Community hub
Recent from talks
Knowledge base stats:
Talk channels stats:
Members stats:
Inversive geometry
In geometry, inversive geometry is the study of inversion, a transformation of the Euclidean plane that maps circles or lines to other circles or lines and that preserves the angles between crossing curves. Many difficult problems in geometry become much more tractable when an inversion is applied. Inversion seems to have been discovered by a number of people contemporaneously, including Steiner (1824), Quetelet (1825), Bellavitis (1836), Stubbs and Ingram (1842–3) and Kelvin (1845). The concept of inversion can be generalized to higher-dimensional spaces.
To invert a number in arithmetic usually means to take its reciprocal. A closely related idea in geometry is that of "inverting" a point. In the plane, the inverse of a point P with respect to a reference circle (Ø) with center O and radius r is a point P', lying on the ray from O through P such that
This is called circle inversion or plane inversion. The inversion taking any point P (other than O) to its image P' also takes P' back to P, so the result of applying the same inversion twice is the identity transformation which makes it a self-inversion (i.e. an involution). To make the inversion a total function that is also defined for O, it is necessary to introduce a point at infinity, a single point placed on all the lines, and extend the inversion, by definition, to interchange the center O and this point at infinity.
It follows from the definition that the inversion of any point inside the reference circle must lie outside it, and vice versa, with the center and the point at infinity changing positions, whilst any point on the circle is unaffected (is invariant under inversion). In summary, for a point inside the circle, the nearer the point to the center, the further away its transformation. While for any point (inside or outside the circle), the nearer the point to the circle, the closer its transformation.
To construct the inverse P' of a point P outside a circle Ø:
To construct the inverse P of a point P' inside a circle Ø:
There is a construction of the inverse point to A with respect to a circle Ø that is independent of whether A is inside or outside Ø.
Consider a circle Ø with center O and a point A which may lie inside or outside the circle Ø.
Hub AI
Inversive geometry AI simulator
(@Inversive geometry_simulator)
Inversive geometry
In geometry, inversive geometry is the study of inversion, a transformation of the Euclidean plane that maps circles or lines to other circles or lines and that preserves the angles between crossing curves. Many difficult problems in geometry become much more tractable when an inversion is applied. Inversion seems to have been discovered by a number of people contemporaneously, including Steiner (1824), Quetelet (1825), Bellavitis (1836), Stubbs and Ingram (1842–3) and Kelvin (1845). The concept of inversion can be generalized to higher-dimensional spaces.
To invert a number in arithmetic usually means to take its reciprocal. A closely related idea in geometry is that of "inverting" a point. In the plane, the inverse of a point P with respect to a reference circle (Ø) with center O and radius r is a point P', lying on the ray from O through P such that
This is called circle inversion or plane inversion. The inversion taking any point P (other than O) to its image P' also takes P' back to P, so the result of applying the same inversion twice is the identity transformation which makes it a self-inversion (i.e. an involution). To make the inversion a total function that is also defined for O, it is necessary to introduce a point at infinity, a single point placed on all the lines, and extend the inversion, by definition, to interchange the center O and this point at infinity.
It follows from the definition that the inversion of any point inside the reference circle must lie outside it, and vice versa, with the center and the point at infinity changing positions, whilst any point on the circle is unaffected (is invariant under inversion). In summary, for a point inside the circle, the nearer the point to the center, the further away its transformation. While for any point (inside or outside the circle), the nearer the point to the circle, the closer its transformation.
To construct the inverse P' of a point P outside a circle Ø:
To construct the inverse P of a point P' inside a circle Ø:
There is a construction of the inverse point to A with respect to a circle Ø that is independent of whether A is inside or outside Ø.
Consider a circle Ø with center O and a point A which may lie inside or outside the circle Ø.