|
In mathematics, a transformation in elementary terms is any of a variety of different functions from geometry, such as rotations, reflections and translations. These can be carried out in Euclidean space, particularly in dimensions 2 and 3. They are also operations that can be performed using linear algebra, and explicitly using matrix theory. For example, in 3D computer graphics, the operations of moving, scaling, or rotating a 3D model are commonly called transformations. In 2D computer graphics, transformations commonly available include rotation, translation, reflection and shear (usually referred to as skew). Euclid, Greek mathematician, 3rd century BC, known today as the father of geometry; shown here in a detail of The School of Athens by Raphael. ...
Partial plot of a function f. ...
Table of Geometry, from the 1728 Cyclopaedia. ...
A sphere rotating around its axis. ...
IT IS KNOWN AS MARK a lunitice insain int gw brain ...
In Euclidean geometry, translation is a transformation of Euclidean space which moves every point by a fixed distance in the same direction. ...
Around 300 BC, the Greek mathematician Euclid laid down the rules of what has now come to be called Euclidean geometry, which is the study of the relationships between angles and distances in space. ...
Linear algebra is the branch of mathematics concerned with the study of vectors, vector spaces (also called linear spaces), linear transformations, and systems of linear equations. ...
Matrix theory is a branch of mathematics which focuses on the study of matrices. ...
A 3D rendering with raytracing and ambient occlusion using Blender and Yafray 3D computer graphics are works of graphic art created with the aid of digital computers and 3D software. ...
2D computer graphics is the computer-based generation of digital images—mostly from two-dimensional models (such as 2D geometric models, text, and digital images) and by techniques specific to them. ...
Rotation -
An arbitrary rotation with a given point fixed is given by the formula for a rotation about an axis through the origin; just add an arbitrary translation to get an arbitrary move of a rigid object. It can be decomposed into rotations about three fixed axes through that point, in terms of flight dynamics pitch, roll, and yaw. See also degrees of freedom (engineering). In the three-dimensional space, the possible moves of a rigid body are rotations and translations. ...
In the three-dimensional space, the possible moves of a rigid body are rotations and translations. ...
Flight dynamics is the study of orientation of air and space vehicles and how to control the critical flight parameters, typically named pitch, roll and yaw. ...
In mechanical engineering, aeronautical engineering and robotics, degrees of freedom (DOF) describes flexibility of motion. ...
Translation -
A translation, or translation operator, is an affine transformation of Euclidean space which moves every point by a fixed distance in the same direction. It can also be interpreted as the addition of a constant vector to every point, or as shifting the origin of the coordinate system. In other words, if v is a fixed vector, then the translation Tv will work as Tv(p) = p + v. In Euclidean geometry, translation is a transformation of Euclidean space which moves every point by a fixed distance in the same direction. ...
In geometry, an affine transformation or affine map (from the Latin, affinis, connected with) between two vector spaces consists of a linear transformation followed by a translation: In the finite-dimensional case each affine transformation is given by a matrix A and a vector b, which can be written as...
Around 300 BC, the Greek mathematician Euclid laid down the rules of what has now come to be called Euclidean geometry, which is the study of the relationships between angles and distances in space. ...
In mathematics, a vector space (or linear space) is a collection of objects (called vectors) that, informally speaking, may be scaled and added. ...
In mathematics, the origin of a coordinate system is the point where the axes of the system intersect. ...
In mathematics as applied to geometry, physics or engineering, a coordinate system is a system for assigning a tuple of numbers to each point in an n-dimensional space. ...
Reflection -
A reflection is a map that transforms an object into its mirror image. For example, a reflection of the small English letter p in respect to a vertical line would look like q. In order to reflect a planar figure one needs the "mirror" to be a line ("axis of reflection"), while for reflections in the three-dimensional space one would use a plane for a mirror. Reflection sometimes is considered as a special case of inversion with infinite radius of the reference circle. IT IS KNOWN AS MARK a lunitice insain int gw brain ...
Partial plot of a function f. ...
This article is about the Twilight Zone episode. ...
