- Source: 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:
b
(
P
)
=
n
→
(
P
)
⋅
v
→
=
cos
φ
{\displaystyle b(P)={\vec {n}}(P)\cdot {\vec {v}}=\cos \varphi }
where
n
→
(
P
)
{\displaystyle {\vec {n}}(P)}
is the unit normal vector of the surface at point P and
v
→
{\displaystyle {\vec {v}}}
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
v
→
.
{\displaystyle {\vec {v}}.}
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).
Determining points of an isophote
= On an implicit surface
=For an implicit surface with equation
f
(
x
,
y
,
z
)
=
0
,
{\displaystyle f(x,y,z)=0,}
the isophote condition is
∇
f
⋅
v
→
|
∇
f
|
=
c
.
{\displaystyle {\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
f
(
x
,
y
,
z
)
=
0
,
∇
f
(
x
,
y
,
z
)
⋅
v
→
−
c
|
∇
f
(
x
,
y
,
z
)
|
=
0
,
{\displaystyle {\begin{aligned}f(x,y,z)&=0,\\[4pt]\nabla f(x,y,z)\cdot {\vec {v}}-c\;|\nabla f(x,y,z)|&=0,\end{aligned}}}
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
x
→
=
S
→
(
s
,
t
)
{\displaystyle {\vec {x}}={\vec {S}}(s,t)}
the isophote condition is
(
S
→
s
×
S
→
t
)
⋅
v
→
|
S
→
s
×
S
→
t
|
=
c
.
{\displaystyle {\frac {({\vec {S}}_{s}\times {\vec {S}}_{t})\cdot {\vec {v}}}{|{\vec {S}}_{s}\times {\vec {S}}_{t}|}}=c\ .}
which is equivalent to
(
S
→
s
×
S
→
t
)
⋅
v
→
−
c
|
S
→
s
×
S
→
t
|
=
0
.
{\displaystyle \ ({\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
S
→
(
s
,
t
)
{\displaystyle {\vec {S}}(s,t)}
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. 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.
External links
Patrikalakis-Maekawa-Cho: Isophotes (engl.)
A. Diatta, P. Giblin: Geometry of Isophote Curves
Jin Kim: Computing Isophotes of Surface of Revolution and Canal Surface
Kata Kunci Pencarian:
- Isophote
- List of largest galaxies
- Galaxy
- Milky Way
- NGC 3109
- Whirlpool Galaxy
- ESO 383-76
- Large Magellanic Cloud
- Messier 49
- Andromeda Galaxy
