View source for Isophote

{{short description|Curve on an illuminated surface through points of equal brightness}}
[[File:Isoph-ellipsoid-nv.svg|300px|thumb|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 {{mvar|b}} is measured by the following [[scalar product]]:
:<math>b(P)= \vec n(P)\cdot \vec v=\cos\varphi </math>
where {{tmath|\vec n(P)}} is the unit [[normal vector]] of the surface at point {{mvar|P}} and {{tmath|\vec v}} the [[unit vector]] of the light's direction. If {{math|1=''b''(''P'') = 0}}, i.e. the light is [[perpendicular]] to the surface normal, then point {{mvar|P}} is a point of the surface silhouette observed in direction {{tmath|\vec v.}} Brightness 1 means that the light vector is perpendicular to the surface. A [[Plane (geometry)|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.
<ref>J. Binney, M. Merrifield: ''Galactic Astronomy'', Princeton University Press, 1998, {{ISBN|0-691-00402-1}}, p. 178.</ref>

== 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 surface]]s 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). 
<gallery widths = 600px heights = 270px>
Isoph-bbb-g1g2.svg| 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
</gallery>

== Determining points of an isophote ==
=== On an implicit surface ===
For an [[implicit surface]] with equation <math> f(x,y,z)=0,</math> the isophote condition is
<math display=block> \frac{\nabla f \cdot \vec v}{|\nabla f|}= c \ .</math>
That means: points of an isophote with given parameter {{mvar|c}} are solutions of the [[nonlinear system]] 
<math display=block>\begin{align}
  f(x,y,z) &= 0, \\[4pt]
  \nabla f (x,y,z)\cdot \vec v -c\;|\nabla f(x,y,z)| &= 0,
\end{align}</math>
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]] <math>\vec x= \vec S(s,t)</math> the isophote condition is

<math display=block> \frac{(\vec S_s\times\vec S_t)\cdot\vec v}{|\vec S_s\times\vec S_t|}=c\ .</math>

which is equivalent to
<math display=block> \ (\vec S_s\times\vec S_t)\cdot\vec v- c\;|\vec S_s\times\vec S_t|=0 \ .</math>
This equation describes an implicit curve in the s-t-plane, which can be traced by a suitable algorithm (see [[implicit curve#Visualization of an implicit curve|implicit curve]]) and transformed by <math>\vec S(s,t)</math> into surface points.

== See also ==
* [[Contour line]]

== References ==
*J. Hoschek, D. Lasser: ''Grundlagen der geometrischen Datenverarbeitung'', Teubner-Verlag, Stuttgart, 1989, {{ISBN|3-519-02962-6}}, p.&nbsp;31.
*Z. Sun, S. Shan, H. Sang et al.: ''Biometric Recognition'', Springer, 2014, {{ISBN|978-3-319-12483-4}}, p.&nbsp;158.
*C.L. Bajaj, C.M. Hoffmann, R.E. Lynch, J.E.H. Hopcroft: ''Tracing Surface Intersections'', (1988) Comp. Aided Geom. Design 5, pp.&nbsp;285–307.
*C. T. Leondes: ''Computer Aided and Integrated Manufacturing Systems: Optimization methods'', Vol. 3, World Scientific, 2003, {{ISBN|981-238-981-4}}, p.&nbsp;209.
<references />

== External links ==
*[http://web.mit.edu/hyperbook/Patrikalakis-Maekawa-Cho/node148.html Patrikalakis-Maekawa-Cho: Isophotes (engl.)]
*[https://www.liverpool.ac.uk/~pjgiblin/papers/34590049.pdf A. Diatta, P. Giblin: ''Geometry of Isophote Curves'']
*[http://bh.knu.ac.kr/~kujinkim/papers/isophote.pdf Jin Kim: ''Computing Isophotes of Surface of Revolution and Canal Surface'']

[[Category:Curves]]
[[Category:Computer-aided design| ]]