What is Projection: Definition and 433 Discussions

In cartography, a map projection is a way to flatten a globe's surface into a plane in order to make a map. This requires a systematic transformation of the latitudes and longitudes of locations from the surface of the globe into locations on a plane.
All projections of a sphere on a plane necessarily distort the surface in some way and to some extent. Depending on the purpose of the map, some distortions are acceptable and others are not; therefore, different map projections exist in order to preserve some properties of the sphere-like body at the expense of other properties. The study of map projections is the characterization of the distortions. There is no limit to the number of possible map projections.
Projections are a subject of several pure mathematical fields, including differential geometry, projective geometry, and manifolds. However, "map projection" refers specifically to a cartographic projection.
Despite the name's literal meaning, projection is not limited to perspective projections, such as those resulting from casting a shadow on a screen, or the rectilinear image produced by a pinhole camera on a flat film plate. Rather, any mathematical function that transforms coordinates from the curved surface distinctly and smoothly to the plane is a projection. Few projections in practical use are perspective.Most of this article assumes that the surface to be mapped is that of a sphere. The Earth and other large celestial bodies are generally better modeled as oblate spheroids, whereas small objects such as asteroids often have irregular shapes. The surfaces of planetary bodies can be mapped even if they are too irregular to be modeled well with a sphere or ellipsoid. Therefore, more generally, a map projection is any method of flattening a continuous curved surface onto a plane.A model globe does not distort surface relationships the way maps do, but maps can be more useful in many situations: they are more compact and easier to store; they readily accommodate an enormous range of scales; they are viewed easily on computer displays; they can be measured to find properties of the region being mapped; they can show larger portions of the Earth's surface at once; and they are cheaper to produce and transport. These useful traits of maps motivate the development of map projections.
The best known map projection is the Mercator projection. Despite its important conformal properties, it has been criticized throughout the twentieth century for enlarging area further from the equator. Equal area map projections such as the Sinusoidal projection and the Gall–Peters projection show the correct sizes of countries relative to each other, but distort angles. The National Geographic Society and most atlases favor map projections that compromise between area and angular distortion, such as the Robinson projection or the Winkel tripel projection

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  1. Miles Behind

    B East-West error at regular points on the Azimuthal Equidistant Map

    Hello. I am conversing with Flat-Earth folks who tend to lean upon the Azimuthal Equidistant (AE) map centered on the North pole. I know that the AE map is a projection of the globe onto a flat surface, and is only accurate in distances north and south along lines of longitude. The east west...
  2. Euge

    POTW Projection Formula for Ringed Spaces

    Show that if ##f : X \to Y## is a morphism of ringed spaces, ##\mathscr{F}## is an ##\mathcal{O}_X##-module and ##\mathscr{E}## is a locally free ##\mathcal{O}_Y##-module of finite rank, then for all ##p \ge 0##, there is an isomorphism $$R^pf_*(\mathscr{F}\otimes_{\mathcal{O}_X} f^*\mathscr{E})...
  3. chwala

    Finding the projection of a Vector

    I am looking at this now; pretty straightforward as long as you are conversant with the formula: anyway i think there is a mistake on highlighted i.e Ought to be ##-\dfrac{15}{37}(i+6j)## just need a confirmation as at times i may miss to see something. If indeed its a mistake then its time...
  4. M

    Optics for High Resolution Laser Imaging: Finding the Best Path

    Objectives: - best path for optics needed to focus and "draw"/project a high resolution image onto a workspace around 500mm square (for a Laser Direct Imaging machine) - where to cost effectively purchase or make the optics necessary to build a prototype The problem: I need to project a high...
  5. B

    A Quantum projection when measuring state population

    Hello! I have two energy levels of opposite parity close by (we can assume they are far from all the other levels in the system) and an off-diagonal term in the 2x2 matrix Hamiltonian that weakly couples them. I initially populate only one parity state, say the positive one, and after a while...
  6. M

    A Sampling Electrons from a 2D Projection: Is There a Functional Form?

