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: where ⁠

⁠ is the unit normal vector of the surface at point P and ⁠

⁠ 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 ⁠

⁠ 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] 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).

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).

For an implicit surface with equation

That means: points of an isophote with given parameter c are solutions of the nonlinear system

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.

In case of a parametric surface

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