blocks|key|1076983|text|如果立方体是由8个唯一顶点定义的，那么法线可能会与每条边形成135度的角度，正如您所提到的。|type|unstyled|depth|inlineStyleRanges|entityRanges|data|1076984|然而，通常使用24个顶点来定义立方体正是出于这个原因。这允许您通过在每个角点处“复制”顶点来获得垂直于每个面的顶点法线。以这种方式定义一个立方体，实际上就是定义6个单独的面，每个面都适当地向外指向。|1076985|entityMap^0|0|0^^$0|@$1|2|3|4|5|6|7|F|8|@]|9|@]|A|$]]|$1|B|3|C|5|6|7|G|8|@]|9|@]|A|$]]|$1|D|3|-4|5|6|7|H|8|@]|9|@]|A|$]]]|E|$]]

If the cube is defined by 8 unique vertices, then the normals will likely be making a 135 degree angle to each edge, as you mentioned.

However, a cube is often defined using 24 vertices for exactly this reason. This allows you to have vertex normals that are perpendicular to each face, by "duplicating" vertices at each corner. Defining a cube this way is, effectively, just defining 6 individual faces, each pointing outwards appropriately.

blocks|key|1077009|text|在平滑具有8个顶点的立方体以使其看起来像球体时，没有点。这样你会得到一个极其丑陋的球体。绘制立方体的唯一合理方法是使用24个唯一的顶点。|type|unstyled|depth|inlineStyleRanges|entityRanges|data|1077010|entityMap^0|0^^$0|@$1|2|3|4|5|6|7|D|8|@]|9|@]|A|$]]|$1|B|3|-4|5|6|7|E|8|@]|9|@]|A|$]]]|C|$]]

There is no point in smoothing the cube with 8 vertices in order to make it look like a sphere. You'll get an extremely ugly sphere this way. The only reasonable way to draw the cube is using 24 unique vertices.

blocks|key|4744209|text|立方体的八个角顶点的中心定向法线实际上将形成125度的角度，每条连接的边需要16分钟。|type|unstyled|depth|inlineStyleRanges|entityRanges|data|4744210|There's+a+good+discussion+of+this+topic+elsewhere+on+SO.|offset|length|4744211|entityMap|0|LINK|mutability|MUTABLE|url|https://stackoverflow.com/questions/2954349/when-should-i-use-indexed-arrays-of-opengl-vertices^0|0|0|1K|0|0^^$0|@$1|2|3|4|5|6|7|N|8|@]|9|@]|A|$]]|$1|B|3|C|5|6|7|O|8|@]|9|@$D|P|E|Q|1|R]]|A|$]]|$1|F|3|-4|5|6|7|S|8|@]|9|@]|A|$]]]|G|$H|$5|I|J|K|A|$L|M]]]]

The center-oriented normals of the eight corner vertices of a cube will actually form an angle of 125 degrees, 16 minutes with each connected edge.

<a href="https://stackoverflow.com/questions/2954349/when-should-i-use-indexed-arrays-of-opengl-vertices">There's a good discussion of this topic elsewhere on SO.</a>

