What is Mapping: Definition and 404 Discussions

Texture mapping is a method for defining high frequency detail, surface texture, or color information on a computer-generated graphic or 3D model. The original technique was pioneered by Edwin Catmull in 1974.Texture mapping originally referred to diffuse mapping, a method that simply mapped pixels from a texture to a 3D surface ("wrapping" the image around the object). In recent decades, the advent of multi-pass rendering, multitexturing, mipmaps, and more complex mappings such as height mapping, bump mapping, normal mapping, displacement mapping, reflection mapping, specular mapping, occlusion mapping, and many other variations on the technique (controlled by a materials system) have made it possible to simulate near-photorealism in real time by vastly reducing the number of polygons and lighting calculations needed to construct a realistic and functional 3D scene.

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  1. D

    If two functions are homotopic must their mapping cylinders be also?

    Hey I feel like I understand the concept of two spaces being homotopic and I can "visualize" the concept because I think of one space kind of continuously morphing into the other. But when it comes to thinking of homotopy between functions, I have a harder time. I was trying to think of a way...
  2. P

    Rigged Hilbert space, separable space, domain of CSCO, mapping

    Suppose that we have rigged Gilbert space Ω\subsetH\subsetΩ\times (H is infinite-dimensional and separable). Is the Ω a separable space? Is the Ω\times a separable space? Consider the complete set of commuting observables (CSCO) which contain both bounded and unbounded operators...
  3. A

    Show Injectivity of Mapping y=Ax for Finite Alphabet w/o Search Algorithm

    Let y=Ax. A is a matrix n by m and m>n. Also, x gets its values from a finite alphabet. How can i show if the mapping from x to y is injective for given A and alphabet (beside a search method)? For example, let A and the alphabet be [1 0 1/sqrt2 1/sqrt2] [0 1 1/sqrt2 -1/sqrt2] and...
  4. P

    MHB Show f is a contraction mapping

    f:[1,infinity)->[1,infinity) $f(x)=x^{0.5}+x^{-0.5}$ I thought about using MVT but it doesn't work and I've tried showing it conventially but i can't reduce it to k|x-y|
  5. T

    A mapping from an integral domain to non-negative integers, Abstract Algebra

    So just had this question as extra credit on a final: Let D be an integral domain, and suppose f is a non-constant map from D to the non-negative integers, with f(xy) = f(x)f(y). Show that if a has an inverse in D, f(a) = 1. Couldn't figure it out in time. I was thinking the way to go...
  6. N

    Limits after mapping in double integral

    Homework Statement I have the double integral, ∫∫sqrt(x^2+y^2) dxdy, and the area D:((x,y);(x^2+y^2)≤ x) Homework Equations The Attempt at a Solution By completing the squares in D we get that D is a circle with origo at (1/2,0), and radius 1/2. Then I tried changing...
  7. N

    Double integral limits after mapping

    I have the double integral, ∫∫sqrt(x^2+y^2) dxdy, and the area D:((x,y);(x^2+y^2)≤ x) By completing the squares in D we get that D is a circle with origo at (1/2,0), and radius 1/2. Then I tried changing the variables to x=r cosθ+1/2, y=r sinθ and J(r,θ)=r which leads to a not so nice...
  8. C

    Conformal Mapping - Can't Prove Analyticity

    Homework Statement We have the conformal map w = f(z) = z + K/z. Prove this mapping is indeed conformal. Homework Equations z = x + iy A map w = f(z) is conformal if it is analytic and df/dz is nonzero. f(z) = u(x,y) + iv(x,y) The Attempt at a Solution df/dz = 1 - Kz^-2 =/= 0 for finite...
  9. M

    Understanding the Mapping Problem on the Unit Circle Bisected by the X-Axis

    Homework Statement Take the unit circle in the x-y plane with center at (0, 0), bisected by the x-axis. Take two maps, the first MS from the circle minus the south pole S to the x-axis that take a point P on the circle to the intersection of the line from the south pole (0, −1) through P with...
  10. P

    Show no non-abelian group G such that Z(G)=Z2 exists satisfying the mapping

    Homework Statement Show that there is no non-abelian group G such that Z(G)=\mathbb{Z}_2, which satisfies the short exact \mathbb{Z}_2\rightarrow G\rightarrow\mathbb{Z}_2^3.The Attempt at a Solution I have knowledge of group theory up through proofs of the Sylow theorems. I know the center is...
  11. W

    Mapping Torus of a Manifold is a Manifold.

