Isophote

${\displaystyle b(P)=\overrightarrow{n}(P)\cdot \overrightarrow{v}=\cos \phi }$

where ⁠${\displaystyle \overrightarrow{n}(P)}$⁠ is the unit normal vector of the surface at point P and ⁠${\displaystyle \overrightarrow{v}}$⁠ 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 ⁠${\displaystyle \overrightarrow{v}.}$⁠ 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.

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).