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--- a/src/blog/the-physics-behind-a-black-hole-shader/index.md
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@@ -14,22 +14,22 @@
 5. [Appendix](#appendix)
 
 Back in August 2023 I had to submit a project for an elective WebGL programming course in university.
-I chose to create an interractive and animated black hole demo.
+I chose to create an interactive and animated black hole demo.
 The primary idea was to show how the gravitational pull of black holes "bends" light and how the result looks from different points of view.
 
-Then in the December of that same year, I sat down and started writing a blogpost, explaning how some of the more complicated parts were made.
+Then in the December of that same year, I sat down and started writing a blog post (as you may notice from the "Created" date), explaining how some of the more complicated parts were made.
 Funnily enough, I started finding errors... a *lot* of errors.
 
 As it turned out, my implementation was completely wrong, but somehow I fudged the formulas around in just the right way to make it *look* correct.
 So the post got delayed and sparsely rejuvenated over the following months.
 
-For this reason I chose to take up two related electives this year: one on general relativity and one on raytracing, with the hopes of gaining enough understanding of the material to rework the original solution into something proper.
+For this reason I chose to take up two related electives this year: one on general relativity and one on ray tracing, with the hopes of gaining enough understanding of the material to rework the original solution into something proper.
 
 This article represents the first of a two-part series about this very shader.
 Here, we'll talk about physics and how to generally model them.
-In the second part, we'll explore raytracing and computer graphics with WebGL.
+In the second part, we'll explore ray tracing and computer graphics with WebGL.
 
-I'll assume you don't know much about physics, but are familiar with mathematics and at the very least vectors.
+I'll assume you don't know much about physics, but are familiar with mathematics and at the very least some linear algebra.
 
 ## What do the physics say?
 
@@ -39,21 +39,23 @@ Using [Einstein's field equations](TODO), we can precisely calculate the curvatu
 Obviously we're going to use [Schwarzschild metric](TODO) on a (Schwarzschild) black hole.
 To further simplify our calculations, we can make approximations using [celestial mechanics](TODO).
 
+This is not *necessary* to understand, but it will make our goal **much** clearer.
+
 Now, let's try and unravel everything.
 
 ### What is gravity?
 
 Everything with mass applies a "pulling" force to everything else with mass.
-It's easy to see this is true, but we get into trouble when trying to find formula which predict/calculate that pulling force.
+It's easy to see this is true, but we get into trouble when trying to find a formula which predicts/calculates that force.
 
 Newton gave it a good shot, with his [3 laws of motion](https://en.wikipedia.org/wiki/Newton%27s_laws_of_motion), which proved to be correct for about a century and a half.
 They even helped the discovery of Neptune, since measurements showed that the orbit of Uranus didn't meet expectations if there wasn't another planet meddling with things.
-The real crux of the issue came when the same problem occured with [Mercury's orbit](https://en.wikipedia.org/wiki/Mercury_%28planet%29#Advance_of_perihelion), but no new planet was found.
+The real crux of the issue came when the same problem occurred with [Mercury's orbit](https://en.wikipedia.org/wiki/Mercury_%28planet%29#Advance_of_perihelion), but no new planet was found.
 
 As it turned out, Newton's formulae had only been approximations, they worked when light (and therefore [causality, or the speed at which changes propagate](https://www.youtube.com/watch?v=msVuCEs8Ydo)) had infinite speed.
 
 Albert Einstein's general theory of relativity showed that gravity wasn't a "force which pulled", it was actually an effect which **bent** space.
-And that only happened because the speed of light wasn't infinite.
+And that only happened because the (**constant**) speed of light wasn't infinite.
 
 ### What does it mean to bend space?
 
@@ -77,47 +79,48 @@ On a flat surface, we can define it as "following a (straight) line segment", on
 Better yet, in geometry there are [geodesics](TODO), the general term for a curve on any surface which is the shortest[^geodesics-are-local] path between any two points.
 So now we can say "going forwards means going along a geodesic".
 
-[^geodesics-are-local]: This actually isn't completely true, a geodesic is the "locally shortest distance", I've skipped the details since they aren't really that important for our purpopses. More info can be found on [wikipedia](https://en.wikipedia.org/wiki/Geodesic#Introduction).
+[^geodesics-are-local]: This actually isn't completely true, a geodesic is the "locally shortest distance", I've skipped the details since they aren't really that important for our purposes. More info can be found on [Wikipedia](https://en.wikipedia.org/wiki/Geodesic#Introduction).
 
