**Are you an EPFL student looking for a semester project?**

Work with us on data science and visualisation projects, and deploy your project as an app on top of GraphSearch.

Lecture# Product of 4 Plane Reflections: Normalization

Description

This lecture covers the concept of normalizing the product of 4 plane reflections, focusing on the decomposition of isometries in space and the construction of a screw motion through the composition of 4 specific plane reflections. The lecture explains the process step by step, emphasizing the order of transformations and the resulting screw motion or half screw motion. It also delves into the configurations of isometries defined by pairs of specific planes in a screw motion, detailing translations and rotations. The fundamental theorem of isometries in space is presented, showing how the product of 4 reflections relative to 4 planes equates to a screw motion defined by orthogonal planes. Various scenarios are explored, illustrating how the screw motion can degenerate into different transformations based on the alignment and angles of the planes involved.

Official source

This page is automatically generated and may contain information that is not correct, complete, up-to-date, or relevant to your search query. The same applies to every other page on this website. Please make sure to verify the information with EPFL's official sources.

In course

Instructor

Related concepts (50)

MATH-124: Geometry for architects I

Ce cours entend exposer les fondements de la géométrie à un triple titre :
1/ de technique mathématique essentielle au processus de conception du projet,
2/ d'objet privilégié des logiciels de concept

In linear algebra, a QR decomposition, also known as a QR factorization or QU factorization, is a decomposition of a matrix A into a product A = QR of an orthonormal matrix Q and an upper triangular matrix R. QR decomposition is often used to solve the linear least squares problem and is the basis for a particular eigenvalue algorithm, the QR algorithm. Any real square matrix A may be decomposed as where Q is an orthogonal matrix (its columns are orthogonal unit vectors meaning ) and R is an upper triangular matrix (also called right triangular matrix).

In numerical analysis and linear algebra, lower–upper (LU) decomposition or factorization factors a matrix as the product of a lower triangular matrix and an upper triangular matrix (see matrix decomposition). The product sometimes includes a permutation matrix as well. LU decomposition can be viewed as the matrix form of Gaussian elimination. Computers usually solve square systems of linear equations using LU decomposition, and it is also a key step when inverting a matrix or computing the determinant of a matrix.

In Euclidean geometry, an angle is the figure formed by two rays, called the sides of the angle, sharing a common endpoint, called the vertex of the angle. Angles formed by two rays are also known as plane angles as they lie in the plane that contains the rays. Angles are also formed by the intersection of two planes; these are called dihedral angles. Two intersecting curves may also define an angle, which is the angle of the rays lying tangent to the respective curves at their point of intersection.

In linear algebra, linear transformations can be represented by matrices. If is a linear transformation mapping to and is a column vector with entries, then for some matrix , called the transformation matrix of . Note that has rows and columns, whereas the transformation is from to . There are alternative expressions of transformation matrices involving row vectors that are preferred by some authors. Matrices allow arbitrary linear transformations to be displayed in a consistent format, suitable for computation.

Rotation or rotational motion is the circular movement of an object around a central line, known as axis of rotation. A plane figure can rotate in either a clockwise or counterclockwise sense around a perpendicular axis intersecting anywhere inside or outside the figure at a center of rotation. A solid figure has an infinite number of possible axes and angles of rotation, including chaotic rotation (between arbitrary orientations), in contrast to rotation around a axis.

Related lectures (184)

Isometries & Orientation in Modern Geometry

Explores true angle magnitude, reflections, isometries, and symmetries in modern geometry, with practical CAD applications.

Symmetry in Modern GeometryMATH-124: Geometry for architects I

Explores modern symmetry in geometry, focusing on the Klein bottle and different types of transformations.

Symmetry in Modern Geometry

Delves into modern geometry, covering transformations, isometries, and symmetries.

Symmetry in Space: RotororeflectionsMATH-124: Geometry for architects I

Explores rotororeflections, normalization of reflections, creation of mirrored pieces, and practical applications in design.

Symmetry in Modern Geometry

Explores the modern definition and practical applications of symmetry in geometry.