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.
According to Einstein, the Big Bang theory did not make sense because he said if you mapped the paths of the galaxies and stars back in time they would not collide at a singularity at the center of the universe, they would miss each other. Since many people accept the Big Bang theory, has it...
i been trying to understand this and basically the answer i got was it makes it easier to solve for product of sums...is this close to being correct? could you explain how it is applied. thank you.
Homework Statement
let T(V)=V be a linear map, where V is a finite-dimensional vector space. Then T^2 is defined to be the composite TT of T with itself, and similarly T^(i+1) = TT^i for all i >=1. Suppose Rank (T) = Rank (T^2)
Homework Equations
a) prove that Im(T) = Im(T^2)
b) for...
Homework Statement
I know that the Mobius transformation:
g(z) = \frac{z-z_1}{z-z_3}\frac{z_2-z_3}{z_2-z_1}
maps a circle (with points z_1, z_2, z_3 somewhere on the circumference) onto a line.
But, i want a general formula for f(z) that maps a circle (z_1,z_2,z_3) ontp a circle...
A complex function f\left(z\right)=\sqrt{z} can be splitted into two branches:
1. Principal branch: f_{1}\left(z\right)=\sqrt{r} e^{i \left(\theta/2\right)}
2. Second branch: f_{2}\left(z\right)=\sqrt{r} e^{i \left[\left(\theta+2\pi\right) /2\right]}
My question is, is there a way to...
Define the mapping torus of a homeomorphism \phi:X \rightarrow X to be the identification space
T(\phi)= X \times I / \{ (x,0) \sim (\phi(x),1) | x \in X \}
I have to identify T(\phi) with a standard space and prove that it is homotopy equivalent to S^1 by constructing explicit maps f:S^1...
Homework Statement
Hi all. We are asked to transform the shaded area in below figure to between two concentric circles, an annulus. Where these circles' center will be is not important, just transform the area to between any two concentric circles. As you see in figure, shaded area is whole...
1. Homework Statement
Find all \mathcal{C}^1 functions f(\mathbf{x}) in \mathbb{R}^3 such that the mapping \psi : \mathbb{R}^3 \to \mathbb{R}^3 also preserves volumes, where
\begin{equation*}
\psi(\mathbf{x}) = \left(
\begin{array}{c}
x_1 \\
x_1^2 + x_2 \\
f(\mathbf{x})...
Homework Statement
Let B be the outside of the unit ball centered at the origin, and let c be a non-zero constant. Consider the mapping where k=1,2,3.
Find the image of the set B under the mapping. (Hint: consider the norm of (y1, y2, y3))
Homework Equations
The unit ball would be 2...
Given:
G is the group of matrices of the form:
1 n
0 1
Where n is an element of Z, and G is a group under matrix multiplication.
I must show that G is isomorphic to the group of integers Z. I do not know how to do this, since all examples we covered gave us the specific mapping...
Let (X, \mathcal{A}), (Y, \mathcal{B}) be measurable spaces. A function K: X \times \mathcal{B} \rightarrow [0, +\infty] is called a kernel from (X, \mathcal{A}) to (Y, \mathcal{B}) if
i) for each x in X, the function B \mapsto K(x,B) is a measure on (Y, \mathcal{B}), and
ii) for each B in...
Homework Statement
I'm trying to find a function that map the exterior of a circle |z|>1 into the interior of a regular hexagon. Homework Equations
The Attempt at a Solution
I have tried mapping the exterior to the interior circle. Then mapping interior circle to the upper plane which then I...
Hi
My book defines one-to-one mapping as
A mapping T is one-to-one on D* if for (u,v) and (u',v') ∈ D*,
T(u,v) = T(u', v') implies that u = u' and v = v'
I don't really understand what they are trying to say, because right now what I'm getting from this information is that only...
Here's an interesting article on the first 'skylight' - opening to a possible underground lava tube or cavern - discovered on the Moon:
http://www.newscientist.com/article/dn18030-found-first-skylight-on-the-moon.html
I'm wondering how it might be possible to map out underground caverns and...
Hi, my question is what mapping to use for the problem in the picture attached.
I need to be able to find the potential distribution etc by mapping from the x-y plane (as pictured) to a straight lines plane capacitor, which would be pretty straightforward, but I can't find this map in any...
Hey guys, new to the forum but hoping you can help.
