Isophotes

Ellipsoid with isophotes (red)

In geometry, an isophoto is a curve on an illuminated surface that connects points of equal brightness. It is assumed that the surface is illuminated with parallel light and the brightness is measured in a surface point with the following scalar product: ${\ displaystyle h}$ ${\ displaystyle P}$

{\ displaystyle h (P) = {\ vec {n}} (P) \ cdot {\ vec {v}} = \ cos \ varphi \.}

It is the unit normal vector at the surface point and the unit light vector. Is , d. H. the direction of light perpendicular to the normal is a point of the surface outline with parallel projection in the direction . Brightness 1 means that the light falls at point P perpendicularly onto the surface. A plane has no isophotes because every point on the plane has the same brightness. ${\ displaystyle {\ vec {n}} (P)}$ ${\ displaystyle P}$ ${\ displaystyle {\ vec {v}}}$ ${\ displaystyle h (P) = 0}$ ${\ displaystyle P}$ ${\ displaystyle {\ vec {v}}}$

In astronomy , an isophot is a curve on a photograph that connects points of equal brightness.

Application with example

Isophotes are used in CAD as test curves to visually assess the quality of surface transitions. Because with a sufficiently differentiable surface (implicit or parameterized) the normal vector of the surface depends on the first derivatives. The differentiability of the isophotes and thus their geometric continuity is reduced by 1 compared to the area. If only the tangential planes are continuous (i.e. -continuous) at one point on the surface , the isophote has a kink at this point (i.e. it is only -continuous). ${\ displaystyle G ^ {1}}$ ${\ displaystyle G ^ {0}}$

In the following example (see picture), two intersecting tensor product Bezier surfaces are rounded using a transition surface . On the left the transition surface only has -contact with the Bezier surfaces, on the right -contact. This difference cannot be seen without the isophotes. Only the isophotes show the difference: on the left the isophots have kinks, on the right they are smooth. ${\ displaystyle G ^ {1}}$ ${\ displaystyle G ^ {2}}$

Isophotes on two Bezier surfaces and a G1 or G2-continuous transition surface: left G1-continuous, right G2-continuous, i.e. H. the isophotes on the left are only G0-continuous (have a kink), on the right the isophotes are smooth

Determination of points of an isophoto

For an implicit surface

If the area is implicitly given by an equation , then the isophotic condition reads ${\ displaystyle f (x, y, z) = 0}$

{\ displaystyle {\ frac {\ nabla f \ cdot {\ vec {v}}} {| \ nabla f |}} = c \.}

In order to determine points of the isophotes belonging to a given one, one has to find solutions to the non-linear system of equations ${\ displaystyle c}$

${\ displaystyle \ f (x, y, z) = 0, \ qquad \ nabla f (x, y, z) \ cdot {\ vec {v}} - c \; | \ nabla f (x, y, z ) | = 0 \}$

determine. This system can be viewed as the intersection of two implicit surfaces and with the tracking algorithm of Bajaj et al. (see literature) calculate a sufficient number of points.

For parameterized areas

If the area is parameterized by given, then the isophotic condition reads ${\ displaystyle {\ vec {x}} = {\ vec {S}} (s, t)}$

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

This equation can be converted into the equation

${\ displaystyle \ ({\ vec {S}} _ {s} \ times {\ vec {S}} _ {t}) \ cdot {\ vec {v}} - c \; | {\ vec {S} } _ {s} \ times {\ vec {S}} _ {t} | = 0}$

reshape. This equation describes an implicit curve in the st plane, for which one can determine a sufficient number of points with a tracking algorithm (see implicit curve ) and then calculate them with area points . ${\ displaystyle {\ vec {S}} (s, t)}$

literature

J. Hoschek, D. Lasser: Fundamentals of geometric data processing , 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.
CL Bajaj, CM Hoffmann, RE Lynch, JEH Hopcroft: Tracing Surface Intersections , (1988) Comp. Aided Geom. Design 5, pp. 285-307.

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

Web links

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