Reflection lines

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 C^2 . 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.

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 C^2 . 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. r = v - 2 (n \cdot v) n. (The vector v is decomposed into its normal part v_n = (n\cdot v) v and tangential part v_t = v - v_n . Upon reflection, the tangential part is kept and the normal part is negated.) For reflection lines we consider the surface M surrounded by parallel lines with direction a , representing infinite, non-dispersive light sources. For each point p on M we determine which line is seen from direction v . The position on each line is of no interest. Define the vector r_p to be the reflection direction r projected onto a plane P that is orthogonal to a : r_p = r - (r \cdot a)a and similarly let v_p be the viewing direction projected onto P : v_p = v - (v \cdot a)a Finally, define v_o to be the direction lying in P perpendicular to a and v_p : v_o = a \times v_p Using these vectors, the *reflection line function* \theta(p): M \rightarrow (-\pi,\pi] is a scalar function mapping points p on the surface to angles between v_p and r_p : \theta = \arctan{( r_p \cdot v_o, r_p \cdot v_p)} where arctan(y,x) is the atan2 function producing a number in the range (-\pi,\pi] . ( v_p and v_o can be viewed as a local coordinate system in P with x -axis in direction v_p and y -axis in direction v_o .) Finally, to render the reflection lines positive values \theta>0 are mapped to a light color and non-positive values to a dark color. \theta = \arctan{( n_a \cdot a^\perp,n_a \cdot x)} where n_a is the surface normal projected on the light source plane P : \hat{n_a}/|\hat{n_a}|, \hat{n_a} = n - (n \cdot a)a The relationship between reflection lines and highlight lines is likened to that between specular and diffuse shading.

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