# Euclidean geometry and transformations pdf

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- Geometric Transformations of the Euclidean Plane
- Euclidean Geometry and Transformations PDF
- Geometric Transformations of the Euclidean Plane

*A Course in Modern Geometries pp Cite as. The presentation of non-Euclidean geometry in Chapter 2 was synthetic ; that is, figures were studied directly and without use of their algebraic representations.*

## Geometric Transformations of the Euclidean Plane

Rotation in mathematics is a concept originating in geometry. Any rotation is a motion of a certain space that preserves at least one point. It can describe, for example, the motion of a rigid body around a fixed point. A clockwise rotation is a negative magnitude so a counterclockwise turn has a positive magnitude. Mathematically, a rotation is a map. All rotations about a fixed point form a group under composition called the rotation group of a particular space.

But in mechanics and, more generally, in physics , this concept is frequently understood as a coordinate transformation importantly, a transformation of an orthonormal basis , because for any motion of a body there is an inverse transformation which if applied to the frame of reference results in the body being at the same coordinates.

For example, in two dimensions rotating a body clockwise about a point keeping the axes fixed is equivalent to rotating the axes counterclockwise about the same point while the body is kept fixed. These two types of rotation are called active and passive transformations. The rotation group is a Lie group of rotations about a fixed point. This common fixed point is called the center of rotation and is usually identified with the origin. The rotation group is a point stabilizer in a broader group of orientation-preserving motions.

A representation of rotations is a particular formalism, either algebraic or geometric, used to parametrize a rotation map. This meaning is somehow inverse to the meaning in the group theory. Rotations of affine spaces of points and of respective vector spaces are not always clearly distinguished. The former are sometimes referred to as affine rotations although the term is misleading , whereas the latter are vector rotations.

See the article below for details. A motion of a Euclidean space is the same as its isometry : it leaves the distance between any two points unchanged after the transformation. But a proper rotation also has to preserve the orientation structure. The " improper rotation " term refers to isometries that reverse flip the orientation. In the language of group theory the distinction is expressed as direct vs indirect isometries in the Euclidean group , where the former comprise the identity component.

Any direct Euclidean motion can be represented as a composition of a rotation about the fixed point and a translation. There are no non- trivial rotations in one dimension. In two dimensions , only a single angle is needed to specify a rotation about the origin — the angle of rotation that specifies an element of the circle group also known as U 1. Composition of rotations sums their angles modulo 1 turn , which implies that all two-dimensional rotations about the same point commute. Rotations about different points, in general, do not commute.

Any two-dimensional direct motion is either a translation or a rotation; see Euclidean plane isometry for details. Rotations in three-dimensional space differ from those in two dimensions in a number of important ways.

Rotations in three dimensions are generally not commutative , so the order in which rotations are applied is important even about the same point. Also, unlike the two-dimensional case, a three-dimensional direct motion, in general position , is not a rotation but a screw operation. Rotations about the origin have three degrees of freedom see rotation formalisms in three dimensions for details , the same as the number of dimensions.

A general rotation in four dimensions has only one fixed point, the centre of rotation, and no axis of rotation; see rotations in 4-dimensional Euclidean space for details. Instead the rotation has two mutually orthogonal planes of rotation, each of which is fixed in the sense that points in each plane stay within the planes. The rotation has two angles of rotation, one for each plane of rotation , through which points in the planes rotate. Rotations in four dimensions about a fixed point have six degrees of freedom.

A four-dimensional direct motion in general position is a rotation about certain point as in all even Euclidean dimensions , but screw operations exist also.

When one considers motions of the Euclidean space that preserve the origin , the distinction between points and vectors , important in pure mathematics, can be erased because there is a canonical one-to-one correspondence between points and position vectors.

The same is true for geometries other than Euclidean , but whose space is an affine space with a supplementary structure ; see an example below. Alternatively, the vector description of rotations can be understood as a parametrization of geometric rotations up to their composition with translations. In other words, one vector rotation presents many equivalent rotations about all points in the space. A motion that preserves the origin is the same as a linear operator on vectors that preserves the same geometric structure but expressed in terms of vectors.

For Euclidean vectors , this expression is their magnitude Euclidean norm. As it was already stated , a proper rotation is different from an arbitrary fixed-point motion in its preservation of the orientation of the vector space.

Thus, the determinant of a rotation orthogonal matrix must be 1. Matrices of all proper rotations form the special orthogonal group. Since complex numbers form a commutative ring , vector rotations in two dimensions are commutative, unlike in higher dimensions.

They have only one degree of freedom , as such rotations are entirely determined by the angle of rotation. The set of all appropriate matrices together with the operation of matrix multiplication is the rotation group SO 3. The matrix A is a member of the three-dimensional special orthogonal group , SO 3 , that is it is an orthogonal matrix with determinant 1. That it is an orthogonal matrix means that its rows are a set of orthogonal unit vectors so they are an orthonormal basis as are its columns, making it simple to spot and check if a matrix is a valid rotation matrix.

Above-mentioned Euler angles and axis—angle representations can be easily converted to a rotation matrix. Another possibility to represent a rotation of three-dimensional Euclidean vectors are quaternions described below. Unit quaternions , or versors , are in some ways the least intuitive representation of three-dimensional rotations.

They are not the three-dimensional instance of a general approach. They are more compact than matrices and easier to work with than all other methods, so are often preferred in real-world applications. A versor also called a rotation quaternion consists of four real numbers, constrained so the norm of the quaternion is 1.

