Zariski tangent space Puns

I've encountered three definitions of tangent spaces for differentiable manifolds (e.g., in Lee's Introduction to Smooth manifolds, Ch. 3), and I am trying to understand which definitions work in which contexts.

Most introductory books focus on the smooth case, a few include the C^r case for finite r, and very few touch on the analytic case. I was wondering if anyone could break down these methods and where they work and fail.

Consider a C^r manifold M (r a positive integer, infinity, or omega). The three methods I've seen of defining the tangent vectors are:

1. As derivations on the space of C^r functions
2. As derivations on the germs of C^r functions
3. As equivalence classes of curves on M

Here's what I think I've gathered so far: For smooth manifolds all three methods work equally well (they are naturally isomorphic). For analytic manifolds, method 1 might fail to be local and thus won't work, method 2 works fine, and I'm not sure about method 3. And for r finite, I know method 3 works, but I have no idea about methods 1 and 2.

Apologies if my understandings in the above paragraph are incorrect. Can anyone summarize how these three methods work in the r finite and the analytic cases? If you can include the reason for the failures of various methods, that would also be helpful. E.g., the lack of analytic bump functions may be the reason method 1 fails for analytic manifolds.

So I’m reading John Stillwell’s Naive Lie theory book for undergrads and there’s this exercise where I’m supposed to give a connected Lie group with a non trivial, non discrete normal subgroup such that it’s tangent space is trivial.

So I pick Z(SL(n,C)) = {w1 |where |w|= 1} as my normal subgroup which I’m pretty sure is non discrete. And since sl (n,C) is simple, and the tangent space transforms normal subgroups to ideals, T_1 (w1) must be trivial.

Qn) Does this work as an example? And if so, can someone please explain this unintuitive result. Because looking at w1 where w is isomorphic to S_1, I can’t intuit why it’s tangent space is {0}

Having a lot of trouble finding something an understandable definition of tangent space of a quotient space.
For simplicity lets assume the quotient is of two vector spaces.

My question is not related to normal mapping.

First, what I'm trying to do:

I have a program which creates cubemap textures for spheres to make them look like gas-giant planets. It works by using curl-noise(pdf) on the surface of the sphere, representing a 3d velocity field as a bunch of 3d vectors in 6 2d arrays like a cubemap. Then I dump a bunch of particles onto the surface of the sphere, and let them roam around on the velocity-field rails, leaving behind alpha-blended fading trails on the surface of a cubemap, by projecting their positions out onto a cube. After awhile, spit out six square images that can be used as a cubemap by some other program (e.g. a space game of some kind.) All that works just fine.

Someone has requested that I allow exporting the velocity field as a "flow map", with x,y components stored in R and G channels of 6 cubemap png images. Sounds simple enough, and like a good idea, as there are some shaders that can use flowmaps and maybe make some cool animations. I've made a couple of failed attempts.

1st attempt: I already have the velocity field in world space. I think I just need to convert it to tangent space. For each vector in worldspace, figure out which cubemap face it's on, compute a rotation from the sphere normal at that point to the cube-normal at that face, and rotate the velocity vector by that same rotation. I believe this is under-constrained, and doesn't really get the right rotation (does not take into account anything like a tangent or bitangent vector). It made something which at first glance looked very promising, but it turned out to be all kinds of wrong.

2nd attempt:

I'm already moving particles around to create a cubemap image. I can just compute the 2D velocity of those particles from their positions as projected onto the cubemap directly. This kind of works, however, there are abrupt changes in speed as the particles cross cubemap face boundaries. I think this is because of the simple projection. e.g. if I have a world-space velocity vector (0.9, 0.1, 0.0) and project it onto the x-y plane, you get (0.9, 0.1) but if you project onto y-z plane, you get (0.1, 0) -- so you're going to get abrupt speed changes at the cubemap edges. (I've simplified the numbers, but hopefully you get the gist.)

