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rogueviz::ads::tour:: added longer explanations to math slides

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Zeno Rogue 2025-04-08 12:25:57 +02:00
parent fa3db0ff0f
commit 1fa346fa9b
1 changed files with 34 additions and 5 deletions
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39
rogueviz/ads/tour.cpp
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@ -500,7 +500,11 @@ slide relhell_tour[] = {
},
{"Euclidean geometry", 999, LEGAL::NONE | QUICKGEO | USE_SLIDE_NAME | NOTITLE,
"explanation",
"OK, so let us think what the Euclidean geometry is.\n\n"
"Let us focus on three-dimensional Euclidean geometry. "
"We need to define what points are in our space, and how to compute distances between them. "
"This, in turns, let us define 'isometries' (rotations, etc.) which are basically transformations of "
"the space that keep the distance.\n\nThis template will be also used in other geometries.",
[] (presmode mode) {
setCanvas(mode, &ccolor::chessboard, [] { set_geometry(gEuclidSquare); set_variation(eVariation::pure); });
latex_slide(mode, defs+R"=(
@ -519,7 +523,14 @@ slide relhell_tour[] = {
}},
{"Minkowski geometry", 999, LEGAL::NONE | QUICKGEO | USE_SLIDE_NAME | NOTITLE,
"explanation",
"The Minkowski geometry is similar to Euclidean geometry, except that in the squared distance formula, "
"the square of the time difference has a different sign. Thus, we have different isometries, which "
"can turn space to time and vice versa, just like Euclidean rotations turned X to Y and vice versa. "
"Because of the different sign, these 'Lorentz transformations' work different -- for example, they are not based on sin and cos, "
"but sinh and cosh.\n\n"
"Just like Euclidean geometry, Minkowski geometry is maximally symmetric: spacetime directions can be classified as space-like (squared distance > 0), "
"light-like (squared distance = 0) and time-like (squared distance < 0), but if we have a point and direction, we have an isometry that "
"takes it into any other point and direction of the same type.",
[] (presmode mode) {
latex_slide(mode, defs+R"=(
{\color{remph}Minkowski spacetime with 2 space and 1 time dimension:}
@ -560,7 +571,12 @@ slide relhell_tour[] = {
}},
{"spherical geometry", 999, LEGAL::NONE | QUICKGEO | USE_SLIDE_NAME | NOTITLE,
"explanation",
"Now, let us discuss how spherical and hyperbolic geometries are obtained. Spherical "
"is quite straightforward: we get the spherical geometry by restricting to the set of points "
"in distance 1 from the chosen center, and also distances are the arc lengths. Just like "
"Euclidean and Minkowski geometry, spherical geometry is maximally symmetric: every point and "
"every direction works the same.\n\n"
"The next slide gives a similar description of hyperbolic geometry.",
[] (presmode mode) {
setCanvas(mode, &ccolor::football, [] { set_geometry(gSphere); });
if(mode == pmStart) {
@ -580,7 +596,14 @@ slide relhell_tour[] = {
}},
{"hyperbolic geometry", 999, LEGAL::NONE | QUICKGEO | USE_SLIDE_NAME | NOTITLE,
"explanation",
"To get hyperbolic geometry, we also restrict to the set of points in the same squared distance, "
"but now we start with Minkowski geometry, and the 'squared radius' is negative (time-like). "
"The obtained maximally symmetric manifold thus loses its time-like dimension and is purely a space.\n\n"
"Therefore, in this model, every point in two-dimensional hyperbolic space is described with three "
"coordinates. This may look scary, but actually is very similar to how spherical geometry works, "
"we just need to use sinh and cosh, not sin and cos. The usual 3D graphics "
"also employ an extra coordinate, and it is straightforward to apply 3D engines to work with "
"spherical and hyperbolic geometry too, using these models.",
[] (presmode mode) {
latex_slide(mode, defs+R"=(
{\color{remph}2-dimensional hyperbolic space (Minkowski hyperboloid model):}
@ -602,7 +625,13 @@ slide relhell_tour[] = {
}},
{"anti-de Sitter spacetime", 999, LEGAL::NONE | QUICKGEO | USE_SLIDE_NAME | NOTITLE,
"explanation",
"Here is how we add a time coordinate to the hyperbolic plane, in order to get 2+1D anti-de Sitter spacetime. "
"As you can see, the construction is quite similar, and again, we get a maximally symmetric spacetime.\n\n"
"Press 5 for an animated visualization of this construction. Initially you see the hyperbolic plane at time 0 (u=0, t>0). "
"First '5' adds the different time slices to the visualization, and the second '5' unwraps it into the universal cover.\n\n"
"Note: the construction is quite similar to that of the Thurston geometry 'universal cover of SL(2,R)' -- in fact, Relative Hell "
"uses the RogueViz implementation of that space. However, the angular coordinate becomes time-like, making our spacetime to be "
"much more symmetric, and the geodesics work in a much more intuitive way.",
[] (presmode mode) {
latex_slide(mode, defs+R"=(
{\color{remph}anti-de Sitter spacetime:}