Isophote

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title: "Isophote" type: doc version: 1 created: 2026-02-28 author: "Wikipedia contributors" status: active scope: public tags: ["curves", "computer-aided-design"] description: "Curve on an illuminated surface through points of equal brightness" topic_path: "general/curves" source: "https://en.wikipedia.org/wiki/Isophote" license: "CC BY-SA 4.0" wikipedia_page_id: 0 wikipedia_revision_id: 0

::summary Curve on an illuminated surface through points of equal brightness ::

::figure[src="https://upload.wikimedia.org/wikipedia/commons/3/3f/Isoph-ellipsoid-nv.svg" caption="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 b is measured by the following scalar product: :b(P)= \vec n(P)\cdot \vec v=\cos\varphi where is the unit normal vector of the surface at point P and the unit vector of the light's direction. If , 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.

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). 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

Determining points of an isophote

On an implicit surface

For an implicit surface with equation f(x,y,z)=0, the isophote condition is \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 \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} 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 \vec x= \vec S(s,t) the isophote condition is

\frac{(\vec S_s\times\vec S_t)\cdot\vec v}{|\vec S_s\times\vec S_t|}=c\ .

which is equivalent to \ (\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 \vec S(s,t) into surface points.

References

• J. Hoschek, D. Lasser: Grundlagen der geometrischen Datenverarbeitung, Teubner-Verlag, Stuttgart, 1989, , p. 31.
• Z. Sun, S. Shan, H. Sang et al.: Biometric Recognition, Springer, 2014, , p. 158.
• C.L. Bajaj, C.M. Hoffmann, R.E. Lynch, J.E.H. Hopcroft: Tracing Surface Intersections, (1988) Comp. Aided Geom. Design 5, pp. 285–307.
• C. T. Leondes: Computer Aided and Integrated Manufacturing Systems: Optimization methods, Vol. 3, World Scientific, 2003, , p. 209.

References

1. J. Binney, M. Merrifield: ''Galactic Astronomy'', Princeton University Press, 1998, {{ISBN. 0-691-00402-1, p. 178.

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