 It's important to note that an arbitrary geodesic doesn't have most properties you might be used to.
 For example, on a flat surface, we can have parallel lines, but on our sphere, great circles will always intersect[^parallels-of-latitude].
 
-[^parallels-of-latitude]: You might think about the latituginal lines on a globe, they are indeed parallel (equidistant), but if you take any two points on the same such line, the distance between them won't be the smallest.
+[^parallels-of-latitude]: You might think about the latitudinal lines on a globe, they are indeed parallel (equidistant), but if you take any two points on the same such line, the distance between them won't be the smallest.
+
+This is what gravity does, it doesn't pull or push, it *bends*, and more precisely, it **bends geodesics**.
+Furthermore, we live in three spacial dimensions, so we can move in three *different*[^colinearity] *directions*.
 
-Another important effect is that every N-dimentional object can be curved in N+1-dimentions.
-Meaning, a 1D line can be bent into a 2D squigle.
-A 2D plane can be bent into a 3D curvature.
-But what that really means is, for the N-dimentional being, it can view the effects of this change, without begin able to percieve the entire change.
+[^colinearity]: TODO
 
-Imagine a flatlander, some creature living in a 2D plane, which can only look at thing on the X and Y coordinates.
-If we fold this flat plane into a 3D sphere[^sphere-plane-homeomorphism], that creature can calculate the change, but it can't understand or imagine what a 3D sphere is, since it only thinks and percieves in 2D.
+The resultant effect is that if you travel perfectly forwards in a spaceship and pass by an object with mass (like a planet) from a close enough distance, your path will be curved, without you steering your ship.
 
-[^sphere-plane-homeomorphism]: In reality, [math shows this cannot be done](https://old.reddit.com/r/math/comments/152kfq/is_it_possible_to_form_a_flat_plane_into_a_sphere/c7iqo6p/) without making important changes to the plane. But this serves as a decent-enough intuitive example.
+Although not yet mentioned, you need to know that **photons** (light) **don't have mass**!
+In effect, in a universe where there is only light bouncing around, Newton's equations still work.
 
-This means the same thing can happen with our 3D space.
-It can get bent into a more-complicated 4D shape, we can calculate the exact object and understand it on a mathematical level, but we cannot truly envision it.
+On that note, the point that light **doesn't** travel at an infinite speed is significant.
+When the opposite is true, General Theory of Relativity collapses into the Newtonian equations!
 
-This is what gravity does, it *somehow* changes the curvature of space.
-We can't really imagine or see it, but we can understand and work with it in practice as the **bending of our geodesics**.
+Going back on track, Einstein gives us the necessary equations for those geodesics, the [Einstein Field Equations](TODO).
 
-The resultant effect is that if you travel perfectly forwards in a spaceship and pass by an object with mass (like a planet) from a close enough distance, your path will be curved, without you steering your ship.
+### How to handle the EFEs?
 
-Einstein gives us the necessary equations for those geodesics, the [Einstein Field Equations](TODO).
+Although they're called equations the result/output we get from solving them isn't a number!
+It's a function, which *more or less* takes the parameters of objects in the universe and returns *how* they will all behave.
+To be more precise, the output isn't *really* a function, it's a "metric", which is a distinctly different mathematical object.
 
-### How to solve the EFEs?
+In the 1910s Schwarzschild solved the equations for a universe where there is only one non-rotating[^schwm-assumptions] particle with mass, then in the 1960s the equations were solved when that particle could rotate, that's it.
+There are plenty of other mathematical solutions, but we have been able to relate the "real" world to only those two.
 
-This problem of not being able to see or imagine 4D space is quite significant, since we have our equations, but it's incredibly difficult to assign them to reality.
-Schwartzchild solved them for a universe where there is only one particle with mass in the 1910s, then in the 1960s the equations were solved with one spinning particle with mass, and that's it.
+[^schwm-assumptions]: He also assumed the body didn't have an electric charge and the [cosmological constant](TODO) is zero.
 
-There are plenty of mathematical solutions, but these are the only ones which have been connected to the physical phenomena.
-Generally we call a solution a metric, since that's the main mathematical "function" we get.
+This may sound unsatisfactory, but in practice, it's quite enough.
+In the solar system, [>99.86% of mass is in the Sun](TODO), so assuming all other objects are massless, we get enough precision to explain Mercury's orbit.
+We can apply the same idea to our simulation, leaving all objects which aren't our black hole without mass.
 
