Ingeometry, anisophote is acurve on an illuminated surface that connects points of equalbrightness. One supposes that the illumination is done by parallel light and the brightnessb is measured by the followingscalar product:

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

where⁠${\displaystyle \overrightarrow{n}(P)}$⁠ is the unitnormal vector of the surface at pointP and⁠${\displaystyle \overrightarrow{v}}$⁠ theunit vector of the light's direction. Ifb(P) = 0, i.e. the light isperpendicular to the surface normal, then pointP 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. Aplane has no isophotes, because every point has the same brightness.

Inastronomy, an isophote is a curve on a photo connecting points of equal brightness.^[1]

Incomputer-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 theirgeometric 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 intersectingBezier 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

For animplicit surface with equation${\displaystyle f(x,y,z)=0,}$ the isophote condition is$${\displaystyle \frac{\mathrm{\nabla }f\cdot \overrightarrow{v}}{|\mathrm{\nabla }f|}=c.}$$That means: points of an isophote with given parameterc are solutions of thenonlinear systemwhich can be considered as theintersection curve of two implicit surfaces. Using the tracing algorithm of Bajaj et al. (see references) one can calculate apolygon of points.

In case of aparametric surface${\displaystyle \overrightarrow{x}=\overrightarrow{S}(s,t)}$ the isophote condition is

$${\displaystyle \frac{({\overrightarrow{S}}_s\times {\overrightarrow{S}}_t)\cdot \overrightarrow{v}}{|{\overrightarrow{S}}_s\times {\overrightarrow{S}}_t|}=c.}$$

which is equivalent to$${\displaystyle ({\overrightarrow{S}}_s\times {\overrightarrow{S}}_t)\cdot \overrightarrow{v}-c|{\overrightarrow{S}}_s\times {\overrightarrow{S}}_t|=0.}$$This equation describes an implicit curve in the s-t-plane, which can be traced by a suitable algorithm (seeimplicit curve) and transformed by${\displaystyle \overrightarrow{S}(s,t)}$ into surface points.

• Contour line

• J. Hoschek, D. Lasser:Grundlagen der geometrischen Datenverarbeitung, Teubner-Verlag, Stuttgart, 1989,ISBN 3-519-02962-6, p. 31.
• Z. Sun, S. Shan, H. Sang et al.:Biometric Recognition, Springer, 2014,ISBN 978-3-319-12483-4, 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,ISBN 981-238-981-4, p. 209.

• Patrikalakis-Maekawa-Cho: Isophotes (engl.)
• A. Diatta, P. Giblin:Geometry of Isophote Curves
• Jin Kim:Computing Isophotes of Surface of Revolution and Canal Surface

• Curves
• Computer-aided design

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