Types of Geometry

An Introduction to Euclidean, Affine, Projective, Inversive, Non-Euclidean, and Spherical Geometries

The Kleinian View of Geometry

by Aditya Mittal - August 19, 2004

In order to understand different types of Geometries we must first understand what a Geometry is and why we study it. First, let us answer the second question of "Why do we study Geometry?" We study geometry so that we can better understand the space around us. Therefore, any study that can help us better understand the space around us is Geometry. According to our belief we perceive space in different dimensions. We define a point as being 0 dimensional space, a line as being 1 dimensional space, a plane as being 2 dimensional space, and Space as being 3 dimensional space which we see around us everyday. Starting with the point we think of creating a series of infinite points to its left and to its right and we get a line. Taking a line and adding lines to its left and its right we get a plane and taking a plane and adding planes to its left and its right we get a Space. What do we get when we take a Space and add more Spaces to its left and right? In mathematics we refer to this by IR⁴as you should know if you are reading this. However, we do not intuitively understand IR⁴. However, the study of Geometry can help us build our intuition and better understand higher "spatial dimensions". We start out by analyzing shapes in the plane because it is easier to understand the properties of shapes in the plane than understanding surfaces in Space.

Now to study the plane there are a number of ways and each of these 'ways' is the study of Geometry. Now to people the easiest way to study the plane was to analyze the properties of lines and shapes in the plane. It is easier to understand the instances than the generalizations. So people started studying about the radii of circles, and lengths, and areas, and properties of parallel lines etc. These initial properties such as that segments have lengths, lines cross at angles one can measure, triangles are equilateral etc. became known as Euclidean Geometry. The Euclidean properties can be summarized within the transformation t(x) = Ux + a, where U is an orthogonal 2X2 matrix and a is an element of IR². Now Klein observed that Affine Geometry given by the transformation t(x) = Ax + b, where A is an invertible 2X2 matrix and b is an element of IR²does not preserve length. He noticed that we can define new Geometries by defining new transformations that do not necessarily preserve length or angle. This gave rise to his definition of Geometry in which a new Geometry can be created simply by defining a new transformation. However, this remained incomplete. He further observed that both affine Geometry and Euclidean Geometry form a "group". See "group theory." That is they are closed (G1), have identity (G2) and inverse (G3) elements, and have associativity (G4). In this sense he discovered that Euclidean Geometry is a Subgroup of Affine Geometry and preserves all the properties of affine transformations on lines, curves, and shapes. In the rest of this document I will mention the rest of the geometries and the transformations defining them, and the relations between them. Much more detailed analysis of each of these geometries is available on the internet and in various books. I shall list a few of these sources at the end of the article.

Euclidean Geometry

Euclidean Geometry preserves length and angle, is in the space IR² and is given by the transformation t(x) = Ux + a, where U is an orthogonal 2X2 matrix and a is an element of IR²

Euclidean Geometry is a subgroup of Affine Geometry.

Euclidean Geometry is what we have studied throughout our elementary Geometry courses.

Affine Geometry

Affine Geometry preserves ratios along parallel lines, is in the space IR² and is given by the transformation t(x) = Ax + b, where A is an invertible 2X2 matrix and b is an element of IR²

Affine Geometry is a subgroup of Projective Geometry.

Affine Geometry is used heavily in the development of computer graphics.

Projective Geometry

Projective Geometry preserves incidence, collinearity, and cross-ratio, is in the space IRIP² and is given by the transformation t(x) = Ax, where A is an invertible 3X3 matrix.

Here IRIP²= IR² + ideal Line

What is an Ideal Line?

In projective geometry projective points are actually lines in Euclidean space, and projective lines are planes in Euclidean figures. An ideal Line is a projective line or a plane in space. The ideal Line in Euclidean Space is a plane through the origin parallel to the plane IR² in consideration. This plane is composed of Euclidean lines known as the Ideal Points of the plane in consideration. It is usually helpful to denote the plane in consideration by a Greek letter so one might speak of "the ideal Line for pi."

Projective Geometry makes the study of conic sections simpler. It also helps us analyze 3D objects in 2D space much better as in artwork meant to give a 3D look.

