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 {displaystyle b} is measured by the following scalar product: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 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 {displaystyle b} is measured by the following scalar product: n → ( P ) {displaystyle {vec {n}}(P)} is the unit normal vector of the surface at point P {displaystyle P} and v → {displaystyle {vec {v}}} the unit vector of the light's direction. In case of b ( P ) = 0 {displaystyle b(P)=0} , i.e. the light is perpendicular to the surface normal, point P {displaystyle P} is a point of the surface silhouette looked in direction v → {displaystyle {vec {v}}} . Brightness 1 means, the lightvector is perpendicular to the surface. A plane has no isophotes, because any point has the same brightness. In astronomy an isophote is a curve on a photo connecting points of equal brightness. 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). For an implicit surface with equation f ( x , y , z ) = 0 {displaystyle f(x,y,z)=0} the isophote condition is That means: points of an isophote with given parameter c {displaystyle c} are solutions of the non linear system