-That's why we'll work with that initial Schwartzschild metric, which assumed a non-rotating single mass.
-And that's what defines Schwartzchild blackholes too.
+Schwarzschild's metric and accompanying black hole properties are the simplest possible solution, so we'll use that.
 
-### What is a black hole
+### What is a black hole?
 
-Let's imagine atoms as a set of marbles, scattered on a table:
+Let's imagine atoms as a set of glass marbles, scattered on a table:
 
 ![](https://upload.wikimedia.org/wikipedia/commons/thumb/0/0e/Marbles_from_Indonesia.jpg/1280px-Marbles_from_Indonesia.jpg)  
 *Image from By Okkisafire on [Wikipedia](https://commons.wikimedia.org/w/index.php?curid=30338734)*
@@ -129,54 +132,56 @@ At that point (assuming they all stay flat on the table), the glass will keep th
 By increasing the force with which you push on them, after a certain threshold, the glass will break, turning the marbles into shards.
 Those shards will still have some fight in them, so after increasing your force by another level, you'll turn all of those shards into tiny glass granules.
 
-A similar occurence happens with atoms.
-Normally atoms are spread apart, by pushing them together, at a certain point (ignoring the orbiting electrons) all nuclei will be "touching" and the [nuclar force](TODO) will keep them together.
-By increasing your pushing force you can overwhelm the nuclear force and "shatter" the nuclei into their subatomic particles.
+A similar occurrence happens with atoms.
+Normally atoms are spread apart, by pushing them together, at a certain point (ignoring the orbiting electrons) all nuclei will be "touching" and the [nuclear force](TODO) will keep them together.
+By further increasing your pushing force you can overwhelm the nuclear force and "shatter" the nuclei into their subatomic particles.
 
-Assuming an enormus explosion doesn't happen, and you can keep increasing your force, it's also possible to overwhelm the "subatomic forces" that keep those subatomic particles "together"[^matter-together].
+Assuming an enormous explosion doesn't happen, and you can keep increasing your force, it's also possible to overwhelm the "subatomic forces" that keep those subatomic particles "together"[^matter-together].
 However, those subatomic particles are more commonly referred to as [elementary particles](https://en.wikipedia.org/wiki/Elementary_particle), meaning there are no "granules" into which the elementary particles can break down.
-It doesn't make sense if subatomic particles wouldn't be "breakable", since that would require dealing with potentially infinite amounts of force and such an assumption would destroy all of humanity's physics.
 
-Then that leaves us with only one solution: laws of matter break.
-If nothing can keep the elementary particles "together", so keep matter "together", then the result is a sort of "collapse", all of that matter just "deflates" into a single, infinetly small point (but the amount of mass isn't infinite, it's the sum of mass from the particles that collapsed).
-That point is called a **black hole**.
+[^matter-together]: Technically, it's more accurate to say "matter apart", since after the collapse matter is as close together as physically possible, and the nuclear forces keep everything in order but also separated. However, considering the intuitive approach I'm striving for, "matter together" is simpler to understand.
+
+It's not possible for those particle to be completely unbreakable, since that would mean they could theoretically sustain infinite amounts of force.
+Such an assumption is unthinkable.
 
-This object is very bizzare, generally it doesn't have too much properties, but the ones it has are really unique.
-In my case of computer graphics, the only important part is it's effect on light.
+Then that leaves us with only one solution: laws of matter "break".
+If nothing can keep the elementary particles "together", so keep matter "together", then the result is a sort of "collapse", all of that matter just "deflates" into a single, infinitely small point (but the amount of mass isn't infinite, it's the sum of mass from the particles that collapsed).
+That point is called a **black hole**.
 
+This object is very bizarre, generally it doesn't have too much properties, but the ones it has are really unique.
+In our case, the only important part is it's effect on gravity, and more precisely, light.
 