That makes

I'm making a hammer throwing simulator to demonstrate centrifugal force. I've been able to get a circle representing a player to move across the screen while another circle representing the hammer orbits the main circle representing the player. I am able to make the player stop moving with the keypressed(key) function where I say

function love.keypressed(key)
if key == "space" and startposX < 0 then speed = 0 end end

I was hoping to update the position of the hammer by telling it to move at a tangent line to the circle at the angle that it is currently sitting with the lines

function love.keypressed(key)
if key == "space" and startposX < 0 then speed = 0 hxpos = hxpos + 1/math.cos(ctheta) -- trying to make the hammer x position the secant of ctheta hypos = hypos + 1/math.sin(ctheta) -- trying to make the hammer y position the cosecant of ctheta end end

This is not working, as the player cjrcle will stop moving when the space key is pressed, however the cjrcle representing the hammer continues to orbit the player cjrcle and will not fly off at a tangent. How can I get the outer cjrcle to fly off at a tangent when the player touches space?

main.lua contents

https://pastebin.com/4FGa3qdv

i've written my own circle() function instead of using the built in circle() function which i've named cjrcle() and included in the file cjrcle.lua

cjrcle.lua contents

https://pastebin.com/s22jaUQL

I am trying to apply Normal Mapping to a fragment shader which is using Triplanar projection.

With (Incorrect) normal mapping:

Without Normal Mapping

With The Normal mapping implemented as per Shader below

It almost looks ok... but something that I really don't understand is that on only a couple of faces... there is a weird flickering issue... I Don't think it is zbuffer because the same issue *should* be present on all faces is it were due to multiple tir's being painted on top of each other (The shader also has depth stencil set to Less rather then LessEqual)

Here is the existing shader:

#version 450
layout(location=0) out vec4 FragColor;

layout(location =0) in vec2 vertUv;
layout(location = 1) in vec3 vertNormal;
layout(location = 2) in vec3 FragPos;

layout(std140, set=0, binding=0) uniform CameraMat
{
	mat4 CamMatrix;
};
layout(std140, set=0,binding=1) uniform ModelMat
{
   mat4 ModelMatrix;
};
layout(set=1,binding=1) uniform texture2D ColorTexture;
layout(set=1,binding=2) uniform sampler ColorSampler;

layout(std140, set=2, binding=0) uniform LightData
{
	vec3 lightPosition;
	vec3 lightColor;
};

vec3 saturate(vec3 v)
{
   return clamp(v.rgb, 0.0, 1.0);
}

vec2 modvec(vec2 a, vec2 b)
{
   return a - (b * floor(a/b));
}

void main()
{
	//for each axis (x,y,z) a 1 means to 'flip' that axis
	vec3 flip = vec3(vertNormal.x < 0.0, vertNormal.y >= 0.0, vertNormal.z < 0.0);

	// Raise each component of normal vector to 4th power.
	vec3 blend = saturate(abs(normalize(abs(vertNormal))) - 0.5);
	blend *= blend;
	blend *= blend;
	// Normalize result by dividing by the sum of its components.
	blend /= dot(blend, vec3(1.0, 1.0, 1.0)); 
   
     //Three sets of UVs based on World Position.
     vec2 yzS = modvec(abs(FragPos.yz),vec2(1));
     vec2 xzS = modvec(abs(FragPos.xz),vec2(1));
     vec2 xyS = modvec(abs(FragPos.xy),vec2(1));

	 //Three samples from the ColorTexture. One for each axis.

What fraction of all points in a Euclidean space lie within (rather than outside of) the n-ball whose center is the orogin and which is tangent to the point p represented by Cartesian coordinates:

vector(θ) =(θ^1, θ^2, θ^3, θ^4, θ^5, ... θ^n )

representing sigmas in the multivariate normal distribution in n dimensions as illustrated at the top of the link?

Platform(s): PC

Genre: SpaceShip shooter, upgrade/rpg, flash/brower game

Estimated year of release: 1995-2010??

Graphics/art style: This was kind of unique, as it had an isometric 3d view of things. You could shoot things slightly off screen, The first world was icy planet at night, somewhat looking similar to the first starcrafts level of graphics. There was a lot of good lighting and shadow mechanics. It was also like a maze of a map, so you had to navigate hidden areas.

Notable characters: Your ship

Notable gameplay mechanics: There were 5 levels, the first one being "free" each level you had to find resources and harvest them to upgrade your ship, and along the way survive aliens attacking you. It was a 3 D isometric game and you would navigate though a maze of different areas, each being revealed from a fog of war.

