Engineers use reflection lines to judge a surface's quality. Reflection lines reveal surface flaws, particularly discontinuities in normals indicating that the surface is not . Reflection lines may be created and examined on physical surfaces or virtual surfaces with the help of computer graphics. For example, the shiny surface of an automobile body is illuminated with reflection lines by surrounding the car with parallel light sources. Virtually, a surface can be rendered with reflection lines by modulating the surfaces point-wise color according to a simple calculation involving the surface normal, viewing direction and a square wave environment map.

Reflection lines visualized on surfaces completed using a biharmonic and triharmonic equation with and surface continuity respectively. Derivative discontinuities near the yellow-purple boundary on the left reveal normal discontinuities. Below are pseudocolor visualizations of curvature.

Mathematical definition

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Consider a point   on a surface   with (normalized) normal  . If an observer views this point from infinity at view direction   then the reflected view direction   is:

 

(The vector   is decomposed into its normal part   and tangential part  . Upon reflection, the tangential part is kept and the normal part is negated.)

For reflection lines we consider the surface   surrounded by parallel lines with direction  , representing infinite, non-dispersive light sources. For each point   on   we determine which line is seen from direction  . The position on each line is of no interest.

Define the vector   to be the reflection direction   projected onto a plane   that is orthogonal to  :

 

and similarly let   be the viewing direction projected onto  :

 

Finally, define   to be the direction lying in   perpendicular to   and  :

 

Using these vectors, the *reflection line function*   is a scalar function mapping points   on the surface to angles between   and  :

 

where   is the atan2 function producing a number in the range  .

(  and   can be viewed as a local coordinate system in   with  -axis in direction   and  -axis in direction  .)

Finally, to render the reflection lines positive values   are mapped to a light color and non-positive values to a dark color.[1]

Highlight lines

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Highlight lines are a view-independent alternative to reflection lines.[2] Here the projected normal is directly compared against some arbitrary vector   perpendicular to the light source:

 

where   is the surface normal projected on the light source plane  :

 

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

References

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  1. ^ Gingold et al. "Shape Optimization Using Reflection Lines"
  2. ^ Beier and Chen. "Highlight-line algorithm for realtime surface-quality assessment"