Reflection lines

Mathematical Definition

Let us consider a point p on a surface M with (possibly non-unit length) normal n . If an observer views this point from infinity at a incoming direction v then the reflected view direction r is:

r = ( 2 / | n | 2 ) ( ( n ⋅ v ) n − v ) .

For reflection lines we consider repeated infinite, non-dispersive light sources parallel to some line a and therefore perpendicular to a plane P . Define the vector d to be the reflection direction r projected onto the plane P :

d = r − ( r ⋅ a ) a

and similarly let v a be the unit viewing direction projected onto P :

v a = v a ^ / | v a ^ | , v a ^ = v − ( v ⋅ a ) a

Finally, define a ⊥ to be the direction lying in P perpendicular to a and v a :

a ⊥ = a × v a

Then the *reflection line function* θ ( p ) : M → ( − π , π ] is a scalar function mapping points on the surface to angles between v a and the projected reflected view direction d :

θ = arctan ⁡ L ( r ⋅ a ⊥ , r ⋅ v a )

where a r c t a n ( y , x ) is the atan2 function producing a number in the range ( − π , π ] .

Finally, to render the reflection lines positive values θ > 0 are mapped to a light color and non-positive values to a dark color.

Highlight lines

Highlight lines are a view-independent alternative to reflection lines. Here the projected normal is directly compared against some arbitrary vector x perpendicular to the light source:

θ = arctan ⁡ ( n a ⋅ a ⊥ , n a ⋅ x )

where n a is the surface normal projected on the light source plane P :

n a ^ / | n a ^ | , n a ^ = n − ( n ⋅ a ) a

The relationship between reflection lines and highlight lines is likened to that between specular and diffuse shading.