Vertex normals for a cube

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开发者_开发问答 side of the cube? The purpose of the vertex normal is smooth shading and lighting in OpenGL ES application.

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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.

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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.

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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.

There's a good discussion of this topic elsewhere on SO.

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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).