Mapping Definition and 404 Threads

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. A

    MHB Quick question about continuous mapping

    When f maps E into a metric space Y: (E is subset of metric space X) Is it eqivalent to say that f is a continuous mapping and that for a subset E of X, to say that for every p element of E, f is continuous at p.? thank you
  2. H

    Proving Complex Mapping: f(z) Maps Real Axis to Circle of Radius 1

    Homework Statement Let a be a complex number for which Im(a) ≠ 0, and f(z) = (z + conj(a))/(z + a). Prove f(z) maps the real axis onto the circle lwl = 1. 2. The attempt at a solution I wrote out f(z) in an a+bi for and then with the Im(a) ≠ 0 I set the equation as f(a+bi) =...
  3. J

    Show that an analytic mapping is an open mapping

    Homework Statement As in in title. Homework Equations Open mapping: maps open sets to open sets. The Attempt at a Solution Not sure.
  4. N

    Complex Analysis - Finding the image through a mapping

    Homework Statement The point 1 + i is rotated anticlockwise through \frac{∏}{6} about the origin. Find its image. The Attempt at a Solution The point 1 + i creates an angle of arctan(1/1) = ∏/4 The rotation is by a further angle β = ∏/6. So the new point w in the w-plane from...
  5. U

    Verifying Linear Polynomial Mapping

    Homework Statement Prove whether the below equations are linear or not. (iii) U = P^2 -> V = P^6; (Tp)(t) = (t^2)p(t^2) + p(1). (iv) U=P^2 -> V =P^6;(Tp)(t)=(t^2)p(t^2)+1. Homework Equations None. The Attempt at a Solution I really don't know. Thanks Tom
  6. T

    Mapping Functions: Is ∅ an Isomorphism?

    Homework Statement Let F be the set of all functions f mapping ℝ into ℝ that have derivatives of all orders. Determine whether the given map ∅ is an isomorphism of the first binary structure with the second. {F,+} with {ℝ,+} where ∅(f)=f'(0) Homework Equations None The Attempt at...
  7. J

    Proving Open Mapping of Canonical Projection in Normed Vector Space

    Homework Statement Let (X,||\cdot||) be a normed vector space and suppose that Y is a closed vector subspace of X. Show that the map ||x||_1=\inf_{y \in Y}||x-y|| defines a pseudonorm on X. Let (X/Y,||\cdot||_1) denote the normed vector space induced by ||\cdot||_1 and prove that the...
  8. K

    Calculating Electric Field Values for Point-Line Plate

    Homework Statement Compute values for the electric field at four different points on the point-line plate. Comment on the validity of your values. Homework Equations E = F/q ΔV = ∫E dot ds The Attempt at a Solution I have attached 3 electric field maps that I did in the lab and...
  9. S

    Solve Karnaugh Map for 2-Bit Binary Product Problem

    Homework Statement I'm working on a problem that implements the product of two 2-bit binary numbers (wx, yz) and produces such as the output. However, I am having a bit of confusion in regards to implementing the Karnaugh Map. So this is what I have: wx and yz can be 00, 01, 10, 11, the output...
  10. C

    Mapping a cylinder onto a sphere

    Homework Statement How might I show that the map (in cylindrical polar coordinates) given by f:(1,\phi,z)\to(\sqrt{1-z^2},\phi,z) does not change the area?Homework Equations The Attempt at a Solution I can see this is like having a sphere in a cylinder and we shine "light" on the cylinder...
  11. W

    Integration of functions mapping into a vector space

    Given a measurable function f that is not real- or complex valued, but that maps into some vector space, what are the necessary conditions for it to be integrable? I've looked through over 20 books on integration and measure theory, but they all only deal with integration of real (or...
  12. S

    A basic Theory Question (Electric Field Mapping)

    If you reverse the polarities of the electrodes when you are "electric field mapping", wouldn't it show the same results on paper? Just the path it takes is reversed, but you would never know the difference wether it's going away or towards correct? I am diving into this "Electrical Field...
  13. S

    Exploring Complex Mapping: Images of i, 1-i, and the Real and Imaginary Axes

    Homework Statement Let the Complex mapping z → f(z) =(1 + z)/(1 − z) 1.What are the images of i and 1 − i and 2.What are the images of the real and the imaginary axes? The Attempt at a Solution For i we have f(i)=(1+i)/(1-i) since i(depending on the power) can be i,-i,1,-1=>0...
  14. H