-Since a black hole is a single infinetly small point, then all of it's mass is concentrated on that point, making it bend a relatively small are of space **a lot**.
-Strong enough in fact, that if you get too close, you will not be able to overwhelm gravity and "merge" into that infinitessimal point (the white lines are our "geodesics"):
+Since a black hole is a single infinitely small point, then all of it's mass is concentrated on that point, making it alter the geodesics in a relatively small area of space **a lot**.
+So much so, if you get too close, you will not be able to overwhelm gravity and "merge" into that infinitesimal point:
 
 ![](https://upload.wikimedia.org/wikipedia/commons/thumb/9/91/CNX_UPhysics_13_07_spacetime.png/800px-CNX_UPhysics_13_07_spacetime.png?20171124170318)  
+*The white lines are our geodesics*  
 *Image by OpenStax University Physics on [Wikipedia](https://commons.wikimedia.org/w/index.php?curid=64307901)*
 
-An interesting effect from gravity is that light (which always travells "forwards", so along a geodesic) also gets "bent", allowing an observer to view otherwise obstructed objects.
-We call that [graviational lensing](TODO).
-Though not unique to black holes, thats when the effect is strongest:
+A fascinating realization is that light always travels along geodesics ("forwards"), therefore this extreme bending allows us to view otherwise obstructed objects.
+We call that effect [gravitational lensing](TODO).
+Though not unique to black holes, that's when the effect is strongest:
 
 ![](https://upload.wikimedia.org/wikipedia/commons/0/02/Gravitational_lens-full.jpg)  
-*(notice how the gray arrows show the path light travels from a galaxy that is behind (obstructed by) the black hole)*
+*Notice how the gray arrows show the path light travels from a galaxy that is behind (obstructed by) the black hole*
 
-I'll abstrain from explaining light, but you may think of it as a small particle, "similar" to an electron, travelling through space.
-In fact, Newton understood light to be just that, and called the underlying particle a "corpiscule".
-For our purpopses, this approximation will serve us well.
+I'll abstain from explaining the specifics of light, but you may think of it as a small particle, "similar" to an electron, travelling through space.
+In fact, Newton understood light to be just that, and called the underlying particle a "corpuscle".
+This approximation will serve us well.
 
-The distinction that gravity doesn't change the direction or speed of an object "directly" is at upmost importance for physics, since this way we preserve two very important properties of light: it's constant speed and straight travelling path.
-However, in our simulation, it would be significantly simpler to assume the opposite, since to an outside observer (like the "player") the difference is indistinguishable.
+### What about speed and time?
 
-Therefore, in our simulation, we'll have light rays, but when they approach a black hole, their speed and direction would change.
-This will be determined by some physics equations.
+Two of the most significant things Einstein's theory shows, which I've left out thus far, is that time is just another point in space (a fourth "spacial" direction in a way) and everything is relative.
+Which also signifies that our geodesics are actually four dimensional, or in other words, gravity also bends the observed passage of time!
 
-## Finding a formula
+This is relevant for light, since it's speed is constant and it always travels along a geodesic, but to an outside observer it doesn't look that way.
+However, in our simulation, we **only** care about how thing look, so it will make our live much simpler if we try to ignore reality wherever we can.
 
-Now, how do we calculate the exact curvature a black hole produces?
-General theory of relativity provides us with the (really complicated) [Einstein field equations](https://en.wikipedia.org/wiki/Einstein_field_equations) for this exact problem.
-[Schwartzschild](TODO) solved these equations, assuming that the body with mass doesn't spin[^schwm-assumptions].
-This solution is called the [Schwarzschild metric](https://en.wikipedia.org/wiki/Schwarzschild_metric), and a black hole for which this assumption is true is called a "Shwartzschild black hole".
+Therefore, to calculate by how much light will move and change, we'll start with the EFEs and then simplify and approximate until we get something useable.
+
+## Finding a formula
 
-The Shwarzschild black hole is the symplest of it's type, so using the Schwarzschild metric should give us the least trouble.
-Now, here is the metric in Schwartzschild coordinates:
+Let's get down to business, our starting point, the [Schwarzschild metric](https://en.wikipedia.org/wiki/Schwarzschild_metric), in Schwarzschild coordinates, looks like this:
 
 ![](./img/Schwarzschild-metric-in-Schwarzschild-coordinates-wikipedia.png)
 
@@ -184,36 +189,42 @@ Now, here is the metric in Schwartzschild coordinates:
 
 The variables themselves aren't that bad: `c` is the speed of light, <code>&tau;</code> is time, <code>&Omega;</code>, `t` and `r` are Schwarzschild coordinates.
 
