Isophote

In geometry, an isophote is a curve on an illuminated surface that connects points of equal brightness. One supposes that the illumination is done by parallel light and the brightness $b$ is measured by the following scalar product:

b(P)={\vec {n}}(P)\cdot {\vec {v}}=\cos \varphi

where ⁠ ${\vec {n}}(P)$ ⁠ is the unit normal vector of the surface at point $P$ and ⁠ ${\vec {v}}$ ⁠ the unit vector of the light's direction. If $b (P) = 0$ , i.e. the light is perpendicular to the surface normal, then point $P$ is a point of the surface silhouette observed in direction ⁠ ${\vec {v}}.$ ⁠ Brightness 1 means that the light vector is perpendicular to the surface. A plane has no isophotes, because every point has the same brightness.

In astronomy, an isophote is a curve on a photo connecting points of equal brightness. ^[1]

Application and example

In computer-aided design, isophotes are used for checking optically the smoothness of surface connections. For a surface (implicit or parametric), which is differentiable enough, the normal vector depends on the first derivatives. Hence, the differentiability of the isophotes and their geometric continuity is 1 less than that of the surface. If at a surface point only the tangent planes are continuous (i.e. G1-continuous), the isophotes have there a kink (i.e. is only G0-continuous).

In the following example (s. diagram), two intersecting Bezier surfaces are blended by a third surface patch. For the left picture, the blending surface has only G1-contact to the Bezier surfaces and for the right picture the surfaces have G2-contact. This difference can not be recognized from the picture. But the geometric continuity of the isophotes show: on the left side, they have kinks (i.e. G0-continuity), and on the right side, they are smooth (i.e. G1-continuity).

Isophotes on two Bezier surfaces and a G1-continuous (left) and G2-continuous (right) blending surface: On the left the isophotes have kinks and are smooth on the right

Determining points of an isophote

On an implicit surface

For an implicit surface with equation $f(x,y,z)=0,$ the isophote condition is ${\frac {\nabla f\cdot {\vec {v}}}{|\nabla f|}}=c\ .$ That means: points of an isophote with given parameter $c$ are solutions of the nonlinear system ${\begin{aligned}f(x,y,z)&=0,\\[4pt]\nabla f(x,y,z)\cdot {\vec {v}}-c\;|\nabla f(x,y,z)|&=0,\end{aligned}}$ which can be considered as the intersection curve of two implicit surfaces. Using the tracing algorithm of Bajaj et al. (see references) one can calculate a polygon of points.

On a parametric surface

In case of a parametric surface ${\vec {x}}={\vec {S}}(s,t)$ the isophote condition is

${\frac {({\vec {S}}_{s}\times {\vec {S}}_{t})\cdot {\vec {v}}}{|{\vec {S}}_{s}\times {\vec {S}}_{t}|}}=c\ .$

which is equivalent to $\ ({\vec {S}}_{s}\times {\vec {S}}_{t})\cdot {\vec {v}}-c\;|{\vec {S}}_{s}\times {\vec {S}}_{t}|=0\ .$ This equation describes an implicit curve in the s-t-plane, which can be traced by a suitable algorithm (see implicit curve) and transformed by ${\vec {S}}(s,t)$ into surface points.

References

J. Hoschek, D. Lasser: Grundlagen der geometrischen Datenverarbeitung, Teubner-Verlag, Stuttgart, 1989, ISBN 3-519-02962-6, p. 31.
Z. Sun, S. Shan, H. Sang et al.: Biometric Recognition, Springer, 2014, ISBN 978-3-319-12483-4, p. 158.
C.L. Bajaj, C.M. Hoffmann, R.E. Lynch, J.E.H. Hopcroft: Tracing Surface Intersections, (1988) Comp. Aided Geom. Design 5, pp. 285–307.
C. T. Leondes: Computer Aided and Integrated Manufacturing Systems: Optimization methods, Vol. 3, World Scientific, 2003, ISBN 981-238-981-4, p. 209.

^ J. Binney, M. Merrifield: Galactic Astronomy, Princeton University Press, 1998, ISBN 0-691-00402-1, p. 178.

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