    Hi, All: I'm trying to show that the Mapping torus of a manifold X is a manifold, and I'm trying to see what happens when X has a non-empty boundary B. Remember that the mapping torus M(h) of a space X by the map h is constructed like this: We start with a homeomorphism h:X-->X (we...
  12. L

    What does the complex number 1/2*(1+i) represent in linear mapping?

    Homework Statement i recently saw a question about complex number, and its answer about the center of a circle is 1/2*(1+i). what does that mean? Homework Equations f:ℂ → ℝ^2 The Attempt at a Solution since we define z=x+yi is an element of ℂ, so by the mapping above, we can say...
  13. STEMucator

    Understanding Mappings: Injective, Surjective, and Homomorphisms

    So I want to clarify if what I'm thinking is correct. Suppose we have a mapping f : A → B and we have a in A and b in B. If f is an injective map, then f(a) = f(b) implies that a = b or conversely a≠b implies f(a)≠f(b). If f is a surjective map, then for b in B, there exists an a in A such...
  14. M

    Conformal Mapping for Transforming Regions: Finding a Function

    Hello folks, I am trying to find a conformal mapping transform function that maps the following region in z-plane into interior of a unit circle in w-plane: |z-i|<\sqrt{2}\text{ ...AND... }|z+i|<\sqrt{2} Many thanks in advance for help & clues. Max.
  15. Pythagorean

    Generalizing recursion in mapping functions

    I have a mapping function: x_{n+1} = \mu (1-x_n) I have some condition that occurs for: \mu (1-x_0) > 1 (1) which is: x_0 < 1- \frac{1}{\mu} but via the map function, there's an initial condition that leads to the above solution: **UNDER CONSTRUCTION, ERROR FOUND**
  16. B

    Mapping of Functions (Complex Analysis)

    Homework Statement Show that the function w = e^z maps the shaded rectangle in Fig X one-to-one onto the semi-annulus in Fig y. Fig x is the rectangle -1<x<1 ; 0<y<(x+pi(i)) Fig y is the semi-annulus such that y>0 and -e<r<-1/e Homework Equations ... The Attempt at a...
  17. C

    Mapping Conditions in Transformational Space

    Hello, My problem is as follows: I want to generate a series of 24 dimensional random numbers to act as the starting population for a genetic algorithm. These numbers need to fully span the space which is limited by a series of nonlinear boundary conditions. The 24 dimensional vector is...
  18. D

    Show that a mapping is continuous

    Homework Statement Show that the mapping f carrying each point (x_{1},x_{2},...,x_{n+1}) of E^{n+1}-0 onto the point (\frac{x_{1}}{|x|^{2}},...,\frac{x_{n+1}}{|x|^{2}}) is continuous. [b]2. Continuity theorems I am given. A transformation f:S->T is continuous provided that if p is a limit...
  19. B

    Proving Existence of Linear Mapping with Kernel in Subspace S | Helpful Guide

    Hey guys, I was wondering if you could help me out with a question I've got, I really don't know where to go or really where to start! Here's the question: Let S be a subspace of a finite dimensional vector space V. Show that there exists a Linear Mapping L: V → V such that the kernel of L is...
  20. D

    Linear Transformation from R^m to R^n: Mapping Scalars to Vectors

    Can we think of a linear transformation from R^m-->R^n as mapping scalars to vectors? Let me say what I mean. Say we have some linear transformation L from R^m to R^n which can be represented by a matrix as follows: L=[ a11x1+a12x2+...+a1mx m a21x1+... . . . anmx1+...+ anmxm...
  21. P

    3D Recipricol Space Mapping of Nanowires by using x-rays

    Hey I have a x-ray setup as in the figure, where alpha is the angles between the incoming x-ray beam and the sample. The x-ray are scattered, and measured by a 2D detector in the two outgoing angles. From this i will get a "slice" of the 3D recipricol map. If a want a 3D map, i think i will get...
  22. C

    Conformal Mapping: Is Non-Analytic Point Conformal?