-However, as you can see, there are two big problems: we have to calculate derivatives (that's what the `d` signifies) and convert from Schwarzschild coordinates to normal (eucledian) coordinates, that the computer understands natively.
-Both of those operations are **incredibly slow**, the average computer doesn't standa a chance at rendering anything in a reasonable amount of time.
+However, as you can see, there are two big problems: we have to calculate derivatives (that's what the `d` signifies) and convert from Schwarzschild coordinates to normal (euclidean) coordinates, that the computer understands "natively".
+This would be **too slow** to render in real time.
 
-Thankfully [Riccardo Antonelli](http://rantonels.github.io/starless/) was able to do all of our coordinate conversions and simplify the living crap out of that monster (using some approximations with celestial mechanics), turning it into something digestible:
+Thankfully [Riccardo Antonelli](https://github.com/rantonels) was able to do all of our coordinate conversions and use Newton's ideas to simplify the living crap out of that monster, turning it into something digestible:
 
 ![](./img/riccardo-antonelli-schwarzschild-black-hole-force-field-formula.png)
 
-I will not be trying to explain why or how the Schwarzschild metric and this formula work, that's way way beyond my elementary understanding and the scope of this post.
+Explaining his steps and the formula is way beyond the scope of my understanding and this post.
+He explains it in more mathematical detail [here](http://spiro.fisica.unipd.it/~antonell/schwarzschild/).
+
+But now, finally, we can get a plan for our simulation:
+
+- represent light as particles
+- move each particle in a direction and amount specified by the EFE-based formula
+- rinse and repeat until the light hits something
 
 Question now is: what do all of those letters represent?
-This turned out to be a much harder problem than it should have, I actually had to refer [Ricardo Antonelli's](https://github.com/rantonels/starless/blob/7a0c3c4efdc0666f53dcfa48d310fb2164be2d1a/tracer.py) and [Dmitry Brant's](https://dmitrybrant.com/2018/12/11/ray-tracing-black-holes) implementation in their projects, alongside copus articles on wikipedia to fully grasp how to put it to work.
+This turned out to be a much harder problem than it should have, I actually had to refer [Ricardo Antonelli's](https://github.com/rantonels/starless/blob/7a0c3c4efdc0666f53dcfa48d310fb2164be2d1a/tracer.py) and [Dmitry Brant's](https://dmitrybrant.com/2018/12/11/ray-tracing-black-holes) implementation in their projects, alongside copious articles on Wikipedia to fully grasp how to put it to work.
 
 ## Applying the formula to "bend" light's path
 
-Our plan for simulating gravitational lensing is to move the corpiscule by a tiny amount in a certain direction by a certain amount, update the direction and amount according to gravity, rince and repeat.
-The direction, alongside the amount by which the particle moves, can be represented with a single vector, which we'll call light's "velocity".
+The direction, alongside the amount by which the particle moves, can be represented with a single vector, which we'll call light's velocity.
 
 Referring to the formula again:
 
 ![](./img/riccardo-antonelli-schwarzschild-black-hole-force-field-formula.png)
 
-The <code>F<sup>&rarr;</sup></code> returns a force vector (`F` for "force" and <code><sup>&rarr;</sup></code> for "vector"): any force acting on an object has a strength (magnitude) and a direction.
+<code>F<sup>&rarr;</sup></code> returns a force vector (`F` for "force" and <code><sup>&rarr;</sup></code> for "vector"): any force acting on an object has a strength (magnitude) and a direction.
 Imagine you are pushing a box: you are applying some force to it in a "forward" direction.
 If you pull the box, you'll apply some force in a "backward" direction.
 
-This force vector conveys how much and in what direction exactly the black hole is "pulling" on the corpiscule.
+This force vector conveys how much and in what direction exactly the black hole is "pulling" on the corpuscle.
 
 Although the function takes in only one argument, `r`, that doesn't mean that `h` or <code>r<sup>\^</sup></code> aren't also arguments:
 
-- `r` is the distance from the center of the black hole (the singularity) to the **current** position of the corpiscule.
+- `r` is the distance from the center of the black hole (the singularity) to the **current** position of the corpuscle.
 
 - `h` is the magnitude of the ["specific angular momentum"](https://en.wikipedia.org/wiki/Specific_angular_momentum) of the light "particle".
   