This constraint limits the degrees of freedom of the quaternion to three, as required. Unlike matrices and complex numbers two multiplications are needed:.

The quaternion can be related to the rotation vector form of the axis angle rotation by the exponential map over the quaternions,. A single multiplication by a versor, either left or right , is itself a rotation, but in four dimensions. Any four-dimensional rotation about the origin can be represented with two quaternion multiplications: one left and one right, by two different unit quaternions.

More generally, coordinate rotations in any dimension are represented by orthogonal matrices. Matrices are often used for doing transformations, especially when a large number of points are being transformed, as they are a direct representation of the linear operator. Rotations represented in other ways are often converted to matrices before being used.

They can be extended to represent rotations and transformations at the same time using homogeneous coordinates. The main disadvantage of matrices is that they are more expensive to calculate and do calculations with.

Also in calculations where numerical instability is a concern matrices can be more prone to it, so calculations to restore orthonormality , which are expensive to do for matrices, need to be done more often. As was demonstrated above, there exist three multilinear algebra rotation formalisms: one with U 1 , or complex numbers , for two dimensions, and two others with versors, or quaternions , for three and four dimensions. In general even for vectors equipped with a non-Euclidean Minkowski quadratic form the rotation of a vector space can be expressed as a bivector.

This formalism is used in geometric algebra and, more generally, in the Clifford algebra representation of Lie groups. It can be conveniently described in terms of a Clifford algebra. For odd n , most of these motions do not have fixed points on the n -sphere and, strictly speaking, are not rotations of the sphere ; such motions are sometimes referred to as Clifford translations.

Affine geometry and projective geometry have not a distinct notion of rotation. One application of this [ clarification needed ] is special relativity , as it can be considered to operate in a four-dimensional space, spacetime , spanned by three space dimensions and one of time. In special relativity this space is linear and the four-dimensional rotations, called Lorentz transformations , have practical physical interpretations.

The Minkowski space is not a metric space , and the term isometry is inapplicable to Lorentz transformation. If a rotation is only in the three space dimensions, i.

But a rotation in a plane spanned by a space dimension and a time dimension is a hyperbolic rotation , a transformation between two different reference frames , which is sometimes called a "Lorentz boost". These transformations demonstrate the pseudo-Euclidean nature of the Minkowski space. The study of relativity is concerned with the Lorentz group generated by the space rotations and hyperbolic rotations.

Rotations define important classes of symmetry : rotational symmetry is an invariance with respect to a particular rotation. The circular symmetry is an invariance with respect to all rotation about the fixed axis. As was stated above, Euclidean rotations are applied to rigid body dynamics.

Moreover, most of mathematical formalism in physics such as the vector calculus is rotation-invariant; see rotation for more physical aspects. Euclidean rotations and, more generally, Lorentz symmetry described above are thought to be symmetry laws of nature.

In contrast, the reflectional symmetry is not a precise symmetry law of nature. These complex rotations are important in the context of spinors.

From Wikipedia, the free encyclopedia. This article needs additional citations for verification. Please help improve this article by adding citations to reliable sources. Unsourced material may be challenged and removed. For other uses, see Rotation. Main article: Rotation matrix.

Main article: Rotations and reflections in two dimensions. Main article: Rotation formalisms in three dimensions. See also: Three-dimensional rotation operator. Main article: Quaternions and spatial rotation. Main article: Lorentz transformation. See: point group.

## Euclidean Geometry and Transformations PDF

Euclidean geometry and transformations pdf Geometry can be defined as set of transformations of space. The Euclidean geometry is specified by including only the transformations. Euclidean transformations: translation and rotation. OpenGL matrix operations and arbitrary geometric transformations. Examples in. This introduction to Euclidean geometry emphasizes both the theory and the practical application of isometries and similarities to geometric transformations. In the third part of this book, we will look at Euclidean geometry from a different perspective, that of Euclidean transformations.

Rotation in mathematics is a concept originating in geometry. Any rotation is a motion of a certain space that preserves at least one point. It can describe, for example, the motion of a rigid body around a fixed point. A clockwise rotation is a negative magnitude so a counterclockwise turn has a positive magnitude. Mathematically, a rotation is a map.

A Course in Modern Geometries pp Cite as. The presentation of non-Euclidean geometry in Chapter 2 was synthetic , that is, figures were studied directly and without use of their algebraic representations. This reflects the manner in which both Euclidean and non-Euclidean geometries were originally developed. They realized that by assigning to each point in the plane an ordered pair of real numbers, algebraic techniques could be employed in the study of Euclidean geometry. This study of figures in terms of their algebraic representations by equations is known as analytic geometry.

This introduction to Euclidean geometry emphasizes both the theory and the practical application of isometries and similarities to geometric transformations.

## Geometric Transformations of the Euclidean Plane

It is concerned with properties of space that are related with distance, shape, size, and relative position of figures. Until the 19th century, geometry was almost exclusively devoted to Euclidean geometry , [a] which includes the notions of point , line , plane , distance , angle , surface , and curve , as fundamental concepts. During the 19th century several discoveries enlarged dramatically the scope of geometry. One of the oldest such discoveries is Gauss ' Theorema Egregium remarkable theorem that asserts roughly that the Gaussian curvature of a surface is independent from any specific embedding in an Euclidean space.

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#### Clayton W. Dodge

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