    A theorm I took down in class says: Consider the analytic function f(z). The mapping w=f(z) is conformal at the point z0 if and only if df/dz at z0 is non-zero. However, if df/dz does not exist at that point z0, is that point still a conformal mapping? That would make the function...
  23. S

    Gnuplot Density Mapping: Tips and Tricks for Efficient Data Plotting

    Hi all, I used below bash command to plot my data (.dat file) #!/bin/bash ( echo 'set term jpeg' echo 'set style data lines' echo 'set yrange [0:200]' echo 'set xrange[0:200]' echo 'set pm3d map' echo 'set palette defined (-2 "yellow", 0 "green", 2 "red")'for f in "$@" do # echo "Processing...
  24. G

    Proof of mapping to and from null set

    Homework Statement Using the precise denition of a function and a little logic, show that, for every set Y , there is exactly one function f from \emptyset to Y . When is f injective? Surjective? Let X be a set. Show that there are either no functions from X to \emptyset or exactly one...
  25. S

    Mapping unit circle from one complex plane to another

    I want to show that if the complex variables ζ and z and related via the relation z = (2/ζ) + ζ then the unit circle mod(ζ) = 1 in the ζ plane maps to an ellipse in the z-plane. Then if I write z as x + iy, what is the equation for this ellipse in terms of x and y? Any help would be...
  26. J

    Why is T(1)=2 in the matrix of a linear mapping?

    Hi, I have the following problem that is solved, but I get lost at one step and cannot find how to do it in the notes. I would really appreciate it if someone could tell me where my teacher gets the result from. The problem says: "Find the matrix of linear mapping T:P_3 → P_3 defined by...
  27. A

    Hi,I need a conformal mapping that changes the superellipse to an

    hi, I need a conformal mapping that changes the superellipse to an easier shape. if anyone send me any helpful thing (relative article, idea) I will be so pleased.
  28. srfriggen

    Linear Algebra: Mapping Question

    Homework Statement From Serge Lang's "Linear Algebra, 3rd Edition", pg 51 exercise 9. Prove that the image is equal to a certain set S by proving that the image is contained in S, and also that every element of S in in the image. 9. Let F:R2→R2 be the mapping defined by F(x,y)=(xy,y)...
  29. J

    Mapping generator to generator in cyclic groups.

    Attached is my attempt at a proof. Please critque! :shy: Thank you!
  30. N

    A simple Complex Analysis Mapping

    Homework Statement http://img684.imageshack.us/img684/779/334sn.jpg The Attempt at a Solution The first part was fairly straightforward, solve for z + 1, and then get w in terms of u + iv, rationalise the denominator, and then we get (x,y) in terms of u and v, which we substitute back...
  31. A

    How Do Conformal Mappings Aid in Solving Laplace's Equation?

    Exam tomorrow and I am lacking understanding of conformal transformations and their applications. Can someone therefore point the main properties of conformal mappings that are used to make the conclusions in the following type of exercises: the mapping f(z) = 1 + 1/z maps the unit circle...
  32. S

    How to verify if a mapping is quotient.

    Prove or disprove that f is a quotient mapping. f:R^3\{(x1,x2,x3):x1=0}--->R^2 defined by (x1,x2,x3)|->(x2/x1,x3/x1)
  33. A

    Mapping function from 2D to 1D

    I have 2D elements distributed in a space of [-4, +4] and want to convert any point in the 2D space to a 1D real-valued number 0~1.0 such that 1st quadrant [+, +] should have higher values (importance) suppose 0.4~1 , 2nd and 3rd quadrant [+, -] and [-, +] should be next 0.2~0.4, and the 4th...
  34. L

    Solving Laplace's equation using conformal mapping

    I'm trying to use conformal mapping to solve for a function u(x,y) satisfying Laplace's equation ∇2u = 0 on the outside of the unit circle (i.e. the complement of the unit disk), with boundary conditions: u = 1 on the unit circle in the first quadrant, u = 0 on the rest of the unit circle...
  35. J

    Mapping contours over normal Riemann surfaces

    Hi, Can someone here help me understand how to illustrate maps of analytically-continuous paths over algebraic functions onto their normal Riemann surfaces? For example, consider w=\sqrt{(z-5)(z+5)} and it's normal Riemann surfaces which is a double covering of the complex plane onto a single...
  36. D

    Mapping Argand Plane to Upper Half Plane

    Homework Statement find linear fractional transformation from D={z:|Arg z| < \alpha}, \alpha≤\pi to the upper half plane Homework Equations The Attempt at a Solution The problem I am having here what exactly D is.. (visualizing it) D is just z such that |Arg z|≤\pi right? so...
  37. R

    Group Operation and True Meaning of Mapping

    Can't find (or maybe recognize when I see it) anything that discusses this question: A group G is a set of members. We normally assign familiar labels on the members such as a five member group with members labeled as 0, .. , 4. Then, a group operation + is defined as GxG -> G so that a look...
  38. B

    Total differential of general function mapping

    I am looking for an explanation and derivation of a total differential of a 2nd order function, i.e. a function that maps one function to another. To be more specific, let's say I have a function l:ℝ^n\to ℝ that I use to define a 2nd order function L:(ℝ^k\to ℝ^n) \to (ℝ^k\to ℝ) as L(f) :=...
  39. N

