Convex curve has at least four vertices, points that are local minima or local maxima of curvature. A curve "C" is convex if and only if there are no three different points in "C" such that the tangents in these points are parallel. Proof: ⇒ If there are three parallel tangents, then one of them, say "L", must be between the other two. This means that "C" lies on both sides of "L", so it cannot be convex. ⇐ If "C" is not convex, then by definition there is point "p" on "C" such that the tangent line at "p" (call it
Focal cloud antennas might benefit from minor deformation, producing a footprint encompassing satellite tidal motion. 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