    Hello! I have some electrons produced from a 3D gaussian source isotropically inside a uniform electric field. The electric field guides them towards a position sensitive detector and I end up with an image like the one below (with more electrons on the edge and fewer as you move towards the...
  7. AimaneSN

    I Finding the orthogonal projection of a vector without an orthogonal basis

    Hi there, I am currently reading a course on euclidian spaces and I came across this result that I am struggling to prove : Let ##F## be a subspace of ##E## (of finite dimension) such that ##F=span(e_1, e_2, ..., e_p)## (not necessarily an orthogonal family of vectors), let ##x \in E## Then...
  8. A

    Engineering Signals & Systems with Linear Algebra

    Hello everyone, I would like to get some help with the above problem on signals and linear projections. Is my approach reasonable? If it is incorrect, please help. Thanks! My approach is that s3(t) ad s4(t) are both linear combinations of s1(t) and s2(t), so we need an orthonormal basis for the...
  9. K

    I Projection postulate and the state of a system

    Quantum Mechanics, McIntyre states the projection postulate as: "After a measurement of ##A## that yields the result ##a_n##,the quantum system is in a new state that is the normalized projection of the original system ket onto the ket (or kets) corresponding to the result of the measurement"...
  10. jisbon

    A Projection of 3D Plane from a certain perspective on the 2D Plane

    Hello all! As seen in the summary, I'm not sure if anyone can understand, but I will try to make this as clear as possible. Working in the 3D Plane: Given that there is a trajectory motion in the 3D Plane, and I have the coordinates of the motion at every 1s interval. This means at t=1s, the...
  11. bobtedbob

    I Linear Bezier curve projection

    I'm looking at the following web page which looks at rendering bezier curves. GPU Gems 3 - Chapter 25 Paper on same topic The mathematics is quite interesting, I was interested to know what the F matrix would look like for for a linear bezier equation. The maths for the quadratic case is in...
  12. D

    B Name of projection to firmamentum (celestial sphere)

    Does the central projection of the position of a star to the firmamentum (celestial sphere) has a special name?
  13. M

    Magnitude of the Projection of force "F" on the u-axis

    So I was watching a YouTube video preparing for a quiz on Wednesday, and I saw something that I would like clarification on. I'm pretty sure I understand what is being explained, but I just want to confirm. The figure above is associated with the problem at hand. So I understand that to get the...
  14. Euge

    MHB Projection Map $X \times Y$: Closure Property

    Here is this week's POTW: ----- Let $X$ and $Y$ be topological spaces. If $Y$ is compact, show that the projection map $p_X : X \times Y \to X$ is closed. -----
  15. A

    Orthogonal Projection Problems?

    Summary:: Hello all, I am hoping for guidance on these linear algebra problems. For the first one, I'm having issues starting...does the orthogonality principle apply here? For the second one, is the intent to find v such that v(transpose)u = 0? So, could v = [3, 1, 0](transpose) work?
  16. A

    3D object parallel projection, filtering of facets

    I'm sorry if the wording is a bit clunky, but this is not a common topic for me. Say you have some 3D object consisting of vertices and facets. There are many tools that can visualize this and they only show its projection on 2D, i.e. on your screen. I presume that you would filter the faces...
  17. L

    Transparent surfaces that become opaque

    Are there any materials that go from transparent to opaque when a bright light is shined on it? In particular I would like something that acts like a window until I project a movie onto it and then is opaque in the regions where the light hits it at certain levels of brightness.
  18. C

    Calculation Involving Projection Tensor in Minkowski Spacetime

    In Minkowski spacetime, calculate ##P^{\gamma}_{\alpha}U^{\beta}\partial_{\beta}U^{\alpha}##. I had calculated previously that ##P^{\gamma}_{\alpha}=\delta^{\gamma}_{\alpha}+U_{\alpha}U^{\gamma}## When I subsitute it back into the expression...
  19. R

    Vector Field associated with Stereographic Projection

    I identified $$(\Phi_{SN})_{*})$$ as $$J_{(\Phi_{SN})}$$ where J is the Jacobian matrix in order to $$(\Phi_{SN})$$, also noticing that $$\frac{\partial}{\partial u} = \frac{\partial s}{\partial u}\frac{\partial}{\partial s} + \frac{\partial t}{\partial u} \frac{\partial}{\partial t} $$, I wrote...
  20. anorlunda

    Could This New Map Projection Revolutionize Our Understanding of the Globe?

    A recent thread Why Grid North doesn't agree with True North on maps had a lot of discussion on map projections. Here is a new contender for best way to project the globe in 2D. I am interested in their claim. "distances across oceans or across poles are both accurate and easy to measure."...
  21. Demystifier

    A Difference Between Collapse and Projection

    [Moderator's note: Spun off from a thread in the QM forum to separate out the interpretation discussion.] Yes, but I was hoping that you could give a precise specification of the difference. (In my opinion, @vanhees71 never specified the difference precisely.)
  22. rannasquaer

    MHB Proving Orthogonal Projection of Triangle V, v'_{1}

    Given the triangle above where V < v'_{1}, prove that the \[ v_{1}=V \cos(\psi)+v'_{1} \cos(\theta - \psi) \] It is said that v_{1} is equal to the sum of the orthogonal projections on v_{1} of V and of v'_{1} and that is precisely the expression that show. But I couldn't see how to make the...
  23. V

    Using Least Squares to find Orthogonal Projection

    I'm a little confused how to do this homework problem, I can't seem to obtain the correct answer. I took my vectors v1, v2, and v3 and set up a matrix. So I made my matrix: V = [ (6,0,0,1)T, (0,1,-1,0)T, (1,1,0,-6)T ] and then I had u = [ (0,5,4,0) T ]. I then went to solve using least...
  24. thaiqi

    I Why is the projection postulate needed?