Inversive Geometry

In inversive geometry we understand a line as a generalized circle connected at both ends at a point infinity in addition to the plane IR² so is in the space IR² U {¥}. The plane with the addition of the point at infinity is called the extended plane. A generalized circle in the extended plane is a set that is either a circle or an extended line.

Let C be a generalized circle in the extended complex plane. Then an inversion of the extended plane with respect to C is a function t defined by one of the following rules:

1) if C is a circle of radius r centered at O, then

t(A) = the inverse of A with respect to C, if A is an element of IR² - {O},

t(A) = ¥, if A = O,

t(A) = O, if A = ¥;

2) if C is an extended line L U {¥}, then

t(A) = the reflection of A in L, if A is an element of IR²

t(A) = ¥, if A = ¥.

Möbius Transformations

Möbius transformations, M are used to express Inversive transformations algebraically. Inversive transformations can take two forms, t(z) = M(z) or t(z) = M(z).

A Möbius transformation transforms from the extended complex plane to the extended complex plane M(z) = (az + b)/(cz + d), where a, b, c, d are elements of the extended complex plane and ad - bc is not zero. If c = 0, then we adopt the convention that M(¥) = ¥; otherwise, we adopt the convention that M(-d/c) = ¥ and M(¥) = a/c.

Möbius transformations can be represented as a 2X2 matrix with rows [a,b] and [c,d] and the determinant of this matrix will be non-zero as defined by ad - bc is not zero. This also ensures that a Möbius transformation is not a constant function since the numerator can not be a multiple of the denominator. Every Möbius transformation is an inversive transformation.

Möbius transformations preserve the magnitude and orientation of angles, and map generalized circles to generalized circles.

Reflections in mirrors are an inversion. Inversive Geometry is used heavily in the study of Optics.

Non-Euclidean

The space in which we deal with Non-Euclidean Geometry is the unit disc in 2 space. When we begin to deal with the "unit sphere" or "unit ball" in three space it becomes a part of Spherical Geometry. Because of our visualization of space Spherical Geometry makes more sense to us than Non-Euclidean Geometry. It is therefore argued that space might actually be Non-Euclidean.

In 2 space we deal with the unit disc. In this unit disc we draw "d-lines." These are lines that are connected to the boundary of the unit disc in 2 places and meet the boundary at right angles. Therefore, all diameters are d-lines. All other d-lines are curved. However, Klein noticed that we can linearize them by representing the right angles where they meet the boundary inaccurately without losing any information.

The linearization map L of the unit disc onto itself consists of the inverse of the stereographic projection map followed by the orthogonal projection map.

The non-linearized non-Euclidean lines in the unit disc are part of the so-called Poincaré model. The new linearized model is called the Beltrami-Klein model, after the Italian mathematician Eugenio Beltrami (1833-1900) who discovered it, and the German mathematician Felix Klein, who connected it to projective geometry.

This geometry can be used to generalize cross ratio from projective geometry to the Euclidean plane.

In this geometry triangles have angles that add up to less than 180 degrees and the sum of the angles of quadrilaterals add up to less than 360 degrees. These triangles and quadrilaterals have edges composed of d-lines in the unit disc. This geometry gives rise to different but corresponding theorems from Euclidean Geometry. Theorems about properties such as areas exist but are different.

This geometry helps us understand the designs created in the Kaleidoscopes if you ever remember playing with one. It also gives rise to neat tessellations between d-lines inside a circle.

Spherical Geometry

As you can guess this geometry deals with spheres. In 2 space with the surface of the sphere. A great circle is the circle cut out on the sphere's surface by a plane through the center of the sphere. We define latitude and longitude on the sphere's surface. Distances between points on the surface are given by the angle they subtend at the center of the sphere. Spherical coordinates are used.

Just like Non-Euclidean geometry spherical geometry has triangles whose angles add up to greater than 180 degrees. The edges of the triangle in spherical geometry is composed of parts of 3 great circles. Great circles are analogous to diameters that are a type of d-lines in Non-Euclidean Geometry. Creating such triangles we get Spherical Trignometry.

We can also connect projective geometry and spherical geometry to get images of projective conics on the surface of the unit ball. These images are the various conic sections. Some more study of projective and spherical geometries together reveals Möbius bands, which illustrate a significant mathematical point: 2-dimensional real projective space IRIP² is not orientable - that is, it cannot be given an everywhere consistent sense of left or right.