps Explained!

Hi, my name is Edwin. I'm a Grasshopper expert at ShapeDiver and in this blog post I will show the difference between standard meshes and b-reps and how this can impact the performance and rendering of your Grasshopper definitions on ShapeDiver. Let's get going!

It's All About Performance

Grasshopper  represents  geometry  in two ways: using  b-reps  or using  meshes . This is something the majority of Grasshopper developers already know, but a deeper understanding of these concepts can help clarify how  they  can  affect  the  performance  of Grasshopper definitions when they  run  on  ShapeDiver .

This performance is especially  crucial  for any ShapeDiver  model  that will be  accessed  or interacted with via a  web browser , where the available internet connection or even the device's own hardware can play a significant role on the UX. At ShapeDiver we  recommend our users to create definitions that typically  load  in  under   5 seconds  as it's proven that the average B2C end user won't wait for longer than this time before heading elsewhere.

In this post I will discuss the  difference  between  b-reps  and  meshes  and I will give you some  tips  and tricks that will make your definitions  load  in the  least   possible   time .

What Is A Mesh?

A  mesh  is the native geometry  representation  your  graphics card  (GPU) needs in order to  display   objects  and it consists of a  collection of vertices  (points) and  faces  (typically triangles and rectangles).

The  more   vertices  and  faces  a mesh has, the  more   time  will be required to  transfer  the mesh to the end  device  and the  more GPU  resources will be used. This amount of vertices and faces is what we need to keep the lowest possible in order to have a definition with a great performance.

What Is A B-Rep?

On the other hand, a  b-rep (or brep) is the short name given to  "Boundary Representation" . It consists of representing  surfaces  through  mathematical equations  and a set of limits in the 3D space. B-Reps are both  more compact  in their definition while retaining  more structural  information about the shape. Therefore, one can define  complex operations  on them  easily  (such as boolean differences or piping) which is challenging to do with meshes.

However,  b-reps need to be converted to meshes  in order to be rendered by a GPU. During this operation, any performance benefit from using b-rep can be cancelled by the meshing algorithm, if it isn't done right. To get precise information on any given b-rep, type "what" in the Rhino command line. This will open a window describing the object you have selected. At the bottom of this window you'll be able to see the size of the mesh that is being rendered to display this object.

What Happens When Displaying B-Reps In ShapeDiver's Viewer?

All  b-reps  output to  ShapeDiver's  viewer are eventually  converted  to  meshes . We try to make this conversion as efficient as possible. However, the  automatic conversion  potentially comes with  two issues : it will  increase  the online  loading times  of the models in a non predictable way, and it can produce  visual artefacts  since the amount of vertices and faces of the meshes can't be controlled. For example, whenever a b-rep contains several faces grouped with hard edges, those faces might be meshed independently and might not appear watertight in the online viewer.

How Can I Handle B-Reps Then?

Optimal performance  on ShapeDiver can be achieved by  balancing  b-rep and mesh operations,  keeping  the b-rep representation  as long as necessary  to perform complex operations in the most efficient way but  switching  to meshes  whenever it makes sense , and definitely  before  outputting  geometry  to the viewer using one of the ShapeDiver display components.

Keep in mind that the  meshing  steps of your definition  increase  the  computation   time , but at least it is a time that  you   can  predictably  control , as opposed to letting our servers do the meshing themselves and possibly output heavy meshes to the viewer.  One   should  always  think  of how relevant and how big each piece of geometry is in the definition, and  optimize their mesh size as much as possible according to it.

For example, let's say we are creating a  house  and we need to have the  3D  model of the  door knobs . The most probably situation is that these door knobs will be  barely seen  in the entire definition as they are a very  small detail . In this case, using a  perfect sphere  to represent the knobs is  not worth it . Instead, improve your definitions by using a sphere with very few polygons. But...

How Does One Convert B-Reps Into Meshes?

The most efficient way is to  use primitive shapes  since Grasshopper already offers direct solutions to create meshes in which you can clearly and exactly define the amount of polygons that you wish to use.

However, in most cases, your geometry will be more complex than a primitive shape, which means you will need to use the  component " MeshBrep "  that offers different settings solutions to play with the amount of faces and vertices your mesh has.

The simplest setting component is called " Settings(Speed) " and will try to make your  mesh as efficient  as possible. However, sometimes it is better to use the " Settings(Custom) " as this one has a variety of options to  optimize  your mesh even more.

Other  components  which need meshes as inputs will also  convert  your b-reps  directly  into meshes, such as the  Texture Mapping  components in the  Human  plugin or our  owncomponent   "ShapeDiverTextureTransform"  (I talked about this tool in my previous blog post), but it is  not recommended  to convert your b-reps in this way as you  won't have  any  control over this conversion.

TIP: If you want to visually check how the wireframe (faces and vertices) of your mesh is, type ctrl+m in Grasshopper, as usually Grasshopper hides the wireframe by default.

* Vertices and faces count difference when converting to mesh directly vs using mesh settings with the "MeshBrep" component.

How Can You Test The Performance Difference Between B-Reps and Meshes?

I have created  two performance test  models on ShapeDiver.  The first one  was created using  meshes  while  the second one  was created using  b-reps . If you push the definitions by drastically changing the settings from the minimum values to the maximum ones, you can  notice  how the  b-rep  definition (the second one)  takes  almost  double the time  to load in comparison to the mesh definition. Sometimes, depending on your device, the b-rep definition  might even crash the browser .

How Can One Be Even More Efficient?

If you have any knowledge using  C# Scripting  or other programming languages available in Grasshopper, you can  create  very  efficient  definitions by  coding  exactly the way a mesh will be constructed from scratch. You can also use different components inside the mesh tab in Grasshopper.

As a demonstration of using the components just mentioned, I would like to share a  cluster that  we  have  created  which allows you to do something as simple as  extruding , but it outputs a  mesh  that can be easily  optimized  using the tolerance parameter.

This example outputs the  same geometry  than the one in the  "MeshBrep"  component example, but as you can see, we could bring the  face count  to  half  of what was obtained in the  best case  with  "MeshBrep" .

Conclusions

The  performance/loading times  of Grasshopper definitions are  not always  the main  preoccupation  of a Grasshopper  designer , but since  ShapeDiver.com  brings the possibility to  share  those definitions  online , we as Grasshopper developers have to  make sure  that  everyone  can have a pleasant  experience  by making the loading  times  as  fast  as possible in the majority of  devices .

I hope that knowing the difference between b-reps and meshes will let you  improve  the  performance  of your present and future projects, but this is  just the beginning . If you want to  learn  other ways of making your definitions as smooth as possible,  visit  our  Forum  and stay tuned for future blog posts.

Was this tutorial helpful? Do you have any comments or feedback? Make sure to visit our  Forum  and start a thread with your doubts! We're always checking for new topics from our users so we'll make sure to give you useful answers.