**Other details: I COULD swear this was perhaps a "Wild Tangent " game, perhaps dark tangent as the name? Dark ....SOMETHING.**It was a very popular online browser game at the time, late 90s early 2000s i beleive.

Edit: I am inputing a voronoi based on UV0, so no problem should be there..

The set sl(n,C)= Tracefree matrices, show that this is the tangent space of SL(n,C) at the identity.

If X is in the tangent space then there is a map f:(-s,s)->SL(n,C) s.t f'(0)=X and f(0)=I.

Then I need to show this matrix is also trace free.

And I need to also show trace free matrices have such paths.

I've been following the excellent Learn OpenGL tutorials and just read the page on normal mapping. I understood everything, including the math since I have a background in engineering, but one thing still bothers me. It seems like everything that we need to calculate in order to do normal mapping is because we are trying to solve a problem that we have created in the first place. It would seem intuitive to me to store normal maps in "object space" and then just transform the normals using the normal matrix like we do when we manually create vertices and vertex normals. Why is that not the standard?

It's not a mirror, it's a clock. So TNG is from 12 to 3, Yesterday's Enterprise is from 3 to 6, The Mirror universe is from 6 to 9, and Deep Space Nine is from 9 to 12. This explains why the mirror universe was never utilized in TNG.

Also Dax is a synth and She cheated on Worf with the wormhole's holosuite representation

Hello everyone,

I'm trying to implement normal mapping in my 3D program. The idea is that each vertex has an associated normal, tangent, and bitangent vector. Each of these vectors is transformed into world space via multiplication with the model matrix. These three vectors form the column vectors of the tangent matrix which is then passed to the fragment shader.

Vertex shader

The fragment shader receives this now interpolated matrix. The normal vector is retrieved from the map and its values are remapped from the range [0,1] to [-1, 1], followed by a transformation into world space by way of multiplication with the tangent matrix. For now, this normal vector is simply output as the color of the fragment.

Fragment shader

Note that in this clip, +X is pointing right, positive Y is pointing straight up (i.e. the sphere's axis is world space Y), and +Z is pointing into the screen. With this setup, it should look like the sphere is being illuminated by a red light from the right, a green light from the top, and a blue light from the back, but, uh, that's not what it looks like.

normals are wrongo

The issue here is either with my shader code or with my custom Blender exporter not outputting the right tan, bitan, and normal vectors, though posting the code of that is I suspect outside the scope of this sub.

Thanks for any insight you may be able to provide.

So am I the only one who absolutely hated their weird tangent on pokémon and the people who like it on the last onion nuggets thing? I just thought it’s out of character for some fat neckbeards who all have their living room plastered with shit from some franchise. Paul literally has different lament configurations standing around but then it’s not okay for adults to enjoy their little imaginary animals.

Idk I just don’t get why, in this fucked up world of misery and suffering, you would tell people it’s childish and “cringe” to enjoy something that might be aimed at children.

(I posted this to the codependency subreddit as well, since I really think this fits in both categories. )

I understand the concept of space is healthy, I know I need it and so does my partner. He’s also codependent. But the idea of him asking for it is horrific for me and I hate that. He doesn’t ask for it, but I worry about the fact that he “should” and when he will if he does. I wish I could be okay with it! But in the beginning of our relationship when he was really distant and had no real intention of true connection (therefore suppressing his own codependency,) he was more comfortable asking for alone time, and he’d usually ask for time to play video games. I remember telling my therapist this, that it bothered me because I wanted him to want to be with me and she said there’s nothing wrong with him doing that and it stung so badly. But she’s right. If he wants to do his own thing he should have the ability to. I don’t understand why I have such a strong need to be needed that badly. I guess I just feel insignificant, and truly feel and felt unimportant to him, even when he tries to show me he loves me.

In all honesty though, I feel this is partly because of how my relationship with him started out. He was super distant and immediately set up these intense boundaries that caused us to kind of miss out on the honey moon phase, which I do truly like and appreciate even if it calms down because it allows me to see this person wants me. So we never got that, and ever since I’ve never felt secure in my relationship with him. I’ve realized that I want more alone time and am accepting and trying to be okay with him wanting it too, but I’m looking for this passion that comes with the honey moon phase I missed out on, in a relationship that feels like we’ve been married for 10 years.

