What is Projections: Definition and 101 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. nomadreid

    I Dot product, inner product, and projections

    In simple Euclidean space: From trig, we have , for u and v separated by angle Θ, the length of the projection of u onto v is |u|cosΘ; then from one definition of the dot product Θ=arcos(|u|⋅|v|/(u⋅v)); putting them together, I get the length of the projection of u onto v is u⋅v/|v|. Then I...
  2. C

    Spectral decomposition of 4x4 matrix

    ## A = \pmatrix{ -4 & -3 & 3 & 3 \\ -3 & -4 & 3 & 3 \\ -6 & -3 & 5 & 3 \\ -3 & -6 & 3 & 5 } ## over ## \mathbb{R}##. Let ## T_A: \mathbb{R}^4 \to \mathbb{R}^4 ## be defined as ## T_A v = Av ##. Thus, ## T_A ## represents ## A ## in the standard basis, meaning ## [ T_A]_{e} = A ##. I've...
  3. B

    B Wave/Particle Duality: Helical Paths & Planar Projections

    With respect to wave/particle duality, is it correct to think of a particle traveling a wave path? And if so, given that no orientation with respect to an observer of that wave is generally considered, is the "wave" that is referred to then more accurately the planar projection of a helical path?
  4. Tesla In Person

    Orthographic projections of object drawings

    Hi, I have this mcq . The image shows the Top and Front view of an object, what is the side view of this object? I am finding it very difficult to even interpret the top and front views. I will get questions like these in my exam so I must be able deduce the 3rd view from 2 given views of the...
  5. A

    Linear algebra projections commutativity

    Textbook answer: "If P1P2 = P2P1 then S is contained in T or T is contained in S." My query: If P1 = \begin{pmatrix} 1 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \\ \end{pmatrix}and P2 =\begin{pmatrix} 0 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 0 \\ \end{pmatrix} as far as I...
  6. Isaac0427

    B Orthogonal Projections: Same Thing or Not?

    Aren't they the same thing? If so, why would textbooks write the former? Ex: https://textbooks.math.gatech.edu/ila/projections.html or http://www.math.lsa.umich.edu/~speyer/417/OrthoProj.pdf or https://en.wikipedia.org/wiki/Projection_(linear_algebra)#Orthogonal_projections Thank you!
  7. Physics lover

    Chemistry Newman projections of meso-2,3-butanediol

    The question and options are-: Ok so what I did was that i converted all of them to their fischer projection.But only P gave me meso-2,3-butanediol.Others were not meso.Please help.
  8. Math Amateur

    MHB Orthogonal Projections .... Garling, Proposition 11.4.3 .... ....

    I am reading D. J. H. Garling's book: "A Course in Mathematical Analysis: Volume II: Metric and Topological Spaces, Functions of a Vector Variable" ... ... I am focused on Chapter 11: Metric Spaces and Normed Spaces ... ... I need some help to fully understand the proof of...
  9. M

    Projections of functions and bases

    Homework Statement On ##L_2[0,2\pi]## where ##e = \{ 1/\sqrt{2 \pi},1/\sqrt{\pi}\sin x,1/\sqrt{2 \pi}\cos x \}##. Given ##f(x) = x##, find ##Pr_e f##. Homework Equations See solution. The Attempt at a Solution I take $$e \cdot \int_0^{2\pi} e f(x) \, dx = \pi - 2 \sin x.$$ Look correct?
  10. 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...
  11. M

    I Proving the Orthogonal Projection Formula for Vector Subspaces

    Hi PF! I've been reading and it appears that the orthogonal projection of a vector ##v## to the subspace spanned by ##e_1,...,e_n## is given by $$\sum_j\langle e_j,v \rangle e_j$$ (##e_j## are unit vectors, so ignore the usual inner product denominator for simplicity) but there is never a proof...
  12. hnnhcmmngs

    Vectors and scalar projections

    Homework Statement Let a and b be non-zero vectors in space. Determine comp a (a × b). Homework Equations comp a (b) = (a ⋅ b)/|a| The Attempt at a Solution [/B] comp a (a × b) = a ⋅ (a × b)/|a| = (a × a) ⋅b/|a| = 0 ⋅ b/|a| = 0 Is this the answer? Or is there more to it?
  13. I

