[{"@context":"http:\/\/schema.org\/","@type":"BlogPosting","@id":"https:\/\/wiki.edu.vn\/en\/wiki12\/isophote-wikipedia\/#BlogPosting","mainEntityOfPage":"https:\/\/wiki.edu.vn\/en\/wiki12\/isophote-wikipedia\/","headline":"Isophote – Wikipedia","name":"Isophote – Wikipedia","description":"before-content-x4 From Wikipedia, the free encyclopedia after-content-x4 Curve on an illuminated surface through points of equal brightness after-content-x4 ellipsoid with","datePublished":"2014-06-19","dateModified":"2014-06-19","author":{"@type":"Person","@id":"https:\/\/wiki.edu.vn\/en\/wiki12\/author\/lordneo\/#Person","name":"lordneo","url":"https:\/\/wiki.edu.vn\/en\/wiki12\/author\/lordneo\/","image":{"@type":"ImageObject","@id":"https:\/\/secure.gravatar.com\/avatar\/c9645c498c9701c88b89b8537773dd7c?s=96&d=mm&r=g","url":"https:\/\/secure.gravatar.com\/avatar\/c9645c498c9701c88b89b8537773dd7c?s=96&d=mm&r=g","height":96,"width":96}},"publisher":{"@type":"Organization","name":"Enzyklop\u00e4die","logo":{"@type":"ImageObject","@id":"https:\/\/wiki.edu.vn\/wiki4\/wp-content\/uploads\/2023\/08\/download.jpg","url":"https:\/\/wiki.edu.vn\/wiki4\/wp-content\/uploads\/2023\/08\/download.jpg","width":600,"height":60}},"image":{"@type":"ImageObject","@id":"https:\/\/upload.wikimedia.org\/wikipedia\/commons\/thumb\/3\/3f\/Isoph-ellipsoid-nv.svg\/300px-Isoph-ellipsoid-nv.svg.png","url":"https:\/\/upload.wikimedia.org\/wikipedia\/commons\/thumb\/3\/3f\/Isoph-ellipsoid-nv.svg\/300px-Isoph-ellipsoid-nv.svg.png","height":"327","width":"300"},"url":"https:\/\/wiki.edu.vn\/en\/wiki12\/isophote-wikipedia\/","wordCount":3309,"articleBody":"     (adsbygoogle = window.adsbygoogle || []).push({});before-content-x4From Wikipedia, the free encyclopedia       (adsbygoogle = window.adsbygoogle || []).push({});after-content-x4Curve on an illuminated surface through points of equal brightness         (adsbygoogle = window.adsbygoogle || []).push({});after-content-x4ellipsoid with isophotes (red)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\u2192(P)\u22c5v\u2192=cos\u2061\u03c6{displaystyle b(P)={vec {n}}(P)cdot {vec {v}}=cos varphi }where n\u2192(P){displaystyle {vec {n}}(P)} is the unit normal vector of the surface at point P and        (adsbygoogle = window.adsbygoogle || []).push({});after-content-x4v\u2192{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\u2192.{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.[1]Table of ContentsApplication and example[edit]Determining points of an isophote[edit]on an implicit surface[edit]on a parametric surface[edit]See also[edit]References[edit]External links[edit]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).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 rightDetermining points of an isophote[edit]on an implicit surface[edit]For an implicit surface with equation f(x,y,z)=0{displaystyle f(x,y,z)=0} the isophote condition is\u2207f\u22c5v\u2192|\u2207f|=c\u00a0.{displaystyle {frac {nabla fcdot {vec {v}}}{|nabla f|}}=c .}That means: points of an isophote with given parameter c{displaystyle c} are solutions of the non linear system\u00a0f(x,y,z)=0,\u2207f(x,y,z)\u22c5v\u2192\u2212c|\u2207f(x,y,z)|=0\u00a0,{displaystyle  f(x,y,z)=0,qquad nabla f(x,y,z)cdot {vec {v}}-c;|nabla f(x,y,z)|=0 ,}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[edit]In case of a parametric surface x\u2192=S\u2192(s,t){displaystyle {vec {x}}={vec {S}}(s,t)} the isophote condition is(S\u2192s\u00d7S\u2192t)\u22c5v\u2192|S\u2192s\u00d7S\u2192t|=c\u00a0.{displaystyle {frac {({vec {S}}_{s}times {vec {S}}_{t})cdot {vec {v}}}{|{vec {S}}_{s}times {vec {S}}_{t}|}}=c .}which is equivalent to\u00a0(S\u2192s\u00d7S\u2192t)\u22c5v\u2192\u2212c|S\u2192s\u00d7S\u2192t|=0\u00a0.{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\u2192(s,t){displaystyle {vec {S}}(s,t)} into surface points.See also[edit]References[edit]J. Hoschek, D. Lasser: Grundlagen der geometrischen Datenverarbeitung, Teubner-Verlag, Stuttgart, 1989, ISBN\u00a03-519-02962-6, p.\u00a031.Z. Sun, S. Shan, H. Sang et al.: Biometric Recognition, Springer, 2014, ISBN\u00a0978-3-319-12483-4, p.\u00a0158.C.L. Bajaj, C.M. Hoffmann, R.E. Lynch, J.E.H. Hopcroft: Tracing Surface Intersections, (1988) Comp. Aided Geom. Design 5, pp.\u00a0285\u2013307.C. T. Leondes: Computer Aided and Integrated Manufacturing Systems: Optimization methods, Vol. 3, World Scientific, 2003, ISBN\u00a0981-238-981-4, p.\u00a0209.^ J. Binney, M. Merrifield: Galactic Astronomy, Princeton University Press, 1998, ISBN\u00a00-691-00402-1, p. 178.External links[edit]     (adsbygoogle = window.adsbygoogle || []).push({});after-content-x4"},{"@context":"http:\/\/schema.org\/","@type":"BreadcrumbList","itemListElement":[{"@type":"ListItem","position":1,"item":{"@id":"https:\/\/wiki.edu.vn\/en\/wiki12\/#breadcrumbitem","name":"Enzyklop\u00e4die"}},{"@type":"ListItem","position":2,"item":{"@id":"https:\/\/wiki.edu.vn\/en\/wiki12\/isophote-wikipedia\/#breadcrumbitem","name":"Isophote – Wikipedia"}}]}]