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  2. Rotations and reflections in two dimensions - Wikipedia

    en.wikipedia.org/wiki/Rotations_and_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 intersection of L 1 and L 2. I.e., angle ∠ POP′′ will measure 2θ.

  3. Axis–angle representation - Wikipedia

    en.wikipedia.org/wiki/Axisangle_representation

    The angle θ and axis unit vector e define a rotation, concisely represented by the rotation vector θe.. In mathematics, the axisangle representation parameterizes a rotation in a three-dimensional Euclidean space by two quantities: a unit vector e indicating the direction of an axis of rotation, and an angle of rotation θ describing the magnitude and sense (e.g., clockwise) of the ...

  4. Rotation formalisms in three dimensions - Wikipedia

    en.wikipedia.org/wiki/Rotation_formalisms_in...

    Using the x-convention, the 3-1-3 extrinsic Euler angles φ, θ and ψ (around the z-axis, x-axis and again the -axis) can be obtained as follows: = ⁡ (,) = ⁡ = ⁡ (,) Note that atan2( a , b ) is equivalent to arctan ⁠ a / b ⁠ where it also takes into account the quadrant that the point ( b , a ) is in; see atan2 .

  5. Isometric projection - Wikipedia

    en.wikipedia.org/wiki/Isometric_projection

    The term "isometric" comes from the Greek for "equal measure", reflecting that the scale along each axis of the projection is the same (unlike some other forms of graphical projection). An isometric view of an object can be obtained by choosing the viewing direction such that the angles between the projections of the x , y , and z axes are all ...

  6. Cartesian coordinate system - Wikipedia

    en.wikipedia.org/wiki/Cartesian_coordinate_system

    The orientation is usually chosen so that the 90-degree angle from the first axis to the second axis looks counter-clockwise when seen from the point (0, 0, 1); a convention that is commonly called the right-hand rule. The coordinate surfaces of the Cartesian coordinates (x, y, z). The z-axis is vertical and the x-axis is

  7. Orientation (geometry) - Wikipedia

    en.wikipedia.org/wiki/Orientation_(geometry)

    Changing orientation of a rigid body is the same as rotating the axes of a reference frame attached to it.. In geometry, the orientation, attitude, bearing, direction, or angular position of an object – such as a line, plane or rigid body – is part of the description of how it is placed in the space it occupies. [1]

  8. Euler's rotation theorem - Wikipedia

    en.wikipedia.org/wiki/Euler's_rotation_theorem

    A spatial rotation is a linear map in one-to-one correspondence with a 3 × 3 rotation matrix R that transforms a coordinate vector x into X, that is Rx = X. Therefore, another version of Euler's theorem is that for every rotation R , there is a nonzero vector n for which Rn = n ; this is exactly the claim that n is an eigenvector of R ...

  9. Reflection (mathematics) - Wikipedia

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

    A reflection through an axis. In mathematics, a reflection (also spelled reflexion) [1] is a mapping from a Euclidean space to itself that is an isometry with a hyperplane as the set of fixed points; this set is called the axis (in dimension 2) or plane (in dimension 3) of reflection.