    Mapping of complex exponential

    Homework Statement Determine the image of the line segment joining e^(i*2*pi/3) to -e^(-i*2*pi/3) under the mapping f = e^(1/2*Log(z)). Homework Equations The Attempt at a Solution The line joining the two points: {z | -0.5 < x 0.5, y = sqrt(3)/2} f = the principle branch of...
  15. P

    Looking for an analytic mapping theorem

    Say we have a complex function f, analytic on some punctured open disk D\{a} where it has a pole at a. Is there some theorem which says something like: f must map D\{a} to a horizontal strip in ℂ of at least width 2π, or something like that?
  16. C

    Mapping Function for (0,1) into Open Unit Square

    Homework Statement Consider the open interval (0,1), and let's S be the set of points in the open unit square that is, S={(x,y):0<x,y<1}. Find a function that maps (0,1) into S. but not necessarily onto. The Attempt at a Solution so I can describe any point in my square with x and y...
  17. H

    Prove function to be one-to-one: Complex Mapping

    Homework Statement Show that f = sin(z) is one-to-one in the set S = { z | -pi < Re(z) < pi, Im(y) > 0}. That is, show that if z1 and z2 are in S and sin(z1) = sin(z2) then z1 = z2. Hint: Try to find the image of S through f. Homework Equations The Attempt at a Solution z = x...
  18. K

    The set of 1-1 Mapping of S Onto itself

    Homework Statement I was reading my textbook and i encountered this...--->> " For instance if f,g,h are in A(S) and fg = fh then g=h " I understand this part... because we can take the the inverse of f both sides and say g=h. then it says--->> " If gf = f^(-1)g but since f ≠ f^(-1)...
  19. H

    Circle Inversion Mapping: Proof of w = 1/z Transforming |z-1| = 1 to x = 1/2

    Homework Statement Show that the inversion mapping w = f(z) = 1/z maps the circle |z - 1| = 1 onto the vertical line x = 1/2. Homework Equations The Attempt at a Solution z = a + ib w = x + yi = a^2/(a^2 + b^2) + ib^2/(a^2 + b^2) |z - 1| = |(a -1) + ib | = 1 (a - 1)^2 + b^2 = 1...
  20. A

    Why Is the Rank of a Matrix Equal to Its Number of Pivot Points?

    How come the rank of a matrix is equal to the amount of pivot points in the reduced row echelon form? My book denotes this a trivial point, but unfortunately I don't see it :(
  21. C

    Proving the Facet Property of Dual Polytopes with a Vertex in the Interior

    Homework Statement Let v be a vertex of a d-polytope P such that 0 \in intP . Prove that P^* \cap \{y \in \mathbb{R}^d \mid\left < y, v\right>=1\ \} is a facet of P^{*} . Thanks Homework Equations The definitions are: P^*=\{ y\in\mathbb{R}^{d}\mid\left < x...
  22. L

    Finding [L] for Two Consecutive Linear Transformations in R3

    Homework Statement A- Find [L] for L : R3 → R3 where L first reflects throught the plane x−z = 0 and then rotates the zy-plane by pi/6 counterclockwise starting from the y-axis. B-Find [L] for L : R3 → R3 where L first rotates the xy-plane by pi/4 counterclockwise starting from the x-axis and...
  23. L

    Bilinear mapping between quotient spaces

    Problem: Let L and M be finite dimensional linear spaces over the field K and let g: L\times M \rightarrow K be a bilinear mapping. Let L_0 be the left kernel of g and let M_0 be the right kernel of g. a) Prove that dim L/L_0 = dim M/M_0. b) Prove that g induces the bilinear mapping g': L/L_0...
  24. T

    Mapping of a Circle in the z-plane to the w-plane

    Homework Statement Consider the mapping w = 1/(z-1) from the z-plane to the w plane. Show that in the z plane the circle (x-2)² + y² = 4 maps to a circle in the w-plane. What is the radius of this circle and where is it's centre. So in the z-plane this is a circle with radius 2 at the point...
  25. V

    Finding the conditions for a particular mapping to be a bijection

    Homework Statement here's the problem: Let A and B be n x n matrix with coefficient in K (any field), let Mn(K) be the set of all n x n matrix with coefficient in K . T is a linear map defined like this T : Mn(K)---> Mn(K) T(Y) = AYB what are the necessary conditions for T to be a...
  26. S

    Proving Contraction Mapping: T^n is a Contraction

    [b]1. prove that,if T is acontraction mapping then T^n is a contraction Homework Equations The Attempt at a Solution
  27. T

    Is a Mapping Between Lie Algebras an Isomorphism if it Takes a Basis to a Basis?