    Image of Circle |z| = 3 under Mapping w = 6/z

    Homework Statement Find the image of the circle |z| = 3 in the complex plane under the mapping a) w = \frac{6}{z} b) w = \frac{6}{z} + 2i The Attempt at a Solution a) w = \frac{6}{3} = 2 So this is a circle in the w-plane of radius 2, centered on the origin? b) w =...
  40. T

    Conformal Mapping: Part II - Finding u and v for Given Values of x and y

    Homework Statement part ii of http://gyazo.com/0754ea00b2a4ea4a4d171906f6bf28bf Answers http://gyazo.com/821f370c502cd20210925f8498d18fa1 Homework Equations I did part i. I had to spot that 1/(x+iy)^2 = 1/(x^2+y^2)^2... (I subbed y = y-1) is this a standard result? Should...
  41. D

    Analytic mapping of unit disc onto itself with two fixed pts.

    Homework Statement let f(z) be a 1-1 analytic mapping of unit disc |z|<1 onto itself with two fixed points in |z|<1 Show that f(z)=z Homework Equations none The Attempt at a Solution I'm thinking there has to be a theorem or something that I am missing for this.. But I'm not...
  42. A

    MHB Prove F(x) Maps [0,1] into Itself and Not Contraction

    Prove that the function F(x) = 4x(1-x) maps [0,1] into itself and it not contraction to prove it is not contraction it is enough to prove that there exist a number in [0,1] such that the first derivative exceed 1 F'(x) = 4(1-x) - 4x = 4 - 8x 4-8x > 1 \Rightarrow \frac{3}{8} > x...
  43. binbagsss

    Matrix Transformation - A plane mapping onto itself.

    Find which planes map onto themselves under the matrx M. M= 1 2 0 0 1 -1 0 2 1 (in enclosed brackets - apologies for the format.). Attempt: Consider a plane ax+by+cz=d [1]. M^-1 : 3/3 -2/3 -2/3 0 1/3 1/3 0 -2/3 1/3 (in enclosed bracket). - use of the inverse so that...
  44. I

    Biholomorphic Mapping: Proving f(z) = z for All z in Ω?

    Suppose f is a biholomorphic mapping from Ω to Ω, if f(a) = a and f'(a) = 1 for some a in Ω, can we prove that f(z) = z for all z in Ω?
  45. D

    MHB Conformal Mapping of Strip -1 < Im(z) < 1

    Describe the image of the strip $\{z: -1 < \text{Im} \ z < 1\}$ under the map $z\mapsto\dfrac{z}{z + i}$ So I know that $-\infty < x < \infty$ and $-1 < y < 1$. Then $$ \frac{x + yi}{x + i(y + 1)} $$ Now if I take the the line y = -1, I have $$ \frac{x-i}{x} $$ Then find out what happens...
  46. D

    MHB Fractional linear transformation--conformal mapping

    Find necessary and sufficient conditions on the real numbers $a$, $b$, $c$, and $d$ such that the fractional linear transformation $$ f(z) = \frac{az + b}{cz + d} $$ maps the upper half plane to itself. I just need some guidance on starting this one since I am not sure on how to begin.
  47. A

    Electrostatic Conformal Mapping Problem

    Homework Statement The transformation z=1/2(w + 1/w) maps the unit circle in the w-plane into the line −1≤x≤1 in the z-plane. (a) Construct a complex potential in the w-plane which corresponds to a charged metallic cylinder of unit radius having a potential Vo on its surface. (b) Use...
  48. A

    Electrostatic Conformal Mapping Problem

    Homework Statement The transformation z=\frac{1}{2}(w + \frac{1}{w}) maps the unit circle in the w-plane into the line −1≤x≤1 in the z-plane. (a) Construct a complex potential in the w-plane which corresponds to a charged metallic cylinder of unit radius having a potential Vo on its surface...
  49. R

    Conformal mapping between two half space

    Hi all, Suppose there is a bump at the origin, is there a conformal mapping between the bumped half-space (y>|b-x|, |x|<b && y>0, |x|>b) and the flat upper half space (y>0)? Anyone has a hint? Thanks in advance. Regards, Tony
  50. J

    Mapping a matrix to the null space

    Homework Statement I am trying to run a model in matlab. D is a 2 by 3 matrix, Knowing that DL=0, which means L is mapped to the null space. Homework Equations How can i find L so that it is a 3 by 3 matrix with all its entries being one times a scalar. The Attempt at a Solution...
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