How do you prove that vector spaces V and U have a linear injective map given V is finite dimensional. I got the linear part but cannot really figure out the injectivity part, although I am thinking that it has to do with the kernel...
Hello,
so i was looking up the defintion of linear mapping and mapping in general and i have seen the technical defintion a few times but i was wondering if someone would mind explaining it to me in more general english. How would you explain it instead of just pointing out the definition...
Homework Statement
Find the Mobieus mapping that maps { z e C, |z| <= 1 } to a disk {z e C, |z - 1| <= 1} in a real axis.
The Attempt at a Solution
I have had an idea that Mobieus mapping is from C to C such that it is a homeomorfism and it has an inverse mapping.
I am not sure how...
Homework Statement
We're supposed to find a bijective mapping from the open unit disk \{z : |z| < 1\} to the sector \{z: z = re^{i \theta}, r > 0, -\pi/4 < \theta < \pi/4 \}.Homework Equations
The Attempt at a Solution
This is confusing me. I tried to find a function that would map [0,1), which...
Homework Statement
I need to find a unitary operator that can map two (two-dimensional) pure states |+\rangle, |-\rangle as follows:
|+\rangle \to \cos\theta |+\rangle + \sin\theta |-\rangle
|-\rangle \to \sin\theta |+\rangle + \cos\theta |- \rangle
For an arbitrary angle 0 \leq...
Homework Statement
Let S = {(x,y,z) be an element of R3: x2+y2+z2=1} be the unit sphere
I have a map that takes coordinates from the xy plane to points on the sphere.
This map is given by:
σ:R2->S\{(0,0,1)}
σ(u,v) = (2u, 2v, u2+v2-1)/1+u2+v2
Now I wish to find the inverse map...
Homework Statement
If i have a sketch of field lines, and equipotential lines which i completed in a lab, and i know the voltage,V, of points located at intervals ,L, how do i find the electric field for the whole plane?
Homework Equations
lEl = DeltaV/DeltaL
The Attempt at a...
Let B={s|s is a function mapping the set of natural numbers to {0,1}}. Is B a countable set-that is, is it possible to find a function \Phi() mapping the set of natural numbers onto B-?
I know that it has to do with infinite binary sequences, but the countability part confuses me. Can...
For an infinitesimal mapping with u = 1,2,3,4:
x ^u \rightarrow x^u + \xi^u(x)
Now suppose we introduce a new set of variables:
x^{'u} = x^{'u}(x)
I would have thought the infinitesimal mapping in terms of the new variables should be written as:
\xi^{'u}(x^{'}) = \frac{\partial...
Hey! I have a really easy question here, but I still can't figure it out.
I have a range of values from 182 to 455. I need a function that gives me back values from 1-50. IE, f(318) = 25. The numbers aren't critical, but I'd love a general equation to use for this kinda stuff. Can anybody...
Homework Statement
Hello. :smile:
I was hoping I could get some help with a homework question.
"Draw the graph of f. State whether f is One to One and also whether it is Onto."
Homework Equations
Equation 1 : f(x) = 1/x^2 if x≤-1
Equation 2 : f(x) = (x+3)/2 if -1≤x≤1
Equation...
In order to build a QE measurement system, I want to confirm the following issues: (1) Is collimated light required for QE characterization of the entire solar cell? How "bad" it could be if having Gaussian beam? (2) It seems LBIC is most common for QE mapping. Is it possible to use...
Homework Statement
Let T: R2 -> R1 be given by T(x,y) = (y^2)x + (x^2)y.
Is T linear? justify your answer
Homework Equations
The Attempt at a Solution
Yes it is a linear mapping because both points map onto one point.
Homework Statement
Hi all.
I have seen a conformal mapping of z = x+iy in MAPLE, and it consists of horizontal and vertical lines in the Argand diagram (i.e. the (x,y)-plane).
On the Web I have read that a conformal map is a mapping, which preserves angles. My question is how this...
Let F: H ->H be a map of a Hilbert plane into itself. For any point A, denote F(A) by A`. Assume that AB is congruent to A'B' for any two points A,B.
How can I prove that this map is in fact a bijection?
In a arbitrary Hilbert plane, one can not be certain that square roots exist, so...
If I take F(x)=\sqrt{1+x^2}, then the derivative is always less than one so this is a contraction mapping from R to R, right?
But there is no fixed point where F(x)=x, where the contraction mapping theorem says there should be.
So where have I gone wrong?