    If a mapping between Lie algebras \varphi : \mathfrak{g} \to \mathfrak{h} takes a basis in \mathfrak{g} to a basis in \mathfrak{h} is it an isomorphism of vector spaces?
  28. A

    A special invertible k-bits to n-bits mapping

    I want a sequence of n bits without all 1's in the output. What is the minimum number of bits k given n that can be used in any linear or non-linear invertible mapping that will produce such a sequence at output. For example, consider n=3. I want to create a mapping that has all 2^3=8 minus 111...
  29. T

    Showing a mapping between lie algebras is bijective

    Homework Statement Show the map \varphi : \mathfrak{g} \to \mathfrak{h} defined by \varphi (aE + bF + cG) = \begin{bmatrix} 0 & a & c \\ 0 & 0 & b \\ 0 & 0 & 0 \end{bmatrix} is bijective. \mathfrak{g} is the Lie algebra with basis vectors E,F,G such that the following relations for...
  30. Somefantastik

    Finding inverse of linear mapping

    so for a mapping f:X->Y where X,Y are Normed Vector Spaces if I have a function f(x) = y such that x in X and y in Y, how do I explicitly find f inverse? I sat down to do this and realize I've only been trained in the Reals where you switch the x,y and then solve for y. But this won't...
  31. S

    Understanding Continuity and the Jacobian Matrix in Multivariable Functions

    Homework Statement a) Let f: RN to RM. Define continuity for mapping f. How does this relate to the notion of metric (norm)? b) Define the Jacobian J of f. Write Taylor series expansion (for f) up to first degree at x = x0. Explain the terms. c) Let y = f(x) \in RM and yj = |f(x)|j = sum...
  32. B

    Mapping Class Group of the Torus.

    Hi, All: I am trying to figure out the mapping class groupof the torus ; more accurately, I am trying to show that it is equal to SL(2,Z). The method: every homeomorphism h: <\tex> T^2 -->T^2<tex> gives rise to, aka, induces an isomorphism g: <\tex> \mathbb pi_1(T^2)-->\mathbb...
  33. B

    Quotients of Mapping Class Group Iso. to Symplectic Group.

    Hi, Again: I am a bit confused about this result: Mg/Mg^(2) ~ Sp(2g,Z) (group iso.) Where: i) Mg is the mapping class group of the genus-g surface, i.e., the collection of diffeomorphisms: f:Sg-->Sg , up to isotopy. ii)Mg^(2) is the subgroup of Mg of maps that...
  34. A

    How Does Conformal Mapping Affect Image Dimensions in Mathematica?

    Hi everybody, I was looking at the following link: http://www.dimensions-math.org/Dim_CH5_E.htm The section 6 deals with conformal mapping of the image for different kinds of transformations. I tried to reproduce them in mathematica for the transformation z \rightarrow z^2. I followed the...
  35. D

    Inverse perspective mapping equations

    Hi . I am making a robot that is supposed to detect a red ball with a camera, and then know where the ball is . The robot has a camera on top of it, that is tilted at an angle ; At each frame the camera detects the red ball and returns 2 coordinates , xp and yp which are the pixel...
  36. X

    Mapping points inside one 2D rectangle into another smaller one

    in some work I'm doing i have a 2D rectangle that can be rotated and/or translated in any direction in 2D space. for example it might look like this: (x=30,y=-10) +-----------+ (x=30,y=2) +-- +y | | |...
  37. N

    Proof of Open Mapping Theorem? (Ash & Novinger)

