Mathematics
Projections
In mathematics, projections refer to the transformation of points or objects onto a lower-dimensional space. This process involves capturing the essential features of the original object while reducing its complexity. Projections are widely used in various mathematical fields, such as linear algebra, geometry, and computer graphics, to simplify and analyze complex structures.
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eBook - PDFEngineering Design Graphics
Sketching, Modeling, and Visualization
- James M. Leake, Molly Hathaway Goldstein, Jacob L. Borgerson(Authors)
- 2022(Publication Date)
- Wiley(Publisher)
CHAPTER 79 5 ❚ PLANAR Projections Introduction Projection is the process of reproducing a spatial object on a plane, curved surface, or line by projecting its points. Common examples of projection include photography, where a 3D scene is projected onto a 2D medium, and map projection, where the earth is projected onto a cylinder, a cone, or a plane in order to create a map. Planar projection figures prominently in both engineering and computer graphics. For our purposes, a projection is a mapping of a three- dimensional (3D) space onto a two-dimensional subspace (i.e., a plane). The word projection also refers to the two-dimensional (2D) image result- ing from such a mapping. Every planar projection includes the follow- ing elements: • The 3D object (or scene) to be projected • Sight lines (called projectors) passing through each point on the object • A 2D projection plane 1 • The projected 2D image that is formed on the projection plane These elements are indicated in Figure 5-1. The projection is formed by plotting piercing points created by the intersection of the projec- tors with the projection plane. By mapping these points onto the projection plane, the 2D image is formed. In effect, three-dimensional information is collapsed onto a plane. The Albrecht Dürer drawing shown in Figure 5-2 further illustrates these projection ele- ments. The lute lying on the table is the object, the piece of string is a (movable) projector, and the frame through which the artist looks is the pro- jection plane. Two movable wires are mounted on the frame, allowing the artist to identify a single piercing point. A piece of paper hinged to the frame serves as the basis for the projected PLANAR Projections AND PICTORIAL VIEWS 1 Although commonly represented as a bounded rectangle, the projection plane is infinite in extent. Object Projector Projected image Projection plane Figure 5-1 Elements of a planar projection CHAPTER 5 PLANAR Projections AND PICTORIAL VIEWS 80 image.
eBook - PDFGraphics and Visualization
Principles & Algorithms
- T. Theoharis, Georgios Papaioannou, Nikolaos Platis, Nicholas M. Patrikalakis(Authors)
- 2008(Publication Date)
- A K Peters/CRC Press(Publisher)
These objects are unified in WCS where they are placed suitably modified; the WCS is essentially used to define the model of a three-dimensional synthetic world. The transition from WCS to ECS, which involves a change of coordinates, is carried out in order to simplify a num-ber of operations including culling (e.g., the specification of the clipping bounds by the user) and projection. Finally, the transition from ECS to CSS ensures that all objects that survived culling will be defined in a canonical space (usually rang-ing from − 1 to 1) that can easily be scaled to the actual coordinates of any display device or viewport and that also maintains high floating-point accuracy. 4.2 Projections In mathematics, projection is a term used to describe techniques for the creation of the image of an object onto another simpler object such as a line, plane, or 4.2. Projections 119 Property preserved Affine Projective Angles No No Distances No No Ratios of distances Yes No Parallel lines Yes No Affine combinations Yes No Straight lines Yes Yes Cross ratios Yes Yes Table 4.1. Properties of affine transformations and projective mappings. surface. A center of projection , along with points on the object being projected, is used to define the projector lines; see Figure 4.3. The intersection of a projector with the simpler object (e.g., the plane of projection) forms the image of a point of the original object. Projections can be defined in spaces of arbitrary dimension. In computer graphics and visualization we are generally concerned with pro-jections from 3D space onto 2D space (the 2D space is referred to as the plane of projection and models our 2D output device). Two such Projections are of interest: • Perspective projection , where the distance of the center of projection from the plane of projection is finite; • Parallel projection , where the distance of the center of projection from the plane of projection is infinite.