    Projections and direct sum

    Homework Statement Let ##V = \mathbb{R}^4##. Consider the following subspaces: ##V_1 = \{(x,y,z,t)\ : x = y = z\}, V_2=[(2,1,1,1)], V_3 =[(2,2,1,1)]## And let ##V = M_n(\mathbb{k})##. Consider the following subspaces: ##V_1 = \{(a_{ij}) \in V : a_{ij} = 0,\forall i < j\}## ##V_2 =...
  14. I

    Proving ideas about projections

    Homework Statement [/B] Let ##V## be a vector space, and let ##U, W## be subspaces of ##V## such that ##V = U \oplus W##. Let ##P_U## be the projection on ##U## in the direction of ##W## and ##P_W## the projection on ##W## in the direction of ##U##. Prove: ##P_U + P_W = Id##, ##P_U P_W = P_W...
  15. U

    Projection Matrix Homework: Equations & Solution

    Homework Statement [/B]Homework EquationsThe Attempt at a Solution The solution is obviously given, but I don't really understand what is done there. What method is being used? so I can understand, because i see how they attained v, but then that vector normalised is not correct is it?
  16. Elissa Damron

    Calculating "population absurdities"

    Homework Statement The urgency of population problems can be emphasized by calculating such "Population absurdities" as the time at which there will be one person per square meter or per square foot of land or the time at which the weight of people will exceed the weight of the earth. Try...
  17. A

    B Ellipse tangent line using projections

    Hi :) The question is in dutch so i'l translate it. on an ellipse E with vertex P and P' on the major axis and vertex Q and Q' on the minor axis. chose R(x1,y1), the projection of R on the major axis is R' and on the minor axis is R''. Define the perpendicular projection of the intrersection...
  18. Buggsy GC

    Charity engineering project water projections

    Kia ora I'm a fist year engineering student in Christchurch NZ, and a few friend and I, are designing a simple engineering report for a water projection system for an organization called gap filler. Were currently thinking of projecting an image of music onto the river Avon so people can read...
  19. Kaura

    I Battle Projections: Predicting Probabilities in Games

    This is a rather odd topic but recently when playing games, mostly first person shooters, I have formed a curiosity about "Battle Projections" or the ability to predict probabilities based on in game variables. For example, if you were spectating a round of no respawn four versus four death...
  20. R

    How Does the IPCC Determine Confidence Levels in Climate Projections?

    Would someone please be able to point me towards the literature that discusses how the IPCC calculate the low, medium and high confidence projections as used in their fifth assessment report? Thanks.
  21. A

    Engineering Free Exercises on Right Angle Projections & Cross Section Views

    I am looking for free solved exercises about: right angle projections, (cross) section views Here are some pictures of my book so you can understand what I am looking for http://1.1m.yt/3cNNDl-.jpg http://1.1m.yt/FekMn6x.jpg http://1.1m.yt/1t3KEer.jpg http://1.1m.yt/urA5V2_.jpg...
  22. A

    Are these projections correct?

    Homework Statement We have to make right angle projections of various objects. Are these projections correct?Homework EquationsThe Attempt at a Solution https://postimg.org/image/tg987f8bx/ https://postimg.org/image/rcyszr8j1/ https://postimg.org/image/64l4ibu25/...
  23. S

    How Do You Calculate Projection and Symmetry in Vector Geometry?

    Homework Statement Find the projection, P', of the point P on the line p, the distance of P from p and the coordinates of the point R symmetric to P with respect to p, where P = [1, 2, 0], and p : X = [3, 0, 0] + t(0, 1, 0), t ∈ IR. *Sorry about all the P's, but this is how the question is...
  24. P

    Not understanding Fischer projections

    This is what's confusing me. Wikipedia says that "All bonds are depicted as horizontal or vertical lines." Then it shows a picture of D glucose chain with diagonal bonds. (the double bond and the hydrogen). Why is it drawn with diagonal bonds if its supposed to be only vert or horiz bonds...
  25. Math Amateur