@@ -226,26 +237,33 @@ Although the function takes in only one argument, `r`, that doesn't mean that `h
   
   Alternatively, you can "launch" something against the train, meaning you can put a second train, moving in the opposite direction of the first train, so they hit head-on.
   Since the second train would also be moving, if it's travelling at the same speed and has the same mass as the first train, upon collision they would both stop[^hitting-trains].
-  If the second travells at twice the speed, then it can have half of the mass of the first to produce the same effect.
+  If the second travels at twice the speed, then it can have half of the mass of the first to produce the same effect.
   
   The property with which we can "prove" why both a larger mass at no speed, and a smaller mass at a faster speed, can stop the train, is the momentum.
   
-  Angular momentum specifies how much "force" at what speed it takes to stop (or change) the speed and directino of a rotating object.
+  Angular momentum specifies how much "force" at what speed it takes to stop (or change) the speed and direction of a rotating object.
   Imagine the train scenario but on a circular track.
   
   The specific angular momentum is the angular momentum of an object, divided by it's mass.
   It's magnitude kind of tells us which property is more prevalent in the momentum, the mass or the speed.
-  For the same angular momentum, a long (big magnitutde) *specific* angular momentum means our object is small and moves fast, while for a short (small magnitutde) *specific* angular momentum out object is large and moves slowly.
-  
-  In our case a light "particle" is being pulled by a stationary object of large mass (a Shwarzschild black hole), so rather than finding the angular momentum[^angular-momentum] and then dividing by the mass[^light-has-no-mass], we'll use another formula:
+  For the same angular momentum, a long (big magnitude) *specific* angular momentum means our object is small and moves fast, while for a short (small magnitude) *specific* angular momentum out object is large and moves slowly.
   
+  Another simplification we need to is **make our black hole stationary**!
+  Then, rather than finding the angular momentum[^angular-momentum] and dividing by the mass[^light-has-no-mass], we'll use another formula:
+
   ![](./img/specific-angular-momentum-wikipedia.png)
   
-  where **in this specific formula only** `r` is the vector from the center of the big mass (the singularity of the black hole) to the light "particle" and `v` is the velocity (direction and amount of speed) the light is travelling with.
+  where **in this specific formula only** `r` is the vector from the singularity to the corpuscle and `v` is light's velocity.
   
-  The cross product returns a vector, so we'll need to take the magnitutde of the result and plug that into the formula.
+  The cross product returns a vector, so we'll need to take the magnitude of the result and plug that into the formula.
   It's also important that **we calculate it only once, in the beginning**.
   So, we shouldn't be calculating the magnitude of the specific angular momentum every time the light "particle" is moved, rather we calculate it once, after it gets close enough to be affected by the black hole.
+
+[^hitting-trains]: Of course, I'm assuming the trains are strong enough they won't break upon the collision.
+[^angular-momentum]: Angular momentum is actually [defined using the cross product](https://en.wikipedia.org/wiki/Specific_angular_momentum#Definition)
+[^light-has-no-mass]: As mentioned, light has no mass, so this version of the formula wouldn't work.
+  You might think that makes no sense, since we cannot divide by zero, so the specific angular momentum shouldn't exist, but then with the cross product we find a value for it.
+  You would be correct: Antonelli does some funky approximations, combining Newtonian physics with General theory of relativity, to make the formula that simple.
   
 - <code>r<sup>&#94;</sup></code> is the normalized ([unit](https://en.wikipedia.org/wiki/Unit_vector), hence the `^` on top) vector from the center of the black hole to the light particle:
   
@@ -255,11 +273,11 @@ Finally, putting all needed variables together in one proper function, we get:
 
 ![Refer to "Antonelli’s formula - explained" in the Appendix](./img/variables-expanded.png)
 
-As you may notice, this can be simplified, if we substitude <code>r<sup>&#94;</sup></code>:
+As you may notice, this can be simplified, if we substitute <code>r<sup>&#94;</sup></code>:
 
 ![Refer to "Antonelli’s formula - simplified" in the Appendix](./img/antonelli-formula-expanded.png)
 
-In effect, we pass the distance from the black hole to the corpiscule and the function returns how light's velocity changes.
+In effect, we pass the distance from the black hole to the corpuscle and the function returns how light's velocity changes.
 
 Finally, we know what to give the formula, and what it will return.
 The task at hand is making an algorithm which utilises it.
@@ -267,13 +285,15 @@ For that we'll need to get even more familiar with black holes.
 
