# 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. ### 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...

39. ### 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...
40. ### 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}...
41. ### 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...
42. ### 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...
43. ### 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...
44. ### 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...
45. ### 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...
46. ### 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...
47. ### 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...
48. ### 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...
49. ### 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...
50. ### 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...