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:
where is the unit normal vector of the surface at point P and 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 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. [1]
Application and example[edit]
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).
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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[edit]
On an implicit surface[edit]
For an implicit surface with equation the isophote condition is
On a parametric surface[edit]
In case of a parametric surface the isophote condition is
which is equivalent to
See also[edit]
References[edit]
- 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.
- ^ J. Binney, M. Merrifield: Galactic Astronomy, Princeton University Press, 1998, ISBN 0-691-00402-1, p. 178.