Halfway Vector

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Historically in graphics, various interpolation lighting models were used to produce continuous shading of surfaces represented by data associated with polygonal meshes.

By lighting model I mean a simplified, non-conserving, "bidirectional reflectance model" (really a smart hueristic) that just uses the vertex's normal, a light and viewing direction.

If the data of each vertex, the corners of each triangle, are interpolated (vertex to vertex, done in a vertex shader) then it's said to be Gouraud shading. If the pixel values between vertices are interpolated (pixel to pixel, done in a fragment shader) then it's said to be Phong shading. (Each of these old school models are named after their inventor.)

This is accomplished by the internal interpolation that happens in a "varying"/ "in"/ "SV_POSITION" variables associated with this vertex.

This turns a discrete vertex normal that was piped in from the vertex shader into a continuous smearing of normals data, "fragment" to "fragment", in a fragment shader. ¹.

This can then be normalized and dotted with the viewing direction to find a diffuse shading weight.

An associated reflection vector can be calculated for approximated specular lighting. See another article I wrote for more: [[../reflection_{andrefractioninaraytracer}/index.org][Reflection and Refraction in a Raytracer

There are also hardware accelerated functions in modern shading languages for finding the reflection vector, e.g. reflect() in glsl.

This works well for diffuse lighting as the dot product between the normal and viewing vectors will only be ever deal with angles of 90 degrees / \(\frac{\pi}{2}\) or less, but this is not the case for the reflection vector.

This is why the "halfway vector" was introduced by Jim Blinn as an extension of the Phong Model -> Blinn-Phong lighting model.

Mathematically, the halfway vector is nothing more than an angle bisector vector with normalized components, but really it's a mapping of how oblique or aligned with the reflection vector the viewing angle is to the normal, thus recovering a better approximation/ hueristic for specular lighting.

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For two vectors (light direction and and viewing vector) \(\vec l\) & \(\vec v\) the bisecting vector between them is: \(\lvert \vec v \rvert \vec l + \lvert \vec l \rvert \vec v\)

See proof wiki for Angle Bisector Vector for a derivation of this expression.

If both vectors have been normalized then this reduces to: \(\vec v + \vec l\)

The halfway vector as presented in literature is just the unit version of this vector:

\[\vec H = \frac{\vec v + \vec l}{\lvert \vec v + \vec l \rvert}\]