    Why is the projection postulate needed? For the measurement result's repetition?
  25. Keth

    A Wolfram Schommers Projection theory

    I'm trying to dig up anything that supports or discounts Schommers' theory as it relates to Projection Theory. Tried some Googling and came up with naught.
  26. DaveC426913

    Projection Maps for Alien World Navigation

    An ocean world navigational map will not have any reason to chop up its navigational maps into artificial pieces to accommodate continents. That disqualifies several projections to start. I want to figure out what kind of a projection they would use. Since they would be doing all their...
  27. M

    MHB Projection Matrix Calculation for Vectors a_k: P_6P_5P_4P_3P_2P_1a_0

    Hey! :o We have the vectors $\displaystyle{a_k=\begin{pmatrix}\cos \frac{k\pi}{3} \\ \sin \frac{k\pi}{3}\end{pmatrix}, \ k=0, 1, \ldots , 6}$. Let $P_k$ be the projection matrix onto $a_k$. Calculate $P_6P_5P_4P_3P_2P_1a_0$. Are the elements of the projection matrix defined as...
  28. H

    How to Find Eigenstates for Value 0 in Projection on Coherent States Paper?

    i am reading this paper . in the definition 19 we have |z><z| = :exp(a-z)^\dagger (a-z): in the extansion the first term is the identity son it is not hart to find an eigenvector for the value 1. it is ok if the vector is annihilated by a. if is the case for the coherent grouns state. how to...
  29. sabenabe

    Water stream projection distance calculation for different nozzle shapes

    Hello, I have built a device and i want to test different types of nozzles. Problem is; the size of the nozzles i need are hard to find in company stock. So i need to make sure before i make any order. I have no background or degree any related area so it is hard to understand the equations...
  30. jk22

    B Question about Mercator's projection

    I read the Wikipedia article : https://en.m.wikipedia.org/wiki/Mercator_projection Section : Mathematics of the Mercator projection. For a map to be conformal should not it be $$k(\phi)=C,h(\phi)=C$$, or the shrinking coefficient shall be not only equal but homogeneous, in order to be...
  31. Cheesycheese213

    Best type of map (projection) for measuring distances?

    Hi! Sorry if this isn't the best question, but I've been trying to do a project where I thought it would be interesting to talk about coastlines. I downloaded a GIS (QGIS) to try and see if I could roughly measure all the continents for myself, but I haven't really done this before, so I wasn't...
  32. A

    B Measurement postulate = von Neumann’s projection postulate?

    I find in the literature frequent reference to the "measurement postulate" and, sometimes, to the "von Neumann’s projection postulate". The difference, if any, seems to me subtle but I can't tell which. However, they are never mentioned both in the same context/paper, so I'm afraid that they may...
  33. Huyanyinglei

    Issue with perspective projection?

    Summary: Perspective projection is often referred to when talking about camera models I have the following problem. Perspective projection is often referred to when talking about camera models(https://en.wikipedia.org/wiki/3D_projection#Perspective_projection). I don’t think I understand it...
  34. S

    I Find the image of a projection

    ## \dfrac{1}{2} \left(\begin{array}{rrrr} 0 & 2 & -2 & 0 \\ 3 & -2 & 2 & -3 \\ 3 & -4 & 4 & -3 \\ -2 & 2 & -2 & 2 \end{array}\right)## The matrix satisfies ##A^2=A##, so it is a projection. To find ##U'##, one can find the ##\text{ker} \ (A-I)=\text{ker} \ (I-A)=\text{im} \ (A)=U'##. Also...
  35. S

    I Proof of ##F## is an orthogonal projection if and only if symmetric

    The given definition of a linear transformation ##F## being symmetric on an inner product space ##V## is ##\langle F(\textbf{u}), \textbf{v} \rangle = \langle \textbf{u}, F(\textbf{v}) \rangle## where ##\textbf{u},\textbf{v}\in V##. In the attached image, second equation, how is the...
  36. P

