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2. Rotations and reflections in two dimensions - Wikipedia

   en.wikipedia.org/wiki/Rotations_and_reflections...

   A rotation in the plane can be formed by composing a pair of reflections. First reflect a point P to its image P′ on the other side of line L 1. Then reflect P′ to its image P′′ on the other side of line L 2. If lines L 1 and L 2 make an angle θ with one another, then points P and P′′ will make an angle 2θ around point O, the ...

3. Point groups in two dimensions - Wikipedia

   en.wikipedia.org/wiki/Point_groups_in_two_dimensions

   The symmetry group of a square belongs to the family of dihedral groups, D n (abstract group type Dih n), including as many reflections as rotations. The infinite rotational symmetry of the circle implies reflection symmetry as well, but formally the circle group S 1 is distinct from Dih(S 1) because the latter explicitly includes the reflections.

4. Plane (mathematics) - Wikipedia

   en.wikipedia.org/wiki/Plane_(mathematics)

   In mathematics, a plane is a two-dimensional space or flat surface that extends indefinitely. A plane is the two-dimensional analogue of a point (zero dimensions), a line (one dimension) and three-dimensional space. When working exclusively in two-dimensional Euclidean space, the definite article is used, so the Euclidean plane refers to the ...

5. Spherical trigonometry - Wikipedia

   en.wikipedia.org/wiki/Spherical_trigonometry

   The angle A (respectively, B and C) may be regarded either as the dihedral angle between the two planes that intersect the sphere at the vertex A, or, equivalently, as the angle between the tangents of the great circle arcs where they meet at the vertex. Angles are expressed in radians.

6. Euclidean plane - Wikipedia

   en.wikipedia.org/wiki/Euclidean_plane

   In mathematics, a Euclidean plane is a Euclidean space of dimension two, denoted or . It is a geometric space in which two real numbers are required to determine the position of each point . It is an affine space , which includes in particular the concept of parallel lines .

7. Circular symmetry - Wikipedia

   en.wikipedia.org/wiki/Circular_symmetry

   In geometry, circular symmetry is a type of continuous symmetry for a planar object that can be rotated by any arbitrary angle and map onto itself. Rotational circular symmetry is isomorphic with the circle group in the complex plane , or the special orthogonal group SO(2), and unitary group U(1).

8. Reflection (mathematics) - Wikipedia

   en.wikipedia.org/wiki/Reflection_(mathematics)

   Step 1 (red): construct a circle with center at P and some fixed radius r to create points A′ and B′ on the line AB, which will be equidistant from P. Step 2 (green): construct circles centered at A′ and B′ having radius r. P and Q will be the points of intersection of these two circles. Point Q is then the reflection of point P through ...

9. Spherical lune - Wikipedia

   en.wikipedia.org/wiki/Spherical_lune

   The surface area of a spherical lune is 2θ R 2, where R is the radius of the sphere and θ is the dihedral angle in radians between the two half great circles. When this angle equals 2π radians (360°) — i.e., when the second half great circle has moved a full circle, and the lune in between covers the sphere as a spherical monogon — the ...