eBook - PDF- D.H. Maling(Author)
- 2013(Publication Date)
- Pergamon(Publisher)
C H A P T E R 5 Some basic ideas about the mathematics of map Projections This is why Elastoplast which stretches is a better fit than ordinary Elastoplast for cuts on knuckles and knees. Jeremy Gray, Ideas of Space, 1979 Introduction A map projection may be defined as: * any systematic arrangement of meridians and parallels portraying the curved surface of the sphere or spheroid upon a plane. For many purposes in the present book, // will suffice to regard the earth as a perfect sphere. This has the advantage of being mathematically simpler to understand without losing sight of any of the salient problems which have to be tackled. The main exceptions to the use of the spherical assumption come in Chapters 15, 16 and 19, where the specialised uses of Projections in surveying and topographical cartography are considered. It was stated in Chapter 2 (p. 28) that every map projection is a form of coordinate representation upon the plane, and that its graticule intersections may be located by means of either cartesian or polar coor-dinates. In other words, each point on the earth's surface, with geo-graphical coordinates (φ, λ) may be reproduced on the plane by a point located in either the (x, y) or (r, Θ) systems of plane coodinates. Functional relationships We may express this idea in the generalized form of functional relation-ships (or functions) and write ^=f^{φΛ) (5.01) y =f2 {φΛ) (5.02) 80 Mathematics of map Projections 81 or τ =/Αφ.λ) (5.03) θ=Μφ,λ) (5.04) These expressions are the mathematical shorthand for statements such as 'x is a function of latitude and longitude', etc. The suffices/1-/4 indicate that these are different functions. Thus we may distinguish between (5.01) and (5.02) by the statement 'whereas χ is one function of both latitude and longitude, y is a different function of these variables'. We can further state that in (5.01), χ is the dependent variable which is a function of two independent variables, φ and A.
eBook - ePub- Ranjan Parekh(Author)
- 2019(Publication Date)
- CRC Press(Publisher)
9Projection
9.1 Introduction
Projection is used to map a higher-dimensional object to a lower-dimensional view i.e. from 2D to 1D or from 3D to 2D. The lower-dimensional entity is called a viewline or viewplane, respectively. In this chapter, we will mostly discuss about projection of 3D objects on to a 2D viewplane but we will introduce the concepts using a 2D projection on a 1D viewline and then extend the concepts for the 3D case.Projection can be of two types: parallel and perspective. In parallel projection, projection lines are parallel to each other. This type of projection produces unrealistic views in the sense that it is not what a viewer in the physical world would see as the apparent size of an object would not depend on its distance from the viewer, and all things near and far would appear at their true sizes. However, parallel projection is useful as it keeps intact the true sizes and angles of objects. In perspective projection, the projection lines appear to converge to a point called projection reference point (PRP). This is actually how we see in the real-world as parallel lines appear to converge at a distance to our eyes. This is known as the perspective effect and happens because the size of objects depend on their distances from the observer—as objects move farther their apparent sizes reduce. Although perspective projection produces realistic views of scenes; however, it distorts the true lengths and angles of lines and surfaces. For 2D projection, points are projected on a line called the viewline, while for 3D projection points are projected on a plane called the viewplane. Parallel projection can again be of two types: orthographic and oblique. In parallel orthographic projection, the projection lines are perpendicular to the view plane. In parallel oblique projection, the projection lines can be oriented at any arbitrary angles to the view plane (Hearn and Baker, 1996) (see Figure 9.1
eBook - PDFComputer Graphics
Theory and Practice
- Jonas Gomes, Luiz Velho, Mario Costa Sousa(Authors)
- 2012(Publication Date)
- A K Peters/CRC Press(Publisher)
We first take a projective trans- formation L in projective space that takes the projection center O to a point at infinity. All the projection lines are transformed by L into parallel lines of affine space. Therefore, the composition L ◦ T is a parallel projection T between the transformed planes L(Π) and L(Π ). The central projection is then given by T = L −1 ◦ T . In other words, it is the com- position between a parallel projection, which is affine, and the projective transformation L −1 , and is therefore a projective transformation. Projections are important in computer graphics because of the virtual camera transfor- mation. The image of the projection is the plane Π , so the projection can be thought of as a transformation of R 3 in R 2 , or, more precisely, of RP 3 in RP 2 . From this point of view, a possible more general projection is a projective transformation T : RP 3 → RP 2 which, in homogeneous coordinates, is given by ⎛ ⎝ y 1 y 2 y 3 ⎞ ⎠ = ⎛ ⎝ a 11 a 12 a 13 a 14 a 21 a 22 a 23 a 24 a 31 a 32 a 33 a 34 ⎞ ⎠ ⎛ ⎜ ⎜ ⎝ x 1 x 2 x 3 x 4 ⎞ ⎟ ⎟ ⎠ . Using this transformation, we have 11 degrees of freedom to define a virtual camera, and several types of cameras are possible, such as the perspective camera (which uses central projection) and the pinhole, affine, weak-perspective, and orthographic cameras. We will study the perspective camera in detail in Chapter 11. 2.10 Comments and References An introduction to the methods of projective geometry in computer graphics can be found in [Roberts 66]. There are various sources on projective geometry. However, to our knowl- edge, there is no book providing an analytical and concise approach to projective geometry stressing its several applications to computer graphics. An attempt in that direction was made in [Penna and Patterson 86], which complements this chapter by providing a large number of examples.