    MHB Infinite Direct Sums and Standard Inclusions and Projections

    I am reading An Introduction to Rings and Modules With K-Theory in View by A.J. Berrick and M.E. Keating (B&K). In Chapter2: Direct Sums and Short Exact Sequences in Sections 2.1.11 and 2.1.12 B&K deal with infinite direct products and infinite direct sums (external and internal). In Section...
  26. Jarvis323

    Lin. Alg. Projections conceptual question

    Homework Statement 16. Suppose P is the projection matrix onto the line through a. (a) Why is the inner product of x with Py equal to the inner product of Px with y? (b) Are the two angles the same? Find their cosines if a = (1;1;¡1), x = (2;0;1), y = (2;1;2). (c) Why is the inner...
  27. R

    Orthogonal Projections vs Non-orthogonal projections?

    Hi everyone, My Linear Algebra Professor recently had a lecture on Orthogonal projections. Say for example, we are given the vectors: y = [3, -1, 1, 13], v1 = [1, -2, -1, 2] and v2 = [-4, 1, 0, 3] To find the projection of y, we first check is the set v1 and v2 are orthogonal...
  28. E

    Econ: Budget cuts and expensive laboratory science projections for 10s

    (I am going to attempt a 2nd, little thread before I finish my earlier thread. I want to see if I can practice using PF Rules & Guidelines and FAQ before I wrap up my thread on Autism in Medical Sciences.) In my humble opinion, the current budget crisis in the news likely augurs deep cuts -...
  29. S

    Proof regarding orthogonal projections onto spans

    Homework Statement Let U be the span of k vectors, {u1, ... ,uk} and Pu be the orthogonal projection onto U. Let V be the span of l vectors, {v1, ... vl} and Pv be the orthogonal projection onto V. Let X be the span of {u1, ..., uk, v1, ... vl} and Px be the orthogonal projection onto X...
  30. A

    Projections in L^2 spaces

    Suppose you have a self-adjoint operator A on L^2(\mathbb R^n) that has exactly one discrete, non-degenerate eigenvalue a with associated normalized eigenfunction f_a. What does the projection onto the associated eigenspace look like? My guess would be: P(f) = (f,f_a)f_a = f_a(x)...
  31. U

    Finding a vector using scalar and vector projections

    Homework Statement Determine the vector(s) whose vector projection on u =< 1,2,2 > is v =< 3,6,6 > and its scalar projection on w =< 1,1,1 > is √3. Homework Equations Vector Projection of b onto a: (|b.a| \ |a|) * (1/ |a|) * a Scalar Projection: (|b.a| \ |a|) The Attempt at a...
  32. NATURE.M

    Vector Projections: A, B & C Explained

    Homework Statement a. Is it possible to have u ↓ v undefined? b. Is it possible to have u ↓ v = v ↓ u ? c. Explain why u ↓( v ↓ w ) = u ↓ w . Homework Equations The Attempt at a Solution I know a is possible if the length of vector v is 0. I think be is false, but not...
  33. M

    Question about Differential Forms as Size of Projections

    Hello, I have a somewhat conceptual question about differential forms. I have been studying differential forms off and on for some time now and things are starting to come together for me. However, there is an irritating gap in my understanding. Regarding the geometric significance or...
  34. P

    Vanishing spinor projections in supergravity

    Hello all I am working on a model in D=5 N=2 supergravity where the metric background is described by a time-dependent three brane, with one extra spatial dimension (a brane-world with bulk sort of set up). The vanishing of fermionic variations gives me the following weird projections...
  35. P

    Understanding Edexcel GCE January 2010 Mechanics M2 QP: Question 8C Explanation

    http://www.scribd.com/doc/26846418/Edexcel-GCE-January-2010-Mechanics-M2-QP http://www.edexcel.com/migrationdocuments/QP%20GCE%20Curriculum%202000/GCE%20January%202010%20-%20MS/6678_01_msc_20100219.pdf Question 8C: The second link is the answers - I don't understand why they set x = 0...
  36. djh101

    Orthogonal Projections: Minimize a^2 + b^2 + c^2

    Homework Statement There are three exams in your linear algebra class and you theorize that your score in each exam will be numerically equal to the number of hours you study. The three exams count 20%, 30%, and 50% and your goal is to score 76% in the course. How many hours, a, b, and c...
  37. G

