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A function f is concave over a convex set if and only if the function −f is a convex function over the set. The sum of two concave functions is itself concave and so is the pointwise minimum of two concave functions, i.e. the set of concave functions on a given domain form a semifield.
An example of a concave polygon. A simple polygon that is not convex is called concave, [1] non-convex [2] or reentrant. [3] A concave polygon will always have at least one reflex interior angle—that is, an angle with a measure that is between 180 degrees and 360 degrees exclusive. [4]
The external angle is positive at a convex vertex or negative at a concave vertex. For every simple polygon, the sum of the external angles is 2 π {\displaystyle 2\pi } (one full turn, 360°). Thus the sum of the internal angles, for a simple polygon with n {\displaystyle n} sides is ( n − 2 ) π {\displaystyle (n-2)\pi } .
In ray tracing, the vergence can then be pictured as the angle between any two rays. For imaging or beams, the vergence is often described as the angle between the outermost rays in the bundle (marginal rays), at the edge (verge) of a cone of light, and the optical axis. This slope is typically measured in radians. Thus, in this case the ...
The term convex is often referred to as convex down or concave upward, and the term concave is often referred as concave down or convex upward. [ 3 ] [ 4 ] [ 5 ] If the term "convex" is used without an "up" or "down" keyword, then it refers strictly to a cup shaped graph ∪ {\displaystyle \cup } .
For every concave kite there exist two circles tangent to two of the sides and the extensions of the other two: one is interior to the kite and touches the two sides opposite from the concave angle, while the other circle is exterior to the kite and touches the kite on the two edges incident to the concave angle. [27] For a convex kite with ...
The interior angle concept can be extended in a consistent way to crossed polygons such as star polygons by using the concept of directed angles.In general, the interior angle sum in degrees of any closed polygon, including crossed (self-intersecting) ones, is then given by 180(n–2k)°, where n is the number of vertices, and the strictly positive integer k is the number of total (360 ...
For a polyhedron, the defect at a vertex equals 2π minus the sum of all the angles at the vertex (all the faces at the vertex are included). If a polyhedron is convex, then the defect of each vertex is always positive. If the sum of the angles exceeds a full turn, as occurs in some vertices of many non-convex polyhedra, then the defect is ...