Cheers
mapping a: S --> T be so that any x ε S has one and only one y &#
What makes it necessary for any mapping a: S --> T be so that any x ε S has one and only one y ε T?
Let L:=\{z:|z-1|<1\} \cap \{z:|z-i|<1\}. Find a Mobius transformation that maps L onto the sector \{z: 0< arg(z) < \alpha \}. What is the angle \alpha?
no idea of how about to set up the problem
The intersection of the two circles forms a lens shaped region L with boundary curves, let's...
Homework Statement
Hi all.
I am given the following discrete mapping: x_{n+1}=f(x_n)=x_n+r-x_n^2 for r>0.
Objective: Find the r, where a period doubling takes occurs.
Attempt: First I find the fixed points: These are x=-\sqrt{r} (which is unstable for all r) and x=\sqrt{r} (which is...
can i know how to map a vector to a vector by preserving the operation if addition and mutiplication ..pls dun use f(x+y)=f(x)+f(y)..
i wan to know how to use in abstract ...
if i do the mapping wat will happens?
Hi all friends,
I am working on tracer in the oil field. In attempt to understand the analytical solution of tracer breakthrough in 5-spot pattern I'm reading the paper of Brigham and Abbasadez. For obtaining of potential field, they used some mapping methods from z plane to w plan, with...
Homework Statement
graph |z-1|=1 and then graph z^2
Homework Equations
z=x+iy
The Attempt at a Solution
well, |z-1|=1 => |(x-1)+iy|=1,
squaring both sides. we get, (x-1)^2+y^2=1. This is a circle. But how am i supposed to get z^2 from this? I don't know what to do with...
hi
if P and Q are 2 permutations of X, their product, P.Q, is the permutation of X (X=1,2,3,4,5), obtained by following the mapping Q with the mapping P. if Q=2 3 4 1 5, and P is 1 2 5 3 4, then how do i find P.Q and Q.P ?
i have tried a few mappings but can never get the same answer as in...
Hi,
I'm trying to write a score function.
The score function is applied to each cell of a grid
and because the grid has many cells ( 400x600 or 800x600 )
If I want to experiment with different score functions to
see which one is best I'd have to see some kind of visual results.
I thought...
If V is a vector space with an inner space <.,.>. V1 is an non empty subset of V. Vector x is contained in V is said to be orthogonal to v1 if <x,y>=0 for all y contained in V1.
1) if v is contained in V and define the mapping f(x)=<x,v>v. Show f is a linear transformation and describe its...
We have a continuous bijection f:X-->Y.
Prove that if f is open, then f inverse is continuous.
I can't figure it out.
"Proof". For V open in Y, there exists W open in X such that f[W] \subseteq V. Where does the f is open definition apply?
Hi, everyone:
I have been trying to show this using the following:
Given f: Y-->X
IF S^n ~ Y_f(x) , then S^n deformation-retracts to Y , and ( not sure of this)
also is homeomorphic to Y (I know Y_f(x) is homotopic to Y ) . But ( so I am
branching out into more...
f(x) is a bijection if and only if f(x) is both a surjection and a bijection. Now a surjection is when every element of B has at least one mapping, and an injection is when all of the elements have a unique mapping from A, and therefore a bijection is a one-to-one mapping.
Let's say that...
If S is a finite set having m > 0 elements, how many mappings are there of s into itelf?
I believe there would be however many mappings there are elements> Any suggestions?
Homework Statement
For X= NxN, Y=N, define the mapping phi: X-->Y as phi(x,y)=x+y. Find the inverse image of phi-inverse (5) of the singleton set {5}. If n: X-->Y is the product operation n(x,y)=xy, find n-inverse (4).
The Attempt at a Solution
I'm not even really sure what the question...
Hi, I'm having some difficulty with this problem. I need to project a point in R2 to the line x2 = x1 (sqrt(3)) and then rotate it 30 degrees clockwise.
I believe the 2x2 matrix to map it is just
sqrt(3) 0
0 1
and to rotate a vector clockwise as opposed to counter clockwise...
Hi everyone,
(I hope I'm posting it in the right place, please feel free to move this thread to the appropriate place)
My high school graduation project is about the application of the theory of complex variables in physics. Specifically, I am learning about the complex potential, its...
Homework Statement
A group of overlapping clones, designated A through F, is isolated from one region of a chromosome. Each of the clones is seperately cleaved by a restriction enzyme and the pieces resolved by agarose gel electrophoresis, with the results shown in the figure below. There are...