Stereographic Definition and 22 Threads

I identified $$(\Phi_{SN})_{*})$$ as $$J_{(\Phi_{SN})}$$ where J is the Jacobian matrix in order to $$(\Phi_{SN})$$, also noticing that $$\frac{\partial}{\partial u} = \frac{\partial s}{\partial u}\frac{\partial}{\partial s} + \frac{\partial t}{\partial u} \frac{\partial}{\partial t} $$, I wrote...

Hello everyone, first post here. I don't need to do this in Grapher.app (the default graphing calculator on every Mac), but it is an elegant humble little app, and I have seen the hypercube projected using it here. I really would love to get the equations and expressions in my own file on my...

So the circle minus the pole is homeomorphic to the number line. Does that mean that the pole itself represents a number that doesn't exist on the real line? After all, the pole certainly exists and yet is the only point that isn't mapped.

Lets assume we are mapping one face of earth. we place a plane touching the Earth at 0 lattitude and 0 longitude. Now we take the plane of projection. suppose that we expand the projection unevenly. The small projectional area of a certain lattitude and longitude is expanded by a factor which is...

Hello Experts, I leaned that using a sterescopic cameras (generates 2 photos) one can measure the distance to any object in the image produced by the camera. For example, if I take a picture of a lamp post using a stereoscopic method, I can determine the Distance from the camera to the lamp...

Homework Statement Show that the equations $$ \delta \phi = \cot \theta \cot \phi \delta \theta, \quad \delta \phi =- \cot \theta \tan \phi \delta \theta$$ represent rotations around the x and y axes respectively of a stereographic sphere. Both these two rotations map the sphere on itself and...

Hi , I came across a problem ,I've search a lot but couldn't exactly find the solution. here is my problem: suppose there is an image ( I call it IMG_A),place IMG_A in the X-Y plane , put a mirror cylinder at the center of IMG_A. what we see in the cylinder mirror is a deform image (I call it...

Let r:R2 →R3 be given by the formula Compute the second fundamental form with respect to this basis (Hint: There’s a shortcut to computing the unit normal n). I can't find thi shortcut, does anyone help me? I'm solving it with normal vector and first and second derivate, but I obtained...

I've noticed that electromagnetic field lines are very similar to stereographic projection of 3D sphere on 2D surface. Pictures below. In such comparison, electric field represents longitude and magnetic field represents latitude. For more visualization see...

We know stereographic projection is conformal but it isn't isometic and in general relativity it can not be used because in this theory general transformations must be isometric. But de sitter in his model (1917) used it (stereographic projection) to obtain metric in static coordinates. How...

In cartesian coordinates (x.y,z) on the and (X,Y) on the plane, the projection and its inverse are given by the following formulae: (X,Y)=(x/1-z,y/1-z) (x,y,z)=(2X/1+X^2+Y^2, 2Y/1+X^2+Y^2, -1+X^2+Y^2/1+X^2 +Y^2) This relates to the field of differntial geo.Anybody have a proof to where thes...

I am computing a stereographic projection in R^4 and i think i am correct in setting x=rcos(x)sin(y) y=rsin(x)sin(y) z=rcos(y) but can't see how to compute r as I do not know to visualise it graphically as was possible in R^3, any help would be greatly appreciated

Hello, if we consider the stereographic projection \mathcal{S}^2\rightarrow \mathbb{R}^2 given in the form: (X,Y) = \left( \frac{x}{1-z} , \frac{y}{1-z} \right) how can I find the metric in X,Y coordinates? -- Should I first express the projection in spherical coordinates, then find...

Homework Statement Show that the map f : R--> S1 given by f(t) =[(t^2-1)/(t^2+1), 2t/(t^2+1)] is a homeomorphism onto S1-{(1, 0)}, where S1 is the unit circle in the plane. I know this is a stereographic projection, but I do not know how to show that it has a continuous inverse. I am...

Homework Statement Hi all - I've been battering away at this for an hour or so, and was hoping someone else could lend a hand! Q: Show that any Mobius transformation T not equal to 1 on \mathbb{C}_{\infinity} has 1 or 2 fixed points. (Done) Show that the Mobius transformation corresponding...

Stereographic Projection for "general" surfaces First off, sorry if this is in the wrong forum. I came across this while studying computer vision, but it's of a somewhat mathematical nature. Please move it if it's in the wrong place. In the book I'm reading*, stereographic projection is used...

Why stereographic projection preserves angles between curves but does not preserve area?

Homework Statement Given the parabolic co-ordinate system defined, given Cartesian coordinates x and y, as $\mu=2xy$ $\lambda = x^2-y^2,$ find the inverse transformation x(\mu, \lambda) and y(\mu,\lambda). Homework Equations None The Attempt at a Solution We compute...

Hello, Can anyone tell me what the triangles, squares, and ovals mean in the projection on page 19? Does this have something to do with the directions for the third axis? On the next page several of the symbols are black. What significance does this have, also? Thanks -scott "[PLAIN...

In any book on differentiable manifolds, the stereographic projection map P from the n-Sphere to the (n-1)-plane is discussed as part of an example of how one might cover a sphere with an atlas. This is usually followed by a comment such as "it is obvious" or "it can be shown" that the inverse...

I know if a cirlce (on S^2) does not contain N (0,0,1) then it is mapped onto the plane H as a circle. Now say the circles on S^2 are lines of latitude. When mapped by the stereographic projection they are cirlces in R^3 on the plane H. Now the only thing I am not sure on is, my claim: When...

So I'm trying to prove that the map f(x,y,z) = \frac{(x,y)}{1-z} from the unit sphere S^2 to R^2 is injective by the usual means: f(x_1,y_1,z_1)=f(x_2,y_2,z_2) \Rightarrow (x_1,y_1,z_1)=(x_2,y_2,z_2) But i can't seem to show it... :frown: I end up with the result that...