blocks|key|3143774|text|135度可以由向外的每个顶点的法向量来直观地解释，并且必须与该顶点所属的每条边共享相同的角度。因为内角是90度，所以这个角向外是270度。因此270度/2+=+135度。|type|unstyled|depth|inlineStyleRanges|entityRanges|data|3143775|每个顶点的法向量被用来计算三角形的法向量。对于你的3d模型是一个平面三角形的集合，只有一个法线来计算它的光照将导致平面着色(如果对象真的是那样的边缘，你就是物理上正确的)。使用顶点法线为三角形的每个点插入“法线”，以提供平滑的照明，重新组装平滑的表面。|3143776|这种方法的问题是，仅使用每个顶点的单个法线会导致立方体具有类似球体的着色，同时仍是立方体。|3143777|这就是为什么人们想要定义一个具有24+(=+6x4)个顶点而不是6个顶点的立方体的原因。这样，一个立方体就可以具有所有面(因此它的每两个三角形)都具有正确(平坦)法线的立方体。|3143778|具有24个顶点，因此具有24条法线，可以为每个三角形/面仅定义面向前的法线，以便法线始终与三角形/面成90度角，从而在每个三角形/面上提供平面着色，这对于立方体更合适，因为它的表面实际上是平的。|3143779|通常，人们不想以平滑连续的方式对像90+(270)度这样的陡峭角度进行着色。法线插值仅用于模拟“有机”/“平滑”曲面。由于这些有机/平滑的表面是规范(想想茶壶或3d图形)，因此决定将顶点法线与位置和UV坐标一起存储，因为它是大多数‘连续’3D表面的规范。通常，可以添加更多三角形来表示模型中的平滑拓扑。因此，拥有顶点法线是一种权衡，以最小化平均模型的信息量。|3143780|因此，具有全部平坦三角形的立方体模型是最坏的情况。这就是为什么每个立方体的角都需要三条顶点法线，每个面都需要一条法线。|3143781|PS:今天，通过使用从更高分辨率模型烘焙的法线贴图，可以进一步定义这些“平滑”表面。使用法线贴图，面中的每个点都会获得自己的法线向量(或者可以从贴图法线贴图提供的法线向量采样中插值每个点的法线向量)。|3143782|entityMap^0|0|0|0|0|0|0|0|0^^$0|@$1|2|3|4|5|6|7|R|8|@]|9|@]|A|$]]|$1|B|3|C|5|6|7|S|8|@]|9|@]|A|$]]|$1|D|3|E|5|6|7|T|8|@]|9|@]|A|$]]|$1|F|3|G|5|6|7|U|8|@]|9|@]|A|$]]|$1|H|3|I|5|6|7|V|8|@]|9|@]|A|$]]|$1|J|3|K|5|6|7|W|8|@]|9|@]|A|$]]|$1|L|3|M|5|6|7|X|8|@]|9|@]|A|$]]|$1|N|3|O|5|6|7|Y|8|@]|9|@]|A|$]]|$1|P|3|-4|5|6|7|Z|8|@]|9|@]|A|$]]]|Q|$]]

The 135 degree can be explained visually by the normal vector of each vertices point outwards and must share the same angle with each edge the vertex is part of. Since the inner angle is 90 degree 270 degrees are outward of this corner. Therefore 270 degree /2 = 135 degree.

The normal vectors of each vertex are used to calculate the normal vector of the triangle. For your 3d model being a collection of flat triangles, having only a single normal to calculate the lighting from it would result in flat shading (thou being physically correct if the object really would be that edgy). Using the vertex normals to interpolate the 'normals' for each point of the triangle gives a smooth lighting reassembling a smooth surface. 

The problem with this approach is using only a single normal per vertex results in the cube to have a shading like a sphere while still being a cube.

That is basically the reason why one want to define a cube with 24 (= 6x4) vertices rather than 6. This way one can have a cube with all faces (and therefore each two of its triangles) to have correct (flat) normals.

Having 24 vertices and therefore 24 normals provide the possibility to define only forward facing normals for each triangle/face so that the normals point always in a 90 degree angle away from the triangle/face and therefore provide a flat shading throughout every triangle/face which is more correct for a cube as its surfaces are really flat.

Usually one does not want to shade a steep angle like 90 (270) degree in a smooth continuous way. The normal interpolation is only used to mimic 'organic'/'smooth' surfaces. Since these organic / smooth surfaces are the norm (think about the tea pot or a 3d-figure) the decision was made to store the vertex normals with the position and UV coordinates as it is the norm in most of the 'continuous' 3d surfaces. Normally you add more triangles to represent a smooth topology in a model. Having vertex normals is, therefore, a trade-off to minimize the amount of information for the average model.

A cube model with its all flat triangles is, therefore, the worst case. This is why each of the cube's corner needs three vertex normals, one for each face it is a vertex of.

PS: Today those 'smooth' surfaces are further defined by using normal maps baked from a higher resolution model. Using normal maps, each point in the face gets its own normal vector (or the normal vector of each point can be interpolated from the normal vector samples provided by the mapped normal map).

A cube has 8 unique vertices. Is it true that each of these 8 vertex normals (unit vectors) is making 135 degree angle to each of the edges which shares that vertex? And the vertex normal pointing outward/out of the cube? Your answer should be technically correct. Or it depends on how the cube is defined (drawn) like using triangle strips or indices that define 2 triangles for each side of the cube? The purpose of the vertex normal is smooth shading and lighting in OpenGL ES application.

Vertex normals for a cube

翻译质量差，导致语言生硬或混乱。

没有提供实际的解决方法或示例。

解答不清晰，无法理解或解决问题。

页面排版不美观，阅读体验差。

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一个立方体有8个唯一的顶点。这8条顶点法线(单位向量)中的每一条都与共享该顶点的每条边成135度角，这是真的吗？以及指向立方体外/外的顶点法线？你的答案在技术上应该是正确的。或者这取决于如何定义(绘制)立方体，就像使用三角形条带或索引，为立方体的每一边定义两个三角形？顶点法线的目的是在OpenGL ES应用程序中平滑着色和照明。

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