    I Quantum Computing - projection operators

    Assume ##P_1## and ##P_2## are two projection operators. I want to show that if their commutator ##[P_1,P_2]=0##, then their product ##P_1P_2## is also a projection operator. My first idea was: $$P_1=|u_1\rangle\langle u_1|, P_2=|u_2\rangle\langle u_2|$$ $$P_1P_2= |u_1\rangle\langle...
  37. R

    I Projection Operators: Pi, Pj, δij in Quantum Mechanics

    In Principles of Quantum mechanics by shankar it is written that Pi is a projection operator and Pi=|i> <i|. Then PiPj= |i> <i|j> <j|= (δij)Pj. I don't understand how we got from the second result toh the third one mathematically.I know that the inner product of i and j can be written as δijbut...
  38. A

    I Is the Chirality Projection Operator Misused in This Scenario?

    Hello everybody! I have a doubt in using the chiral projection operators. In principle, it should be ##P_L \psi = \psi_L##. $$ P_L = \frac{1-\gamma^5}{2} = \frac{1}{2} \begin{pmatrix} \mathbb{I} & -\mathbb{I} \\ -\mathbb{I} & \mathbb{I} \end{pmatrix} $$ If I consider ##\psi = \begin{pmatrix}...
  39. JD_PM

    Why the projection of S on the xy plane cannot be used?

    I don't understand why we could not use the 1/4 of the circle lying on the xy plane as R. In the exercise it is not explained. The idea would be taking the arc length. I know it is not easier than making your projection on the xz plane, but just wondering if this is possible. I guess it is not...
  40. V

    Finding the projection on matlab

    Homework Statement Let u = (3, -8, 5, 5) and v = (-9, -4, -7, 2). Find ||u − projvu||. Homework EquationsThe Attempt at a Solution >> u=[3 -8 5 5] u = 3 -8 5 5 >> v=[-9 -4 -7 2] v = -9 -4 -7 2 >> proj_u_v = dot(u,v)/norm(v)^2*v proj_u_v = 1.2000...
  41. D

    I Vacuum projection operator and normal ordering

    I've been reading this book, in which the author expresses the vacuum projection operator ##\vert 0\rangle\langle 0\vert## in terms of the number operator ##\hat{N}=\hat{a}^{\dagger}\hat{a}##, where ##\hat{a}^{\dagger}## and ##\hat{a}## are the usual creation and annihilation operators...
  42. arivero

    A Charge in a Lie Group.... is it always a projection?

    Given a representation of a Lie Group, is there a equivalence between possible electric charges and projections of the roots? For instance, in the standard model Q is a sum of hypercharge Y plus SU(2) charge T, but both Y and T are projectors in root space, and so a linear combination is. But I...
  43. T

    B Question about finding the force using vector projections

    In my pre-calculus textbook, the problem states: A 200-pound cart sits on a ramp inclined at 30 degrees. What force is required to keep the cart from rolling down the ramp? The gravitational force can be represented by the vector F=0i-200j In order to find the force we need to project vector...
  44. bardia sepehrnia

    I Wrong solution in the book? Calculating force projection.

    The question asks to calculate the determine the projection of the resultant force of F1 and F2 onto b-axis. However instead, the solution is a projection of F1 on b axis plus F2. Shouldn't the solution involve the projection of R which is 163.4 on to be axis? and for answering that, don't we...
  45. jk22

    B Dimension of the metric of a projection of a sphere

    Let ##(x_1,x_2,x_3)=\vec{r}(\theta,\phi)## the parametrization of a usual sphere. If we consider a projection in two dimension ##(a,b)=\vec{f}(x_1,x_2,x_3)## Then I don't understand how to use the metric, since it is ##g_{ij}=\langle \frac{\partial\vec{f}}{\partial...
  46. Math Amateur

    MHB Orthogonal vector projection and Components in Orthogonal Directions ....

    I am reading Miroslav Lovric's book: Vector Calculus ... and am currently focused n Section 1.3: The Dot Product ... I need help with an apparently simple matter involving Theorem 1.6 and the section on the orthogonal vector projection and the scalar projection ...My question is as follows: It...
  47. N

    Fischer Projection to Haworth/Chair Conformation

    Can you tell if the ring will be beta or alpha in the haworth projection, chair conformation by just looking at the Fischer projection, or must you be told first before creating the two from the Fischer? If so, could someone explain how? I've seen a bunch of websites and they all seem to say...
  48. twoski

    Vector projection to other vector

    Let's say i have 2 arbitrary vectors in a 3d space. I want to project Vector A to Vector B using a specified normal. edit: better image A is green, B is red, C is red arrow. Blue is result. In this case, i want to project green vector to red vector in the red direction. This would give me...