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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]
A rectilinear polygon has corners of two types: corners in which the smaller angle (90°) is interior to the polygon are called convex and corners in which the larger angle (270°) is interior are called concave. [1] A knob is an edge whose two endpoints are convex corners. An antiknob is an edge whose two endpoints are concave corners. [1]
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 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 } .
If every internal angle of a simple polygon is less than a straight angle (π radians or 180°), then the polygon is called convex. In contrast, an external angle (also called a turning angle or exterior angle) is an angle formed by one side of a simple polygon and a line extended from an adjacent side. [1]: pp. 261–264
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 ...
In plane geometry, a lune (from Latin luna 'moon') is the concave-convex region bounded by two circular arcs. [1] It has one boundary portion for which the connecting segment of any two nearby points moves outside the region and another boundary portion for which the connecting segment of any two nearby points lies entirely inside the region.
Bretschneider's formula generalizes Brahmagupta's formula for the area of a cyclic quadrilateral, which in turn generalizes Heron's formula for the area of a triangle.. The trigonometric adjustment in Bretschneider's formula for non-cyclicality of the quadrilateral can be rewritten non-trigonometrically in terms of the sides and the diagonals e and f to give [2] [3]