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. T

    Complex Analysis Mapping With Principal Branch

    1. Verify that f(z) = Sqrt(z^2 - 1) maps the upper half plane I am z > 0 onto the upper half plane I am w > 0 slit along the segment from 0 to i. [Hint: use the principal branch] 2. Homework Equations We studied factional linear transformations with T(z) = (a z + b) / (c z + d) , but I...
  2. S

    Conformal Mapping of Upper Half Plane from Unit Half Disc | Homework Solution

    Homework Statement Find a conformal mapping f of the set V to the upper half plane H+ = {z | Im(z) > 0 where V = {z: |z| < 1 and Im(z) > 0} Homework Equations None, really. It's worth noting that V is the unit half disc. The Attempt at a Solution I have a Mobius transformation S...
  3. F

    Support Mapping of an Arbitrary Ellipsoid

    In this context, the support mapping of any convex geometry is any point on the geometry which results in the largest dot product to some direction vector. I would appreciate some help in computationally finding the support mapping of an arbitrary ellipsoid (some arbitrary orthonormal basis...
  4. A

    Find any fixed points for the following mapping:

    Find any fixed points for following mapping f(z) = z2 - z + 1 Map onto the same point gives: z = z2 - z + 1 0 = z2 - 2z + 1 Therefore z = 1 and z = 1
  5. B

    What is somatic cell hybrid mapping and how is it used in genetic research?

    Hello, I have an upcoming exam on human genetics and genetic techniques and I am trying to learn about this procedure. I have searched the internet but come up with more studies that have used this technique rather than a plain 'for dummies' explanation of what it is. If someone could offer...
  6. T

    Hair brained, universal mapping theory (nonsense ?)

    Firstly, Hi everyone :) Secondly, please forgive my virtualy non existent knowledge of any theories that this relates to. I'm not a Dr,MD, or even a graduate. I'm not even studying anything anywhere. I have a few 'problems' (chronic anxiety, depression, yada yada yada) not looking for sympathy...
  7. B

    Surjection Between Mapping Class Grp. and Symplectic Matrices

    Hi, Everyone: I am reading a paper that refers to a "natural surjection" between M<sub>g</sub> and the group of symplectic 2gx2g-matrices. All I know is this map is related to some action of M<sub>g</sub> on H<sub>1</sub>(S<sub>g</sub>,Z). I think this action is...
  8. G

    Complex analysis: mapping a hyperbola onto a line

    Homework Statement We want to create a map from (x,y) to (u,v) such that the right side (positive x) of the hyperbola x^2 - y^2 = 1 is mapped onto the line v = 0 AND all the points to the left of that hyperbola are mapped to above the line. The mapping should be one-to-one and conformal...
  9. T

    Topology: Homeomorphic Mapping

    Homework Statement Let X = [0,2 PI] x [0,2 PI] and let D be the partition of X that contains: -{(x,y)} when x =/= 0,2 PI and y =/= 0,2 PI -{(0,y),(2 PI,y)} for 0 <= y <= 2 PI -{(x,0),(x,2 PI)} for 0 <= x <= 2 PI Let U be the quotient topology on D induced by the natural map p: X ->...
  10. S

    Exploring Bijective Holomorphic Maps on Open Sets and the Unit Circle

    Homework Statement Let f and g be bijective holomorphic maps from an open set A to the unit circle. Let a \in A and c=f(a) and d=g(a). Find a relation between f and g that involves a,c,d,f'(a),g'(a).Homework Equations The Attempt at a Solution If we also assumed that the open set is...
  11. S

    What is the Mod 2 Degree of a Mapping?

    Solved, made a silly mistake. Thanks for reading.
  12. I

    Automorphism groups and determing a mapping

    1. Suppose that Ø:Z(50)→Z(50) is an automorphism with Ø(11)=13. Determine a formula for Ø(x). this is the problem I am getting, its chapter 6 problem 20 in Gallian's Abstract Algebra latest edition (you can find it on googlebooks) Am i wrong in thinking there's something wrong with the problem...
  13. S

    Confusion about defn. of Surjective mapping in WIKI.

