Normal Generation

What are Normals

A normal describes the gradient because they are the same thing at right angles. Perhaps the light equation uses a first order approximation - the normal is the first order.

Consider a surface as a smooth continuous sheet, the normal at a point completely describes the gradient at that point. The gradient is used in the lighting equation.

Now consider a surface as a collection of continuous faceted triangles, joined together in a pattern. If we do not know what the normal is then we will have to construct it from the surface. This is really a non-uniqueness issue as there are infinitely many different possibilities by choosing different normals at each of the points.

We could even give points not unique normals. For example faceted triangles (bright shinny reflections) give the same normal for each of the triangles points. Since different triangles border the same points there is no unique normal for a point.

Alternatively you could specify that the normal must be unique at each point, for a sphere this renders it much smoother and better looking.

Faceted Normals

The normal perpendicular to each triangles plane is used to render the triangle's points.

This is the most primitive normal generation. The facets are visually easy to see.

Computing Vertex Normals from Facet Normals

$Nml = \sum_{i = 1}^{n - 1} \frac{e_{i} \times e_{i + 1}}{|| e_{i} ||^{2} || e_{i + 1} ||^{2}}$

Real Time Rendering, page 453, Max Nelson.

Newell

$n_{x} = \sum_{i = 1}^{N - 1} (y_{i} - y_{i + 1}) (z_{i} + z_{i + 1})$
$n_{y} = \sum_{i = 1}^{N - 1} (z_{i} - z_{i + 1}) (x_{i} + x_{i + 1})$
$n_{z} = \sum_{i = 1}^{N - 1} (x_{i} - x_{i + 1}) (y_{i} + y_{i + 1})$

This works for convex polygons. For triangles the method generates faceted triangles. This is good because the algorithm is general - its simplest behavior is the first order approximate - the faceted triangle method.

To show their equivalence I computed the faceted normal in the x component using Newells method, then compared it with the faceted normal and found they were the same.

A computational advantage of Newells method is that its easy to write computation of the faceted normal. Rather than calculate the cross products, just plug in his formula.

Central Differences

The marching methods use central differences to calculate the tangent.

Marching Squares
$f_{x} = f (i + 1, j) - f (i - 1, j)$
$f_{y} = f (i, j + 1) - f (i, j - 1)$
$\frac{df}{dX} = (f_{x}, f_{y})$
$Nml = (- f_{y}, f_{x})$

Marching Cubes
$f_{x} = f (i + 1, j, k) - f (i - 1, j, k)$
$f_{y} = f (i, j + 1, k) - f (i, j - 1, k)$
$f_{z} = f (i, j, k + 1) - f (i, j, k - 1)$
$\frac{df}{dX} = (f_{x}, f_{y}, f_{z})$

The central differences are for uniform meshes where all the points are equispaced or ordered in some way. For general meshes this is not the case and other methods such as computing vector normals from facet normals could be used.

Here is a more general way to calculate the gradient. Consider in 2D a special case where the mesh is of diamonds (square rotated 45 degrees and possibly streached) and each diamond is divided into four triangles by a central point. The following gradient calculation gives the same result as if central differences were used in this case. This lends argument to this gradient calculation being general as a trivial case defaults to a known and sucessful algorithm.

Let n be the dimension of the space
Let $p_{i}$ be the point about which the gradient is being calculated and associated with the edge with the same index.
$k = 0 ... n - 1$
$\frac{df}{dX} [k] = \max (p_{i} [k]) - \min (p_{i} [k])$

Tangent in 3D

Computing the vertex normal from the facet normals is very general as it approximates the normal by averaging the cross products of the points surrounding facets.

However the gradient methods calculate the tangent at a point in space. In 3D there is then the issue of obtaining the normal from the tangent, as the cross product of two tangents generates the normal.

Consider a particle moving across a surface. The tangent can be used to calculate the surface by estimating the next surface point. But each particle is on its own path and does not know of other particles which together make the surface.

Consider now the problem of knowing one of the tangents to the normal and having surrounding points on the surface of the mesh. How can the normal be calculated?

Here are some ideas I have cooked up.

$Nml = \sum_{i = 0}^{n - 1} \frac{e_{i} \times \frac{df}{dX}}{|| e_{i} ||^{2} || \frac{df}{dX} ||^{2}}$

Let $A_{i}$ be the area between the edges i and i+1. Let $A_{T}$ be the total area of the triangles surrounding the point.
$Nml = \sum_{i = 1}^{n - 1} \frac{A_{i}}{A_{T}} \frac{e_{i} + e_{i + 1}}{|| e_{i} + e_{i + 1} ||^{2}} \times \frac{df}{dX}$