In
geometry
Geometry (; ) is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space such as the distance, shape, size, and relative position of figures. A mathematician who works in the field of geometry is c ...
, an isophote is a
curve
In mathematics, a curve (also called a curved line in older texts) is an object similar to a line (geometry), line, but that does not have to be Linearity, straight.
Intuitively, a curve may be thought of as the trace left by a moving point (ge ...
on an illuminated surface that connects points of equal
brightness
Brightness is an attribute of visual perception in which a source appears to be radiating or reflecting light. In other words, brightness is the perception elicited by the luminance of a visual target. The perception is not linear to luminance, ...
. One supposes that the illumination is done by parallel light and the brightness is measured by the following
scalar product
In mathematics, the dot product or scalar productThe term ''scalar product'' means literally "product with a scalar as a result". It is also used sometimes for other symmetric bilinear forms, for example in a pseudo-Euclidean space. is an algebra ...
:
:
where is the unit
normal vector
In geometry, a normal is an object such as a line, ray, or vector that is perpendicular to a given object. For example, the normal line to a plane curve at a given point is the (infinite) line perpendicular to the tangent line to the curve at ...
of the surface at point and the
unit vector
In mathematics, a unit vector in a normed vector space is a vector (often a spatial vector) of length 1. A unit vector is often denoted by a lowercase letter with a circumflex, or "hat", as in \hat (pronounced "v-hat").
The term ''direction vecto ...
of the light's direction. If , i.e. the light is
perpendicular
In elementary geometry, two geometric objects are perpendicular if they intersect at a right angle (90 degrees or π/2 radians). The condition of perpendicularity may be represented graphically using the ''perpendicular symbol'', ⟂. It can ...
to the surface normal, then point is a point of the surface silhouette observed in direction Brightness 1 means that the light vector is perpendicular to the surface. A
plane
Plane(s) most often refers to:
* Aero- or airplane, a powered, fixed-wing aircraft
* Plane (geometry), a flat, 2-dimensional surface
Plane or planes may also refer to:
Biology
* Plane (tree) or ''Platanus'', wetland native plant
* ''Planes' ...
has no isophotes, because every point has the same brightness.
In
astronomy
Astronomy () is a natural science that studies astronomical object, celestial objects and phenomena. It uses mathematics, physics, and chemistry in order to explain their origin and chronology of the Universe, evolution. Objects of interest ...
, an isophote is a curve on a photo connecting points of equal brightness.
[J. Binney, M. Merrifield: ''Galactic Astronomy'', Princeton University Press, 1998, , p. 178.]
Application and example
In
computer-aided design
Computer-aided design (CAD) is the use of computers (or ) to aid in the creation, modification, analysis, or optimization of a design. This software is used to increase the productivity of the designer, improve the quality of design, improve c ...
, 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
In mathematical analysis, the smoothness of a function is a property measured by the number of continuous derivatives it has over some domain, called ''differentiability class''. At the very minimum, a function could be considered smooth if i ...
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
In mathematics, an implicit surface is a surface in Euclidean space defined by an equation
: F(x,y,z)=0.
An ''implicit surface'' is the set of zeros of a function of three variables. ''Implicit'' means that the equation is not solved for o ...
with equation
the isophote condition is
:
That means: points of an isophote with given parameter
are solutions of the non linear system
*
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 A parametric surface is a surface in the Euclidean space \R^3 which is defined by a parametric equation with two parameters Parametric representation is a very general way to specify a surface, as well as implicit representation. Surfaces that oc ...
the isophote condition is
:
which is equivalent to
*
This equation describes an implicit curve in the s-t-plane, which can be traced by a suitable algorithm (see.
implicit curve
In mathematics, an implicit curve is a plane curve defined by an implicit equation relating two coordinate variables, commonly ''x'' and ''y''. For example, the unit circle is defined by the implicit equation x^2+y^2=1. In general, every implic ...
) and transformed by
into surface points.
See also
*
Contour line
A contour line (also isoline, isopleth, or isarithm) of a function of two variables is a curve along which the function has a constant value, so that the curve joins points of equal value. It is a plane section of the three-dimensional grap ...
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, {{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''
Curves