    Reference: http://en.wikipedia.org/wiki/Bijection,_injection_and_surjection Consider the two sets X & Y connected by a the relation y^2=x^2. (For simplicity we can take X={-2,2} and Y={-2,2}).Then can we call the mapping from X to Y to be surjective? From the definition of WIKI, the answer...
  14. M

    Show mapping is an automorphism

    Show that the mapping Phi(a+bi)=a-bi is an automorphism of the group of complex numbers under addition. I have this as of now: Let (c+di) and (a+bi) be elements in group G. 1) phi(a+bi) is function phi from G to G, by assumption. Therefore the function is a mapping. 2) 1-1: assume...
  15. E

    Double Check My Conformal Mapping Steps for Quarter-Plane

    Homework Statement Can someone double check my mapping steps I've taken? The domain includes the quarter-plane Re(z)>0 and Im(z)>0, with a branch cut from the origin to the point \sqrt{3}exp(pi*i/4) The Attempt at a Solution I want to map the region from the quarter-plane to the...
  16. S

    Define the nonlinear function mapping

    Homework Statement Let P2 be the vector space of real polynomials of degree less or equal than 2. Define the (nonlinear) function E : P2 to R as E(p)=integral from 0 to 1 of ((2/pi)*cos((pi*x)/2)-p(x))^2 dx where p=p(x) is a polynomial in P2. Find the point of minimun for E, i.e. find...
  17. F

    Finding Center and Radius of Circle in Conformal Mapping

    Can you tell me is my solution true of the next problem. Find center w_0 and radius R of the circle k, in which the transformation w=\frac{z+2}{z-2} converts the line l:\text{Im} z+\text{Re} z=0. Solution: 2 \to\infty -2i=(2)^*\to w_0...
  18. P

    Smooth Mapping Between Unit Circle and Curve in R^2?

    Hi, I have been told that in R^2 the unit circle {(x,y) | x^2 + y^2 = 1} is smoothly mappable to the curve {(x,y) | x^4 + y^2 = 1}. Can someone please tell me what this smooth map is between them? I can only think of using the map (x,y) --> (sqrt(x), y) if x is non-negative and (sqrt(-x), y)...
  19. W

    Preserving Distance: Exploring Metric Requirements for Mapping

    Can someone please exPlain to me what the phrase. Which metric do we have to impose in order that the mapping preserves distance means. The example I have is ((-),phi)--->(x,y) = (2a tan(theta/2)cos(phi) , 2a tan(theta/2)sin(phi)) thanks
  20. W

    Help for proving a mapping is a diffeomorphism

    Hi, does anyone have any idea how to prove the mapping R^2->R^2 (x/(x^2+y^2), y/(x^2+y^2) is a diffeomorphism, and if it is not restrict the values so it is one I am fairly sure it is not over R^2 as it is not continuous at 0, but I don't know what values to restrict it over. I have...
  21. W

    How to show a mapping is a diffeomorphism

    mapping
  22. J

    Mapping a circle to an ellipse

    How does the function f(z) = z + 1/z take a circle of radius g.t. 1 to an ellipse? How do I think about it geometrically ? (i.e., how should I be able to look at the complex function and tell straight away)
  23. Pengwuino

    Holographic principle and entropy mapping of a BH

    I've just started reading up on the holographic principle and eventually want to work my way to figuring out what Verlinde has proposed using it for. One thing I've noticed in a couple of papers is the mapping of the entropy of a black hole onto a holographic screen. Why are families of light...
  24. B

    Mapping Class Group and Path-Component of Id.

    Hi, everyone: Given a smooth, orientable manifold X, we turn Aut(X) the collection of all self-diffeos. of X into a topological space, by using the compact-open topology. Aut(X) is also a group under composition. The mapping class group M(X) of X is defined as the quotient: M(X):=...
  25. B

    What is the connection between the mapping class group of a torus and Gl(2,Z)?

    Hi, everyone: I am trying to understand why the mapping class group of the torus T^2 (i.e., the group of orientation-preserving self- diffeomorphisms, up to isotopy) is (iso. to) Gl(2,Z) ( I just realized this is the name of the group of orientation-preserving...
  26. Q

    Bijective Mapping of Cartesian Products: X^{m} \times X^{n} to X^{m + n}?

