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    Two Dimensional Geometric Model: Understanding and Applications in Computer Vision
    Two Dimensional Geometric Model: Understanding and Applications in Computer Vision
    Two Dimensional Geometric Model: Understanding and Applications in Computer Vision
    Ebook102 pages1 hour

    Two Dimensional Geometric Model: Understanding and Applications in Computer Vision

    By Fouad Sabry

    ()

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    About this ebook

    What is Two Dimensional Geometric Model


    A 2D geometric model is a geometric model of an object as a two-dimensional figure, usually on the Euclidean or Cartesian plane.


    How you will benefit


    (I) Insights, and validations about the following topics:


    Chapter 1: 2D geometric model


    Chapter 2: Dimension


    Chapter 3: Euclidean geometry


    Chapter 4: Topology


    Chapter 5: Vector graphics


    Chapter 6: 2D computer graphics


    Chapter 7: Geometric primitive


    Chapter 8: Discrete geometry


    Chapter 9: Constructive solid geometry


    Chapter 10: Geometric modeling


    (II) Answering the public top questions about two dimensional geometric model.


    (III) Real world examples for the usage of two dimensional geometric model in many fields.


    Who this book is for


    Professionals, undergraduate and graduate students, enthusiasts, hobbyists, and those who want to go beyond basic knowledge or information for any kind of Two Dimensional Geometric Model.

    LanguageEnglish
    PublisherOne Billion Knowledgeable
    Release dateMay 5, 2024
    Two Dimensional Geometric Model: Understanding and Applications in Computer Vision

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      Book preview

      Two Dimensional Geometric Model - Fouad Sabry

      Chapter 1: 2D geometric model

      A 2D geometric model is a representation of an object in two dimensions, typically on the Euclidean or Cartesian plane.

      Even though all material items are three-dimensional, a 2D geometric model is typically sufficient for flat objects such as paper cutouts and sheet metal machine parts. Other examples include circles meant to represent thunderstorms, which, when viewed from above, appear flat.

      simple geometric shapes

      boundary representation

      Boolean operations applied to polygons

      {End Chapter 1}

      Chapter 2: Dimension

      The dimension of a mathematical space (or object) is defined informally in physics and mathematics as the smallest number of coordinates required to specify any point within it. Consequently, a line has one dimension (1D) since only one coordinate is required to identify a point on it, such as the point at 5 on a number line. A surface, such as the boundary of a cylinder or sphere, has a dimension of two (2D) because two coordinates are necessary to specify a point on it — for instance, a latitude and longitude are needed to find a point on the surface of a sphere. A two-dimensional Euclidean space is a plane-based two-dimensional space. The interior of a cube, cylinder, or sphere is three-dimensional (3D) because locating a point within these areas requires three coordinates.

      Space and time are distinct entities in classical physics and refer to absolute space and time. This picture of the world is a four-dimensional space, but not the one required to explain electromagnetism. The four dimensions (4D) of spacetime consist of events that are not geographically nor temporally absolute, but rather are known relative to an observer's motion. The pseudo-Riemannian manifolds of general relativity describe spacetime with matter and gravity. Minkowski space first approximates the universe without gravity. The superstring theory has 10 dimensions (6D hyperspace + 4D), supergravity and M-theory have 11 dimensions (7D hyperspace + 4D), and the state-space of quantum mechanics is an infinite-dimensional function space.

      The dimension idea is not limited to physical items. There are frequent occurrences of high-dimensional spaces in mathematics and the sciences. They may be Euclidean spaces or more general parameter spaces or configuration spaces, as in Lagrangian or Hamiltonian mechanics; these are abstract spaces distinct from the physical space in which we reside.

      The dimension of an object in mathematics is, approximately speaking, the number of degrees of freedom of a moving point on the object. In other words, the dimension is the number of independent parameters or coordinates required to define the position of a confined point on the object. For instance, the dimension of a point is zero; the dimension of a line is one since a point can only move along a line in one direction (or its opposite); the dimension of a plane is two, etc.

      The dimension of an object is an intrinsic property in the sense that it is independent of the dimension of the space in which the thing is or can be embedded. Curves, such as circles, have one dimension because the position of a point on a curve is determined by its signed distance along the curve from a fixed point on the curve. This is independent of the fact that, unless it is a line, a curve cannot be embedded in a Euclidean space of dimension less than two.

      The dimension of Euclidean n-space En is n.

      When attempting to generalize to various sorts of spaces, it is important to include, one is faced with the question what makes En n-dimensional? One answer is that to cover a fixed ball in En by small balls of radius ε, one needs on the order of ε−n such small balls.

      This insight results in the definition of the Minkowski dimension and a more complex variation, the Hausdorff distance, However, there are additional responses to that question.

      For example, the boundary of a ball in En looks locally like En-1 and this leads to the notion of the inductive dimension.

      While these notions agree on En, When one examines more broad spaces, they turn out to be different.

      A tesseract is an example of an object with four dimensions. Mathematicians typically express this as The tesseract has dimension 4 or The dimension of the tesseract is 4 or 4D, although outside of mathematics the term dimension is typically used as A tesseract has four dimensions..

      Although the notion of higher dimensions goes back to René Descartes, The significant development of higher-dimensional geometry did not commence until the 19th century, by the efforts of Arthur Cayley, Henry Rowan Hamilton, Ludwig Schläfli and Bernhard Riemann.

      1854's Habilitationsschrift by Riemann, Schläfli's 1852 Theorie der vielfachen Kontinuität, and John T. Hamilton's discovery of quaternions.

      Graves' 1843 discovery of octonions signaled the start of higher-dimensional geometry.

      This part continues with a discussion of some of the most important mathematical definitions of dimension.

      A vector space's dimension is the number of vectors in any basis for the space, or the number of coordinates required to express any vector. This concept of dimension (the cardinality of a basis) is frequently referred to as the Hamel dimension or algebraic dimension to differentiate it from other concepts of dimension.

      For the non-free situation, this generalizes to the concept of a module's length.

      It is possible to calculate the uniquely specified dimension of every connected topological manifold. Locally, a connected topological manifold is homeomorphic to Euclidean n-space, where n is the dimension of the manifold.

      At any point, the dimension of linked differentiable manifolds is also the dimension of the tangent vector space.

      Within the field of geometric topology, Manifold theory is defined by the relative simplicity of dimensions 1 and 2, The high-dimensional examples with n > 4 are simplified by providing additional work space; Moreover, instances n = 3 and 4 are in some ways the

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