📐Geometric Algebra Unit 8 – Reflections and Inversions

Key Concepts and Definitions

• Reflection involves transforming points, lines, or shapes across a line of reflection resulting in a mirror image
• Inversion transforms points with respect to a circle or sphere, mapping points inside the circle to points outside and vice versa
• The line of reflection acts as a perpendicular bisector for any line segment connecting a point to its reflected image
• Inversive geometry studies properties and transformations that are preserved under inversion, such as angles and circles
• The center of inversion is the center of the reference circle or sphere used for the inversion transformation
• The inverse of a point $P$ with respect to a circle with center $O$ and radius $r$ is the point $P'$ on the ray $\overrightarrow{OP}$ such that $OP \cdot OP' = r^2$
• Möbius transformations are a generalization of inversions in the complex plane, preserving circles and angles

Geometric Representation

• Reflections can be visualized as flipping a shape across a line, creating a mirror image on the opposite side
  • For example, reflecting a triangle across a vertical line results in a new triangle that appears to be a horizontal flip of the original
• Inversions can be represented geometrically using a reference circle or sphere
  • Points inside the circle are mapped to points outside the circle, while points outside the circle are mapped to points inside
• The line of reflection is perpendicular to any line segment connecting a point to its reflected image, intersecting at the midpoint
• Compass and straightedge constructions can be used to perform reflections and inversions geometrically
• Reflections preserve distance, meaning the distance between any two points is equal to the distance between their reflected images
• Inversions preserve angles, so the angle between any two curves is equal to the angle between their inverted images
• In three dimensions, reflections can occur across a plane, while inversions are performed with respect to a sphere

Properties of Reflections

• Reflections are isometric transformations, preserving distance and angle measures between points and lines
• The composition of two reflections across parallel lines results in a translation, with the translation vector perpendicular to the lines of reflection
• The composition of two reflections across intersecting lines results in a rotation, with the center of rotation at the intersection point and the angle of rotation twice the angle between the lines
• Reflections are involutive, meaning that reflecting a point or shape twice across the same line returns it to its original position
• Reflections are self-inverse transformations, as the inverse of a reflection is the reflection itself
• The fixed points of a reflection are the points lying on the line of reflection, as they remain unchanged by the transformation
• Reflections preserve orientation in even-dimensional spaces (2D, 4D, etc.) but reverse orientation in odd-dimensional spaces (3D, 5D, etc.)

Properties of Inversions

• Inversions are conformal transformations, preserving angles between curves but not necessarily distances
• The inversion of a line not passing through the center of inversion is a circle passing through the center, and vice versa
• The inversion of a circle not passing through the center of inversion is another circle, while a circle passing through the center is mapped to a line not passing through the center
• Inversion is an involutive transformation, as the inverse of the inverse of a point is the original point itself
• The center of inversion is the only fixed point under an inversion, as it remains unchanged by the transformation
• Inversions preserve the cross-ratio of four collinear points or four concyclic points
• The composition of two inversions with respect to concentric circles is equivalent to a dilation centered at the common center of the circles

Algebraic Formulations

• The reflection of a point $(x, y)$ across the line $y = mx + b$ is given by the formula $(x', y') = (x - \frac{2(y - mx - b)}{m^2 + 1}, y - \frac{2m(y - mx - b)}{m^2 + 1})$
• The reflection of a point $(x, y)$ across the x-axis is given by $(x, -y)$, while the reflection across the y-axis is $(-x, y)$
• The inversion of a point $(x, y)$ with respect to a circle centered at the origin with radius $r$ is given by $(\frac{r^2x}{x^2 + y^2}, \frac{r^2y}{x^2 + y^2})$
• The general formula for the inversion of a point $(x, y)$ with respect to a circle centered at $(a, b)$ with radius $r$ is $(\frac{r^2(x-a)}{(x-a)^2 + (y-b)^2} + a, \frac{r^2(y-b)}{(x-a)^2 + (y-b)^2} + b)$
• In complex notation, the inversion of a point $z$ with respect to a circle centered at $c$ with radius $r$ is given by $c + \frac{r^2}{\overline{z-c}}$
• Möbius transformations in the complex plane can be represented algebraically as $f(z) = \frac{az+b}{cz+d}$, where $a, b, c, d$ are complex constants with $ad-bc \neq 0$

Applications in Geometry

• Reflections are used in the study of symmetry, as symmetric shapes can be generated by reflecting a fundamental region across various lines of reflection
• Inversions are employed in the study of non-Euclidean geometries, such as hyperbolic geometry, where lines are represented by circles orthogonal to a fixed reference circle
• Compass and straightedge constructions often involve reflections and inversions to create geometric figures with specific properties
• In physics, reflections are used to model the behavior of light and sound waves, as well as the motion of objects undergoing elastic collisions
• Inversions are applied in the study of conformal maps, which are used to model complex physical phenomena such as fluid dynamics and electromagnetism
• Möbius transformations, a generalization of inversions, are used in complex analysis to study analytic functions and their properties
• Reflections and inversions are utilized in computer graphics to create realistic images and animations, such as reflections in mirrors or water surfaces

Connections to Other Mathematical Topics

• Reflections and inversions are closely related to the concept of isometries, which are distance-preserving transformations in geometry
• The composition of reflections leads to the study of reflection groups, which are important in abstract algebra and crystallography
• Inversions are a fundamental tool in the study of circle packings and circle configurations, which have applications in discrete geometry and graph theory
• Möbius transformations, a generalization of inversions, are central to the study of hyperbolic geometry and complex analysis
• Reflections and inversions are used in the study of knots and links, as they can be employed to simplify and classify these topological objects
• The properties of reflections and inversions are exploited in the study of minimal surfaces, which are surfaces with zero mean curvature
• Inversions are connected to the theory of quadratic differentials, which are used to study the geometry of surfaces and the behavior of dynamical systems

Problem-Solving Strategies

• When solving problems involving reflections, identify the line of reflection and use the properties of reflections to determine the position of the reflected points or shapes
• For problems involving inversions, identify the center and radius of the reference circle, and use the inversion formula to map points inside and outside the circle
• Utilize the geometric properties of reflections and inversions, such as the preservation of distances and angles, to simplify and solve problems
• Break down complex problems into simpler sub-problems by applying a sequence of reflections or inversions to transform the given configuration into a more manageable form
• Exploit the connections between reflections, inversions, and other geometric transformations, such as rotations and translations, to solve problems efficiently
• Use algebraic formulations of reflections and inversions to compute the coordinates of transformed points or to determine the equations of reflected or inverted curves
• Visualize and sketch the problem setup to gain insight into the geometric relationships and to guide the problem-solving process
• Apply the properties of reflections and inversions in conjunction with other mathematical tools, such as trigonometry, complex numbers, or vectors, to tackle advanced problems