|
In Euclidean geometry, the inversion of a point X in respect to a point P is a point X* such that P is the midpoint of the line segment with endpoints X and X*. In other words, the vector from X to P is the same as the vector from P to X*. Euclid Euclidean geometry is a mathematical system due to the Hellenistic mathematician Euclid of Egypt. ...
A spatial point is an entity with a location in space but no extent (volume, area or length). ...
In mathematics, a line segment is a part of a line that is bounded by two end points. ...
In physics and in vector calculus, a spatial vector is a concept characterized by a magnitude, which is a scalar, and a direction (which can be defined in a 3-dimensional space by the Euler angles). ...
The formula for the inversion in P is
- x*=2a−x
where a, x and x* are the position vectors of P, X and X* respectively.
This mapping is an isometric involutive affine transformation which has exactly one fixed point, which is P. Partial plot of a function f. ...
In mathematics, an isometry, isometric isomorphism or congruence mapping is a distance-preserving isomorphism between metric spaces. ...
In mathematics, an involution is a function that is its own inverse, so that f(f(x)) = x for all x in the domain of f. ...
In geometry, an affine transformation or affine map (from the Latin, affinis, connected with) between two vector spaces consists of a linear transformation followed by a translation: In the finite-dimensional case each affine transformation is given by a matrix A and a vector b, which can be written as...
In mathematics, a fixed point of a function f is an argument x such that f(x) = x; see fixed point (mathematics). ...
In odd-dimensional Euclidean space it does not preserve orientation, it is an indirect isometry. In mathematics, Euclidean space is a generalization of the 2- and 3-dimensional spaces studied by Euclid. ...
In mathematics, an orientation on a real vector space is a choice of which ordered bases are positively oriented (or right-handed) and which are negatively oriented (or left-handed). ...
In mathematics, the Euclidean group is the symmetry group associated with Euclidean geometry. ...
Geometrically in 3D it amounts to rotation about an axis through P by an angle of 180°, combined with reflection in the plane through P which is perpendicular to the axis; the result does not depend on the orientation (in the other sense) of the axis. Notations for the type of operation, or the type of group it generates, are  , Ci, S2, and 1×. The group type is one of the three symmetry group types in 3D without any pure rotational symmetry, see cyclic symmetries with n=1. Rotation of a plane, seen as the rotation of the terrain relative to the plane (exposure time 1. ...
Changing orientation is the same as moving the coordinate axes. ...
The symmetry group of an object (e. ...
Rotational symmetry is symmetry with respect to some or all rotations in m-dimensional Euclidean space. ...
This article deals with the four infinite series of point groups in three dimensions (n≥1) with n-fold rotational symmetry about one axis (rotation by an angle of 360°/n does not change the object), and no other rotational symmetry (n=1 covers the cases of no rotational symmetry...
The following point groups in three dimensions contain inversion: A discrete point group in 3D is a finite symmetry group in 3D that leaves the origin fixed. ...
- Cnh and Dnh for even n
- S2n and Dnd for odd n
- Th, Oh, and Ih
Closely related to inverse in a point is reflection in respect to a plane, which can be thought of as a "inversion in a plane". IT IS KNOWN AS MARK a lunitice insain int gw brain ...
In mathematics, a plane is the fundamental two-dimensional object. ...
Inversion with respect to the origin
Inversion with respect to the origin corresponds to additive inversion of the position vector, and also to scalar multiplication by −1. The operation commutes with every other linear transformation, but not with translation. "Inversion" without indicating "in a point", "in a line" or "in a plane", means this inversion, also called parity transformation. The additive inverse, or opposite, of a number n is the number which, when added to n, yields zero. ...
In mathematics, scalar multiplication is one of the basic operations defining a vector space in linear algebra (or more generally, a module in abstract algebra). ...
In mathematics, a linear transformation (also called linear map or linear operator) is a function between two vector spaces that preserves the operations of vector addition and scalar multiplication. ...
In Euclidean geometry, translation is a transformation of Euclidean space which moves every point by a fixed distance in the same direction. ...
In physics, a parity transformation (also called parity inversion) is the simultaneous flip in the sign of all spatial coordinates: A 3×3 matrix representation of P would have determinant equal to –1, and hence cannot reduce to a rotation. ...
See also |
Feeds |
Contact