eBook - PDF- Narayan Panigrahi(Author)
- 2014(Publication Date)
- CRC Press(Publisher)
4 Basics of Map Projection This chapter focuses on the key concept of map projection. The sequence of mathematical operations which leads to preparation of maps is explained through a sequence diagram. The important map transformations or map Projections and their characteristics are explained. Some of the important ap-plications of map Projections are tabulated along with their properties. The classification of map Projections from different mathematical perspectives and cartographic aspects is given. This chapter deals with the basic concepts of map projection and answers generic queries such as ‘What is map projection?’, ‘Why is map projection required?’, ‘Why there are so many Projections de-vised?’, ‘Which map Projections are suitable for a particular application?’ etc. How map Projections are designed and developed for different regions of the Earth’s surface is explained. Classification of different map Projections using different development surfaces, perspective positions and positions of tangent surface is explained. The categorization of map Projections under different categories such as development surface, usage and characteristics is discussed. The mathematical formulae for forward and reverse map Projections are de-rived ab initio. The chapter ends with a discussion on how to choose a map projection for a specific purpose and for a specific region of the world. All the concepts introduced here are supplemented by illustrations and key notes. 4.1 What Is Map Projection? Why Is It Necessary? A map is a two-dimensional piece-wise representation of the Earth’s crust on a paper surface (Snyder 1989; 1994). The shape of the Earth cannot be equated to any conventional geometric shapes i.e. it cannot be represented or modeled to any standard geometric shape like a sphere, ellipse etc.
eBook - PDFSpatial Augmented Reality
Merging Real and Virtual Worlds
- Oliver Bimber, Ramesh Raskar(Authors)
- 2005(Publication Date)
- A K Peters/CRC Press(Publisher)
4 Geometric Projection Concepts In this chapter, we review the fundamentals of image formation and projec-tion using a projector. Traditionally, a projector is used to create flat and usually rectangular images. In this sense, the three-dimensional computer graphics rendering algorithms used for a CRT or a flat LCD panel can be directly used for a projector without modification. However, a projector and display surface can be used in a variety of geometric configurations. In this chapter, we introduce a general framework that allows image synthe-sis under these geometric variations. The framework leads to a rendering framework and a better understanding of the calibration goals. We describe the issues involved and how the proposed algorithms are used in various types of display environments. 4.1 Geometric Model Consider the conceptual framework for a camera in computer vision. A camera model defines the relationship between three-dimensional points and two-dimensional pixels. These are extremely useful geometric abstrac-tions based on simple approximations, e.g., based on a pin-hole camera model that ignores optical or quantization issues. We introduce a simi-lar conceptual framework for projector-based environments to express the geometric relationship between the display components. Let us consider the problem of rendering images of three-dimensional virtual objects using a projector. There are various possibilities. The user could be moving or be static. The projector can be in an arbitrary position illuminating the display surface in a front-projection setup or rear-93 94 4. Geometric Projection Concepts projection setup. The display surface can be planar or non-planar. The displayed virtual object can be in front of, behind, or on the display sur-face. The proposed framework can be used under any one of the display configurations. 4.1.1 Geometric Relationship Consider the components in Figure 4.1.
- Hongyu Guo(Author)
- 2014(Publication Date)
- WSPC(Publisher)
In computer graphics, the 3-D world scene is modeled and stored in the computer and rendered to the 2-D screen through computation of central projection utilizing the fast matrix processing power of the graphics hardware, the GPU. Hence projective geometry is very important in computer graphics and computer vision applications.§2 Projective Planes
Mathematicians took over the study of perspective drawing and central projection from the artists. They carried it further and developed the theory of projective geometry. Johannes Kepler and Girard Desargues introduced the concept of point at infinity. Desargues is also known for an importance theorem named after him. Jean-Victor Poncelet developed much of synthetic projective geometry in a Russian prison after he was captured while serving as a French officer in Napoleon’s campaign against Russia. August Möbius and Julius Plücker introduced homogeneous coordinates. Michel Chasles, Karl von Staudt, Arthur Cayley and Felix Klein are among other major contributors in this field.Intuitively the projective plane is the picture plane, or the Dürer frame, or the artist’s canvas onto which the 3-D world is projected. It is a 2-D plane but it is not an ordinary 2-D plane. It is related to the 3-D world. It is the image plane of the 3-D world. Sometimes it is the illusion of the 3-D world. The illusion is caused by the visual ambiguity—many points (all the points on a straight line passing through the COP) are projected to the same point on the picture plane, or the projective plane.The projective plane can be defined using different models. All the models are equivalent to each other. We start with the extended Euclidean plane model. (Precisely it should be extended affine plane, but we avoid using the term affine plane for now just to make things simple. Simply put, an affine plane is a Euclidean plane where we refrain from speaking about distances.)