I struggle deeply with distinguishing between codependency and healthy needs. Maybe that’s me justifying my codependency. I just wish I felt wanted, and he does struggle showing me that I’m wanted, but I don’t know that there’s anything he could do at this point to help me. And I feel that’s because of my cptsd. I think about my last relationship and how I felt truly loved by him, but looking back I understand it’s because he was this mix of being a place holder for the hole I felt due to my ptsd, and a source of more trauma. There was super unhealthy codependency going on, and it’s got me deeply afraid I need unhealthy attachment styles in order to feel loved and passion wi

For context, my renderer first does an ambient pass, and then individiual lighting passes with additive blending.

I don't even know where to begin, to be honest. Here's my vertex shader and fragment shader. Both of them are compiled twice, for the ambient and directional light pass which has the DIRECTIONAL_LIGHT macro defined by the rendering engine.

Firstly, notice how I ignore the normal mapping, even though the shaders work in tangent space. This is just to be sure that the normal mapping isn't the problem. The UVs aren't the problem either, so this must be some basic shitty mistake I've done in the diffuse part.

Here's a video which showcases what happens: https://streamable.com/5a685. See how some areas are just black and only light up in very weird orientations, when the light is hardcoded to come from the +X direction in the shader.

I also tried not doing the ambient pass, and immediately doing the lighting pass right after clearing the buffers (still with additive blending), which also produced something strange: https://streamable.com/ezgfc. Stare at the right half of the ball and see four different quarters of the right half be different (transparent black, transparent white, solid black, solid white).

https://i.redd.it/vdzbd4bi5a151.gif

Note: It is not production-ready. I am taking a break from this project, might as well share it for anyone looking to make a better motion trail system

Based on the Disney paper, Tangent Space Optimization of Controls for Character Animation: https://studios.disneyresearch.com/2019/07/12/tangent-space-optimization-of-controls-for-character-animation/

Code and project feature is here https://github.com/JiahuiCai/Blender_Interactive_Motion_Path

https://preview.redd.it/9p8nvc7m4dc81.png?width=557&format=png&auto=webp&s=6a1eaf86413875e973510b1630ae15addee33d73

I understand the concept of space is healthy, I know I need it and so does my partner. He’s also codependent. But the idea of him asking for it is horrific for me and I hate that. He doesn’t ask for it, but I worry about the fact that he “should” and when he will if he does. I wish I could be okay with it! But in the beginning of our relationship when he was really distant and had no real intention of true connection (therefore suppressing his own codependency,) he was more comfortable asking for alone time, and he’d usually ask for time to play video games. I remember telling my therapist this, that it bothered me because I wanted him to want to be with me and she said there’s nothing wrong with him doing that and it stung so badly. But she’s right. If he wants to do his own thing he should have the ability to. I don’t understand why I have such a strong need to be needed that badly. I guess I just feel insignificant, and truly feel and felt unimportant to him, even when he tries to show me he loves me.

In all honesty though, I feel this is partly because of how my relationship with him started out. He was super distant and immediately set up these intense boundaries that caused us to kind of miss out on the honey moon phase, which I do truly like and appreciate even if it calms down because it allows me to see this person wants me. So we never got that, and ever since I’ve never felt secure in my relationship with him. I’ve realized that I want more alone time and am accepting and trying to be okay with him wanting it too, but I’m looking for this passion that comes with the honey moon phase I missed out on, in a relationship that feels like we’ve been married for 10 years.

I struggle deeply with distinguishing between codependency and healthy needs. Maybe that’s me justifying my codependency. I just wish I felt wanted, and he does struggle showing me that I’m wanted, but I don’t know that there’s anything he could do at this point to help me. And I feel that’s because of my cptsd. I think about my last relationship and how I felt truly loved by him, but looking back I understand it’s because he was this mix of being a place holder for the hole I felt due to my ptsd, and a source of more trauma. There was super unhealthy codependency going on, and it’s got me deeply afraid I need unhealthy attachment styles in order to feel loved and passion with someone.

I love my boyfriend, but ever since I’ve really tried to tackle my trauma and heal it, I’ve be