# Isophote

ellipsoid with isophotes (red)

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 ${\displaystyle b}$ is measured by the following scalar product:

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

${\displaystyle {\vec {n}}(P)}$ is the unit normal vector of the surface at point ${\displaystyle P}$ and ${\displaystyle {\vec {v}}}$ the unit vector of the light's direction. In case of ${\displaystyle b(P)=0}$, i.e. the light is perpendicular to the surface normal, point ${\displaystyle P}$ is a point of the surface silhouett looked in direction ${\displaystyle {\vec {v}}}$. Brightness 1 means, the lightvector is perpendicular to the surface. A plane has no isophotes, because any 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).

## Determining points of an isophote

### on an implicit surface

For an implicit surface with equation ${\displaystyle f(x,y,z)=0}$ the isophote condition is

${\displaystyle {\frac {\nabla f\cdot {\vec {v}}}{|\nabla f|}}=c\ .}$

That means: points of an isophote with given parameter ${\displaystyle c}$ are solutions of the non linear system

• ${\displaystyle \ f(x,y,z)=0,\qquad \nabla f(x,y,z)\cdot {\vec {v}}-c\;|\nabla f(x,y,z)|=0\ ,}$

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 ${\displaystyle {\vec {x}}={\vec {S}}(s,t)}$ the isophote condition is

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

which is equivalent to

• ${\displaystyle \ ({\vec {S}}_{s}\times {\vec {S}}_{t})\cdot {\vec {v}}-c\;|{\vec {S}}_{s}\times {\vec {S}}_{t}|=0\ .}$

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 ${\displaystyle {\vec {S}}(s,t)}$ into surface points.