Two intersecting planes in three-dimensional space In mathematics, a plane is a fundamental two-dimensional object. ...
In mathematics, inversive geometry is the geometry of circles and the set of transformations that map all circles into circles, where by a circle one may also mean a line (a circle with infinite radius). ...
Glide translation -
 Example of a glide reflection A glide reflection is a type of isometry of the Euclidean plane: the combination of a reflection in a line and a translation along that line. Reversing the order of combining gives the same result. Depending on context, we may consider a reflection a special case, where the translation vector is the zero vector. Example of a glide reflection In geometry, a glide reflection is a type of isometry of the Euclidean plane. ...
Image File history File links Glide_reflection. ...
Image File history File links Glide_reflection. ...
In mathematics, an isometry, isometric isomorphism or congruence mapping is a distance-preserving isomorphism between metric spaces. ...
In mathematics and astronomy, Euclidean space is a generalization of the 2- and 3-dimensional spaces studied by Euclid. ...
IT IS KNOWN AS MARK a lunitice insain int gw brain ...
In Euclidean geometry, translation is a transformation of Euclidean space which moves every point by a fixed distance in the same direction. ...
Scaling -
Uniform scaling is a linear transformation that enlarges or diminishes objects; the scale factor is the same in all directions; it is also called a homothety. The result of uniform scaling is similar (in the geometric sense) to the original. In Euclidean geometry, uniform scaling is a linear transformation that enlarges or diminishes objects; the scale factor is the same in all directions; it is also called a homothety. ...
In mathematics, a linear transformation (also called linear map or linear operator) is a function between two vector spaces that preserves the operations of vector addition and scalar multiplication. ...
A scale factor is a number which scales some quantity. ...
In mathematics, a homothety (or homothecy) is a transformation of space which dilates distances with respect to a fixed point called the origin. ...
// Two geometrical objects are called similar if one is congruent to the result of a uniform scaling (enlarging or shrinking) of the other. ...
More general is scaling with a separate scale factor for each axis direction; a special case is directional scaling (in one direction). Shapes not aligned with the axes may be subject to shear (see below) as a side effect: although the angles between lines parallel to the axes are preserved, other angles are not.
Shear -
Shear is a transform that effectively rotates one axis so that the axes are no longer perpendicular. Under shear, a rectangle becomes a parallelogram, and a circle becomes an ellipse. Even if lines parallel to the axes stay the same length, others do not. In this shear transformation of an image of the Mona Lisa, the picture was deformed in such a way that its central vertical axis was not modified. ...
In geometry, a rectangle is defined as a quadrilateral where all four of its angles are right angles. ...
A parallelogram. ...
Circle illustration In Euclidean geometry, a circle is the set of all points in a plane at a fixed distance, called the radius, from a fixed point, the centre. ...
The ellipse and some of its mathematical properties. ...
More generally -
More generally, a transformation in mathematics is one facet of the mathematical function; the term mapping is also used in ways that are quite close synonyms. A transformation is, most often, an invertible function from a set X to itself; but this is not always assumed. In a sense the term transformation only flags that a function's more geometric aspects are being considered (for example, with attention paid to invariants). In mathematics, a linear transformation (also called linear map or linear operator) is a function between two vector spaces that preserves the operations of vector addition and scalar multiplication. ...
Partial plot of a function f. ...
In mathematics and related technical fields, the term map or mapping is often a synonym for function. ...
In mathematics, an inverse function is in simple terms a function which does the reverse of a given function. ...
In mathematics, an invariant is something that does not change under a set of transformations. ...
A strong non-linear transformation applied to a plane through the origin |
|
| | Before | After | In mathematics, a linear transformation (also called linear map or linear operator) is a function between two vector spaces that preserves the operations of vector addition and scalar multiplication. ...
Image File history File links Download high-resolution version (842x716, 28 KB) This image is part of a collection of 2 plots of mathemical function (other: image:Transformation_(after). ...
Image File history File links Download high-resolution version (996x798, 27 KB) This image is part of a collection of 2 plots of mathemical functions (other: image:Transformation_(before). ...
See also |
Feeds |
Contact