Inversion transformations are a powerful tool in Geometric Algebra, flipping points inside out relative to a sphere or plane. They're like a geometric magic trick, turning lines into circles and preserving angles while warping space in fascinating ways.

These transformations connect to the broader theme of reflections by extending the concept to curved surfaces. They're essential for solving complex geometric problems and understanding non-Euclidean geometries, making them a key player in this chapter's exploration of spatial manipulations.

Inversion Transformations in Geometric Algebra

Definition and Properties

• Inversion is a non-linear transformation that maps points to their inverse with respect to a given sphere or a plane in a Geometric Algebra space
• The inversion transformation is defined using the geometric product and the inverse of a vector in the Geometric Algebra framework
  • Example: Given a vector $a$ and a sphere with center $p$ and radius $r$, the inversion of $a$ with respect to the sphere is given by $a' = p + \frac{r^2}{(a - p)^2}(a - p)$
• Inversion transformations are involutory, meaning that applying the same inversion twice results in the original point or object
  • Example: If $a'$ is the inversion of $a$ with respect to a sphere, then the inversion of $a'$ with respect to the same sphere is $a$
• The inversion transformation preserves angles between curves and surfaces, making it a conformal transformation
  • Example: If two curves intersect at an angle $\theta$ before inversion, they will intersect at the same angle $\theta$ after inversion
• Inversion transformations are not affine transformations, as they do not preserve lines or planes in general
  • Example: A line that does not pass through the center of the inversion sphere will be transformed into a circle after inversion

Geometric Algebra Formulation

• The inversion transformation can be expressed using the language of Geometric Algebra, which provides a unified framework for describing geometric transformations
• In Geometric Algebra, vectors are treated as fundamental objects, and the geometric product is used to combine vectors and perform transformations
  • Example: The geometric product of two vectors $a$ and $b$ is given by $ab = a \cdot b + a \wedge b$, where $a \cdot b$ is the inner product and $a \wedge b$ is the outer product
• The inverse of a vector in Geometric Algebra is defined using the reverse operation, which allows for the formulation of the inversion transformation
  • Example: The inverse of a vector $a$ is given by $a^{-1} = \frac{a}{a^2}$, where $a^2$ is the squared magnitude of $a$

Properties of Inversion Transformations

Mapping of Geometric Objects

• Inversion transformations map spheres and planes to spheres and planes, respectively, in the Geometric Algebra space
  • Example: A sphere that does not pass through the center of the inversion sphere will be transformed into another sphere after inversion
• Inversion transformations map circles and lines to circles or lines, depending on their relationship to the inversion sphere or plane
  • Example: A circle that lies on a plane perpendicular to the inversion sphere's radius at the center will be transformed into a line after inversion
• The center of the inversion sphere or the point of intersection with the inversion plane remains fixed under the inversion transformation
  • Example: If the inversion sphere has center $p$, then the point $p$ remains unchanged after inversion

Compositional Properties

• Inversion transformations are self-inverse, meaning that the inverse of an inversion transformation is itself
  • Example: If $I$ is an inversion transformation, then $I^{-1} = I$
• The composition of two inversion transformations with respect to different spheres or planes results in a conformal transformation, such as a translation, rotation, or dilation
  • Example: The composition of two inversions with respect to spheres of equal radius results in a translation
• Inversion transformations preserve the cross-ratio of four points, which is a projective invariant
  • Example: If $a$, $b$, $c$, and $d$ are four points in the Geometric Algebra space, then the cross-ratio $(a, b; c, d) = \frac{(a - c)(b - d)}{(a - d)(b - c)}$ remains invariant under inversion

Applying Inversion Transformations

Transforming Geometric Objects

• Apply inversion transformations to points, lines, planes, circles, and spheres in the Geometric Algebra framework
  • Example: To invert a point $a$ with respect to a sphere with center $p$ and radius $r$, use the formula $a' = p + \frac{r^2}{(a - p)^2}(a - p)$
• Analyze the properties of the transformed objects, such as their position, orientation, and relationship to the original objects
  • Example: After inverting a circle with respect to a sphere, determine whether the resulting object is a circle or a line, and find its center and radius (if applicable)
• Determine the inversion sphere or plane given the original and transformed objects
  • Example: Given a pair of circles that are transformed into each other by an inversion, find the center and radius of the inversion sphere

Problem Solving with Inversions

• Use inversion transformations to solve geometric problems, such as finding the intersection of circles or spheres, or constructing tangent lines or planes to curves or surfaces
  • Example: To find the intersection points of two circles, invert both circles with respect to a sphere centered at one of their intersection points, solve the resulting problem for the transformed circles, and then invert the solution back to the original space
• Apply inversion transformations to study the properties of non-Euclidean geometries, such as hyperbolic and elliptic geometry
  • Example: In hyperbolic geometry, inversion transformations with respect to hyperbolic lines or circles can be used to study the properties of hyperbolic triangles and other geometric objects

Inversion Transformations vs Other Transformations

Composing with Other Transformations

• Compose inversion transformations with translations, rotations, dilations, and reflections to create complex geometric transformations
  • Example: To create a Möbius transformation, compose an inversion with respect to a sphere, followed by a translation, rotation, and another inversion with respect to a different sphere
• Use the combination of inversion transformations and other geometric transformations to simplify and solve geometric problems
  • Example: To find the image of a point under a Möbius transformation, first apply the inversion, translation, rotation, and second inversion separately, and then combine the results

Conformal Transformations

• Apply the composition of inversion transformations and other geometric transformations to study the properties of Möbius transformations and their applications in complex analysis and conformal field theory
  • Example: Möbius transformations are used to study the properties of meromorphic functions and to solve problems in complex analysis, such as finding the image of a region under a complex function
• Investigate the relationship between inversion transformations and other conformal transformations, such as the Liouville transformations and the Darboux transformations
  • Example: Liouville transformations, which are conformal transformations in Riemannian geometry, can be studied using the language of Geometric Algebra and inversion transformations
• Use inversion transformations in combination with other geometric transformations to study the properties of minimal surfaces and their applications in physics and engineering
  • Example: Minimal surfaces, such as the catenoid and the helicoid, can be analyzed using inversion transformations and other conformal transformations to study their geometric and physical properties