    Homework Statement Find a bijective map g : X^{m} \times X^{n} \rightarrow X^{m + n} Homework Equations The Attempt at a Solution I don't even know where to begin. How would I map X^{m} \times X^{n} in the first place? How could I map X^{m + n}?
  27. M

    Mapping cones and Homology groups

    Let W be the mapping cone of the map f: S^{1} \rightarrow S^{1} defined by f(z)= z^{p} How do you compute the homology groups of W? What about the homology groups of the Universal covering of W? I know that the mapping cone C_f of f:X\rightarrow Y, is defined to be the quotient of the...
  28. C

    Mapping Points on the Bloch Sphere

    Hey guys, I'm attempting to map some discrete points on the surface of the Bloch sphere: For instance, the full spectrum of ranges for variable theta is 0 < theta < pi. However, my goal is to limit that range from some theta_1 < theta < theta_2. I was going to use a spherical harmonic...
  29. A

    Find Array Mapping Homework Solution

    Homework Statement I don't know if this has ties to linear algebra, so sorry in advance if I'm posting in a wrong section. We have an n*n matrix A, n is an odd number, and "the matrix's sides are 0" meaning: we'll call non-zero elements as 1's for now.1st line 1111...1111 2nd line 0111...1110...
  30. K

    Contraction Mapping Theorem: Proving Continuity and Convergence of a Sequence

    Homework Statement Let f be a function defined on all of R and assume there is a constant c such that 0<c<1 and |f(x)-f(y)<c|c-y| a) Show f is continuous on all of R b)Pick some point y1 in R and construct the sequence (y1,f(y1),f(f(y1)),...) In general if y_(n+1)=f(yn) show that the...
  31. G

    Solving Mapping Problem: Let D = {x ∈ R, -3 ≤ x ≤ 5, x ≠ 0}

    Let D = {x \in R : -3\leq x \leq 5 and x \neq0} and define g(x) = [cos(x) - 1]/x + sqrt(x+3)(5-x) Find G: R ->R such that G is continuous everywhere and g(x)=G(x) when x\inD I'm not really sure how to start this and I've been looking at it for quite a while now. I might just need a push...
  32. M

    Is Every Continuous Open Mapping Monotonic?

    Homework Statement So I'm going through baby Rudin. Problem 15 in chapter 4 has us trying to prove that every continuous open mapping is monotonic. I'm trying to see how this is the case. So, I'm considering f(x) = sin(x). Let V = (Pi/3, 7*Pi/12) be an open set. Then, f(V) = (\sqrt{3}/2, (1...
  33. O

    How to Calculate the Mapping of Points from a Circle to a Tangent?

    Assume you're given a circle with the line AB containing its center O, such that A and B are on the circle (OA=OB=radius). A tangent t is drawn on the point A, and I should calculate the mapping of certain points (a,b,c,d...) of the circle to the points on the tangent (at, bt, ct, dt, ...) such...
  34. T

    Element in a ring mapping one prime to the next

    Homework Statement Let {p_n}n>0 be the ordered sequence of primes. Show that there exists a unique element f in the ring R such that f(p_n) = p_n+1 for every n>0 and determine the family I_f of left inverses of f. Homework Equations The ring R is defined to be: The ring of all maps...
  35. F

    Left invertible mapping left inverse of matrix

    Homework Statement relation from R^2-->R^2 ( R is real line) (y1) [0 1] (x1) (y2) =[-1 1] (x2) is this left invertible? if so what is the left inverse? y1,y2 are element in a 2by 1 matrix, same with x1, x2. the elemenst 0,1,-1,1 are in a 2x2 matrix. I did no know how to...
  36. radou

    Continuity of a mapping in the uniform topology

    Homework Statement Let (a1, a2, ...) and (b1, b2, ...) be sequences of real numbers, where ai > 0, for every i. Let the map h : Rω --> Rω be defined with h((x1, x2, ...)) = (a1x1 + b1, a2x2 + b2, ...). One needs to investigate under what conditions on the numbers ai and bi h is continuous...
  37. A

    Mapping intervals to sets which contain them

    I have recently been extremely bothered by the fact that we can construct a bijection from [0,1] onto the entire two-dimensional plane which itself contains [0,1]. Similarly, I have been bothered by the fact that we can construct a bijection from (0,1) to all real numbers. Indeed we do so...
  38. Z