    Hello, In class we're using the free course on complexe analysis by Ash & Novinger, legally downloadable online. I'm stuck on their proof of the open mapping theorem. More specifically: http://www.math.uiuc.edu/~r-ash/CV/CV4.pdf (page 15) proposition (d) is f(\Omega) \textrm{ is open} and its...
  38. B

    Python Python code for mapping numbers n text file

    Dear Programmers I am having text file having huge data. From the text file i would like to map the two numbers that are present in the same brackets and the values should not considered if they are present in different numbers. say for example I want to map number "4758 & 3895" that are in...
  39. Y

    The mapping to alternating tensors

    I'm wondering why 1/k! is needed in Alt(T), which is defined as: \frac{1}{k!}\sum_{\sigma \in S_k} \mbox{sgn}\sigma T(v_{\sigma(1)},\cdots,v_{\sigma(k)}) After removing 1/k!, the new \mbox{Alt}, \overline{\mbox{Alt}}, still satisfies...
  40. Y

    Discontinuous linear mapping between infinite-dimension vector space

    It is known that any linear mapping between two finite dimensional normed vector space is continuous (bounded). Can anyone give me an example of a linear mapping between two infinite dimensional normed vector space that is discontinuous? Thanks
  41. S

    Mapping functions and bijections

    Homework Statement Hello! I am stuck, having wondered about this question for quite some time now and I am not too sure how to solve it Denote the xy-plane by P. Let C be some general curve in P defined by the equation f(x,y) = 0, where f(x,y) is some algebraic expression involving x...
  42. A

    Solving for b^2+c^2 with Function and Mapping

    The function is one-one find the value of b^2+c^2[f(x)=x^3+3x^2+4x+bsinx+ccosxI tried the approach of monotonocity as a one one function will be strictly inc or decreasing in it's domain but I'm not being able to figure out b^2+c^2
  43. A

    Conformal mapping application- electrostatics

    Homework Statement Consider the transformation: w = i[(1-z)/1+z)] Find the electrostatic potential V in the space enclosed by the half circle x^2 + y^2 = 1, y =>0 and the line y = 0 when V = 0 on the circular boundary and V = 1 on the line segment [-1,1]. Homework Equations w = u +...
  44. A

    Mapping a general curve onto a bijection.

    Homework Statement Denote the xy plane by P. Let C be some general curve in P defined by the equation f ( x , y ) = 0 where f ( x , y ) is some algebraic expression involving x and y. Verify carefully that if B : P -> P is any bijection then B( C ) is defined by the equation f ( B^-1...
  45. A

    Conformal Mapping: How Do I Map the Region Above the x-axis?

    What I'm trying to do is to apply conformal mapping and map the area bounded by the x-axis and a line at 60 degrees to the x-axis to the region above the x-axis. I think the basic goal of what I'm trying to do is to map \pi/3 to \pi. My problem is I really have no idea where to go from there...
  46. Y

    Differentia of function-valued mapping

    I have difficulty in understanding why the differential d\varphi of a function \varphi: \eta \rightarrow F(\cdot, \eta) on \Re can be written as: [\varphi'(b)](\xi)=\frac{\partial F}{\partial y}(\xi,b) (F is defined on RxR) according to the following theorem. the differential d\varphi_{b}...
  47. A

    Capacitance calculation by using conformal mapping

    Hi all How to calculate the capacitance of a sphere-plane system by using conformal mapping? Thanks
  48. C

    Squeeze Mapping: Transform Circle to Quarter Moon Shape

    Hello! I am trying to transform a circle into a "quarter moon shape". This is that every point in the circle is mapped into the "quarter moon shape" - therefore squeezed. In particular I am looking for a expression that relates x x' and y y' ... Can any bright mind help me :) ? Thank you
  49. Y

    Is every bilinear mapping bounded?

    In a book I'm reading, it defines a bounded bilinear mapping \omega: X\times Y\rightarrow W, where X,Y and W are all normed linear spaces as \left\| \omega(\xi,\eta)\right\| \leq b \left\| \xi \right\| \left\| \eta \right\| So it uses \left\| \xi \right\| \left\| \eta \right\| as a norm on...
  50. C

    Conformal Mapping: Transform a Circle to a Rectangle

    Hello! Please I need some help with this: Is it possible to transform a circle into a rectangle? If so what would be the expressions of x' and y' in terms of x and y. Thank you in advance!
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