eBook - PDF- Hartmut Prautzsch, Wolfgang Boehm(Authors)
- 2018(Publication Date)
- A K Peters/CRC Press(Publisher)
PART TWO Images and Projections Technical and scientific illustrations are supposed to convey exact informa tion as to the shape and size of certain objects. For this reason physical objects are most commonly represented by their Projections into a plane using either parallel rays or a family of rays emanating from a center. While central Projections can be realized physically by a camera, parallel projec tions can not. Regardless, both kinds of images can be constructed directly without a physical device. Moreover, one can reconstruct the shape and size of an object from a pair of such Projections. Q ~ Taylor & Francis Taylor & Francis Group http://taylorandfrancis.com 4 Parallel Projections Although parallel Projections cannot be realized physically, the concept of parallel rays has several advantages. Parallel Projections are simple to construct and allow for an easy reconstruction of the measurements of an object from its image. Convincing examples of parallel Projections include the military and cavalier Projections which were first used by G. Monge (1746-1818) to construct scaled drawings of fortifications. L iterature: Hohenberg, Penna-Patterson, Rehbock 4.1 Pohlke’s Theorem It is intriguing to observe that a parallel projection of two parallel lines pro duces a pair of parallel lines and that two parallel distances have the same ratio as their two parallel images. These properties, which are illustrated in Figure 4.1, are summarized in the fundamental theorem: Parallel Projections preserve parallelism and ratios. 32 Part II Images and Projections Figure 4.1: Parallelism and ratio. On employing just these two simple properties as drawing rules, one can draw any parallel projection. This is a consequence of a theorem given by Pohlke in 1853 and illustrated below. Pohlke’s Theorem asserts that a two-dimensional figure of a cube is a parallel projection of a cube if parallel edges have parallel images.
eBook - PDFMatrix Theory
From Generalized Inverses to Jordan Form
- Robert Piziak, P.L. Odell, Zuhair Nashed, Earl Taft(Authors)
- 2007(Publication Date)
- Chapman and Hall/CRC(Publisher)
Chapter 8 Projections idempotent, self-adjoint, projection, the approximation problem 8.1 Orthogonal Projections We begin with some geometric motivation. Suppose we have two nonzero vectors x and y and we hold a flashlight directly over the tip of y . We want to determine the shadow y casts on x . The first thing we note is y → z → x → → α x Figure 8.1: Orthogonal projection. the shadow vector is proportional to x , so must be of the form x for some scalar . If we can discover the scalar , we have the shadow vector, more formally the orthogonal projection of y onto x . The word “orthogonal” comes from the fact that the light was held directly over the tip of y , so the vector z forms a right angle with x . Note, x + z = y , so z = y − x and z ⊥ x . Thus, 0 = < z | x > = < y − x | x > = < y | x > − < x | x > so = < y | x > < x | x > . This is great! It gives a formula to compute the shadow vector. DEFINITION 8.1 Let x , y be vectors in C n with x = −→ 0 . We define the orthogonal projection of y onto x by P x ( y ) = < y | x > < x | x > x . 291 292 Projections First, we note that the formula can be written as P x ( y ) = x ∗ y x ∗ x x = x x ∗ y x ∗ x = ( xx + ) y . Here comes the Moore-Penrose (MP) inverse again! This suggests that P x can be viewed as the matrix xx + , and orthogonal projection of y onto x can be achieved by the appropriate matrix multiplication. The next thing we note is that P x is unchanged if we multiply x by a nonzero scalar. Thus P x depends on the “line” (i.e., one dimensional subspace) span ( x ) and not just x . LEMMA 8.1 For any nonzero scalar , P x ( y ) = P x ( y ) . PROOF P x ( y ) = < y | x > < x | x > x = < y | x > < x | x > x = < y | x > < x | x > x = P x ( y ) . So, from now on, we write P sp ( x ) instead of P x , indicating the connection of P sp ( x ) with the one dimensional subspace span ( x ). Next, to solidify this connection with sp ( x ), we show the vectors in sp ( x ) are exactly those vectors left fix ed by P sp ( x ) .
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