Convex curve itself. A closed regular plane simple curve "C" is convex "if and only if" its curvature is either always non-negative or always non-positive—i.e., if and only if the "turning angle" (the angle of the tangent to the curve) is a weakly monotone function of the parametrization of the curve. Proof: ⇐ If "C" is not convex, then by the parallel tangents lemma there are three points {"p","q1","q2"} such that the tangents at these points are parallel. At least two must have their signed tangents pointing in the same direction. Without loss of generality, assume that these points are "q1" and
Focal cloud Focal cloud A focal cloud is the collection of focal points of an "imperfect" lens or parabolic reflector whether optical, electrostatic or electromagnetic. This includes parabolic antennas and lens-type reflective antennas of all kinds. The effect is analogous to the circle of confusion in photography. In a perfect lens or parabolic reflector, rays parallel to the device's axis striking the lens or reflector all pass through a single point, the focal point. In an imperfectly constructed lens or reflector, rays passing through different parts of the element do not converge to a single point but have different focal points. The