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Internal angle (degrees) 90° (for square and rectangle) In geometry a quadrilateral is a four-sided polygon, having four edges (sides) and four corners (vertices). The word is derived from the Latin words quadri, a variant of four, and latus, meaning "side".
Equivalently, a convex quadrilateral is cyclic if and only if each exterior angle is equal to the opposite interior angle. In 1836 Duncan Gregory generalized this result as follows: Given any convex cyclic 2n-gon, then the two sums of alternate interior angles are each equal to (n-1). [4]
Quaternion to Euler angles (in 3-2-1 sequence) conversion. A direct formula for the conversion from a quaternion to Euler angles in any of the 12 possible sequences exists. [ 2 ] For the rest of this section, the formula for the sequence Body 3-2-1 will be shown. If the quaternion is properly normalized, the Euler angles can be obtained from ...
Ptolemy's theorem. Ptolemy's theorem is a relation among these lengths in a cyclic quadrilateral. In Euclidean geometry, Ptolemy's theorem is a relation between the four sides and two diagonals of a cyclic quadrilateral (a quadrilateral whose vertices lie on a common circle). The theorem is named after the Greek astronomer and mathematician ...
The Euler or Tait–Bryan angles (α, β, γ) are the amplitudes of these elemental rotations. For instance, the target orientation can be reached as follows (note the reversed order of Euler angle application): The XYZ system rotates about the z axis by γ. The X axis is now at angle γ with respect to the x axis.
A spatial rotation around a fixed point of θ radians about a unit axis X Y Z that denotes the Euler axis is given by the quaternion C X S Y S Z S , where C cos θ and S sin θ . Compared to rotation matrices, quaternions are more compact, efficient, and numerically stable. Compared to Euler angles, they are simpler to compose.
The sum of the internal angle and the external angle on the same vertex is π radians (180°). The sum of all the internal angles of a simple polygon is π (n −2) radians or 180 (n –2) degrees, where n is the number of sides. The formula can be proved by using mathematical induction: starting with a triangle, for which the angle sum is 180 ...
Let θ 1 be the angle spanned by one side of the cyclic polygon as viewed from the center of the circumscribing circle. Similarly define the central angles θ 2, ..., θ n for the remaining n − 1 sides. Every Heronian triangle and every Brahmagupta quadrilateral has a rational value for the tangent of the quarter angle, tan θ k /4, for every ...