    Polynomial Mapping: Linear Algebra Basics for Beginners

    Hi I'm a new student and don't have any basics for linear algebra. thanks Homework Statement Q1. Let X be the vector space of polynomials of order less than or equal to M. a) show that the set B = {1,x,...x^M} b) consider the mapping T from X to X as defined as : f(x)=T...
  39. S

    Recombination Freq.: Genetic Mapping Hwk Problem

    Homework Statement The recombination frequency between gene “Q” and gene “Z” is found to be 23.5%. The recombination frequency between “Q” and a third gene “R” is 10%. The frequency of recombination between “Z” and “R” is 13.5%. Which one of the following is NOT true concerning these three...
  40. S

    Biology Genetics: Mapping and Recombinant Frequency

    Homework Statement Two homozygous plants are crossed and all offspring (F1) are yellow with long stems. A cross of the offspring (F1)produces the following phenotypes in the F2 generation: 1,263 yellow, long stem 1,196 white, short stem 205 yellow, short stem 195 white, long stem...
  41. P

    Distnace between points and isometric mapping

    We define isometric mapping so that its tangent mapping preserves the scalar product of vectors from tangent space (the definition doesn't refer explicite to notion of distance in the manifold). Distance between two points of manifold is the length of geodesics which joins them. I wonder...
  42. N

    Contraction Mapping Theorem Question

    Homework Statement Consider the function g : [0,∞) → R defined by g(x) = x + e−2x. Given |g(x2) − g(x1)| < |x2 − x1| for all x1, x2 ∈ [0,∞) with x1 ≠ x2. Is g a contraction on [0,∞)? Why? Homework Equations I think we are intended to use the given equation and the CMT CMT states that...
  43. E

    How can I get help with my MATLAB coursework?

    I Have just received this coursework (see the two attachments below) from my Applications of Computing module and was looking to be able to discuss some questions I have as I go about completing the coursework. firstly by simplifying the formula given I have the following Xn+1 =...
  44. P

    Is the Mapping T Uniformly Continuous on [0,1] x [0,1]?

    Homework Statement Suppose X = [0,1] x [0,1] and d is the metric on X induced from the Euclidean metric on R^2. Suppose also that Y = R^2 and d' is the Euclidean metric. Is the mapping T: [0,1] x [0,1] \rightarrow R^2, T(x,y) = (xy, e^(x.y)) uniformly continuous? Explain your answer...
  45. E

    Complex variables conformal mapping trig identity

    Homework Statement map the function \begin{equation}w = \Big(\frac{z-1}{z+1}\Big)^{2} \end{equation} on some domain which contains z=e^{i\theta}. \theta between 0 and \pi Hint: Map the semicircular arc bounding the top of the disc by putting $z=e^{i\theta}$ in the above formula. The...
  46. P

    Mapping Elements in Quotient Group G/N to Isomorphism f:G/N--H

    Can an element of a quotient group G/N in an isomorphism f:G/N--H map back onto itself if it does not have a corresponding element in H? The example I am looking at is the quotient group of the group of symmetries of an n-gon, G, where n is an even number and N equal to the normal subgroup...
  47. A

    An open mapping is not necessarily a closed mapping in functional analysis

    We know that a linear operator T:X\rightarrowY between two Banach Spaces X and Y is an open mapping if T is surjective. Here open mapping means that T sends open subsets of X to open subsets of Y. Prove that if T is an open mapping between two Banach Spaces then it is not necessarily a closed...
  48. C

    Analysis: fixed point, contraction mapping

    Let p,q : \mathbb{C} \to \mathbb{C} be defined by \begin{align*} p(z) =& z^7 + z^3 - 9z - i, \\ q(z) =& \frac{z^7 + z^3 - i}{9} \end{align*} 1. Prove that p has a zero at z_0 if and only if z_0 is a fixed point for q. If z_0 is a fixed point for q then \begin{align*} q(z_0) =...
  49. Y

    Schools Mapping of Physics for high school

    I would like you to share mapping of Physics for high school. Please help me .:smile:
  50. E

    Describing Biholomorphic Self Maps of Punctured Plane

    how do we describe the biholomorphic self maps of the multiply puncture plane onto itself? I mean C\{pi,p2,p3..pn} Plane with n points taken away. I wanted to generailze the result for the conformal self maps of the punctured plane, but I do feel these are quite different animals. I...
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