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 ) = n → ( P ) ⋅ v → = cos ⁡ φ {\displaystyle b(P)={\vec {n}}(P)\cdot {\vec {v}}=\cos \varphi }

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

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 , {\displaystyle f(x,y,z)=0,} the isophote condition is ∇ f ⋅ v → | ∇ f | = c . {\displaystyle {\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 f ( x , y , z ) = 0 , ∇ f ( x , y , z ) ⋅ v → − c | ∇ f ( x , y , z ) | = 0 , {\displaystyle {\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 x → = S → ( s , t ) {\displaystyle {\vec {x}}={\vec {S}}(s,t)} the isophote condition is

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

which is equivalent to ( S → s × S → t ) ⋅ v → − c | S → s × S → t | = 0 . {\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 S → ( s , t ) {\displaystyle {\vec {S}}(s,t)} into surface points.

Isophote

Application and example

Determining points of an isophote

On an implicit surface

On a parametric surface

See also

External links