    Line maximizing orthogonal projections

    Say I have a set of points in 2D space. How would I find a line that maximizes the sum orthogonal projection of the points onto the line. The line does not have to go through the origin.
  38. P

    Projections of complex vectors

    I need some clarification on projections of complex vectors. If I have a nxm matrix of complex numbers V and a mx1 matrix s, and I want to find the projection of s onto any column of V. The formula to do this is c = <V, s>/||V(j)||^2 where V(j) is the column of V to be used. My question is...
  39. A

    Angle projections to Euler angles

    Consider a vector in 3D. Its projections on two planes, say YX and YZ planes, makes some angle with the vertical axis ( the y-axis in this case). I know these two angles (I call them projected angles). This is the only information I have about the vector. I need Euler angles which when...
  40. P

    Calculating Velocity and Distance with Simple Projections

    A particle is projected with velocity vector (10i + 15j)ms^-1 where i and j are unit vectors in the horizontal and upward vertical directions respectively. Find its velocity vector 2 seconds later and its distance from the point of projection at this time. Well first of all, could anyone...
  41. G

    Projections from Tubular Neighborhoods

    Hello! Could anybody give me some hint with the following problem? Consider a smooth, compact embedded submanifold M = M^m\subset \mathbb{R}^n, and consider a tubular neighborhood U = E(V)\supset M, where E: (x, v) \in NM \mapsto x + v \in M is a diffeomorphism from a open subset of the normal...
  42. I

    Do projections of lines which are not perpendicular correspond to FLTs?

    A math question about projections of lines: Say we have two straight lines which we consider as number lines (\mathbb{R}). I've learned that a projection of one line onto another is of the form ax + b for a,b\in \mathbb{R} when the two lines are parallel. If we allow the possibility that the...
  43. N

    Planar geometry, orthognal projections of a piece of a plane

    Homework Statement I have a piece of a plane in 3 dimensions (imagine holding an enevelope in the air), and two orthognal projections which form quadrilaterals, one on the xy plane (i.e. looking at the enevelope from above) and one on the xz plane (looking at it from the side). We know the...
  44. T

    Linear Algebra: Orthogonal Projections

    Hey first time poster here. Homework Statement Find the orthogonal projection projWy u1 = [-1; 3; 1; 1], u2 = [3; 1; 1; -1], u3 = [-1; -1; 3; -1], y = [1; 0; 0; 1] where {u1, u2, u3} is an orthogonal basis. Homework Equations yhat = [dot(y,U1)/dot(U1,U1)]U1 + ...
  45. Z

    Projection onto Column Space of A and its Perpendicular

    Homework Statement Some of the details in this question are based off the use of matlab. If it's needed I can show the matrices that MATLAB creates. Let A = magic(8); A = A(:,1:3) and let S be the Column Space of A. For b = [1:8]' compute the projection of b onto the Column Space of A...
  46. S

    Finding the Projection onto Subspaces

    Homework Statement See attachment The Attempt at a Solution How should I approach these questions? By using the projection formula?
  47. J

    Can someone check my Fischer Projections?

    Homework Statement 1. R, 2-Bromo-2-Chlorobutane 2. S, NH2-CH-C2H5 ...| .....OH ......O ......|| 3. R,R and R,S of CHDCl-C-CHDNH2 Homework Equations ...knowledge of fischer projections? I can post the video-tutorial i attempted to follow, if needed. The Attempt at a Solution...
  48. T

    Proving Characteristics of Projections

    Can anyone think of a trivial proof for the fact that the projection of any polynomial onto the xy-plane itself gives a polynomial? My professor and I were speculating about this, but could not discover a trivial proof for the fact. Does such a thing exist?
  49. B

    Projections onto a plane and parallel to a vector

    Hi. I'm preparing the Linear algebra 1 Exam for the first year of Physics University. I find very difficult to understand projections. Here's an example: Consider the projection P: R^3 --> R^3 onto the plane U of equation 2x1 - 3x2 + x3 = 0 and parallel (the projection) to the vector...
  50. O

    Projections in Geometry (Question)

    nevermind i figured it out