 ### Properties of a black hole
 
-As stated, we'll be working with non-rotating stationary black holes, Schwarzschild black holes.
-From the [no-hair conjecture](https://en.wikipedia.org/wiki/No-hair_theorem) we know that such a black hole has only three distinct properties: mass, electric charge and angular momentum.
-The last one doesn't apply for our case, and the second is irrelevant.
+To summarize, we're working with a stationary Schwarzschild black hole.
+Meaning our black hole has only one property: mass[^no-hair-theorem].
+Everything else can be derived from it.
+
+[^no-hair-theorem]: The [no-hair theorem](https://en.wikipedia.org/wiki/No-hair_theorem) states that stationary black holes are characterized by only their mass, angular momentum and electric charge. By definition of a Schwarzschild black hole, the last two are null.
 
-Meaning, from the mass alone, we can derive the gravitational "forces", alongside three radii, which will be the backbone of our black hole: the event horizon, the photon sphere and [ISCO](https://en.wikipedia.org/wiki/Innermost_stable_circular_orbit).
+There are three radii, which will be the backbone of our black hole: the event horizon, the photon sphere and [ISCO](https://en.wikipedia.org/wiki/Innermost_stable_circular_orbit).
 
-The event horizon, also called the Schwarzschild radius, is the distance from the center of the black hole (singularity), which once passed, cannot be passed again (you cannot escape the black hole).
+The event horizon, also called the Schwarzschild radius, is the distance from the singularity, which once passed, nothing can return and escape.
 The formula for it is:
 
 ![r_s = (2GM)/(c^2)](./img/Schwarzschild-radius-wikipedia.png)
@@ -285,12 +305,14 @@ Formula:
 
 ![r = r_s * 3/2](./img/photon-sphere-wikipedia.png)
 
-where `G` and `M` are the same as before, `c` is the speed of light in vacuum and <code>r<sub>s</sub></code> is the afroementioned Schwarzschild radius.
+where `G` and `M` are the same as before, `c` is the speed of light in vacuum and <code>r<sub>s</sub></code> is the aforementioned Schwarzschild radius.
 
 Finally, the innermost stable circular orbit (ISCO), formally it's the distance, before which you can orbit the black hole, but once passed, you'll get sucked in and either get shot out or be forever lost past the event horizon.
-For our purposes, it's fine to approximate it as the maximum radius after which spacetime isn't bent[^isco] and objects aren't affected.
+For our purposes, it's fine to approximate it as the maximum radius after which space-time isn't bent[^isco] and objects aren't affected.
 Formula:
 
+[^isco]: It's not really true that gravity stopps affecting spacetime after the ISCO, even though things can orbit Earth that doesn't mean that Earth doesn't bend spacetime. However, for the purposes of a simple shader, this assumption produces perfectly useable results.
+
 ![](./img/isco-wikipedia.png)
 
 Putting them all together, we should expect light to travel like this:
@@ -299,35 +321,43 @@ Putting them all together, we should expect light to travel like this:
 
 ## Light-bending algorithm
 
-Finally, now we have everything we need to divise an algorithm for our light.
-For light we'll store two vectors: one for it's position and one for it's velocity.
-For a black hole we'll store a vector and a quantity: the position of the black hole's center and it's mass.
-For convinience, we could just store the Schwarzschild radius, since all of our radii depend only on it.
+Finally, now we have everything we need to devise an algorithm for our simulation.
+
+- For light we'll store two vectors: one for it's position and one for it's velocity.
+- For the black hole we'll store a vector and a quantity: the position of the black hole's center and it's mass.
+  For convenience, we could just store the Schwarzschild radius, since all of our radii can be expressed only by it.
 
-Once the distance between the light and the center is less than or equal to the ISCO, we can begin augmenting light's velocity.
+Once the distance between a corpuscle and the center is less than or equal to the ISCO, we can begin augmenting light's velocity.
 On small steps, we move the light forward a bit with the velocity and then update the velocity with the black hole's pull.
 
 If light enters the event horizon, then we know it cannot escape, and stop moving it.
 If it passes the ISCO then it has left the black hole and will never approach it again.
 
-If light enters the photon sphere, then we'll have to "track" how long it's been there, if it decides to orbit in a stable manner, then it will stay there indefinetly.
+If light enters the photon sphere, then we'll have to "track" how long it's been there, if it decides to orbit in a stable manner, then it will stay there indefinitely.
 The simplest way to "track" this is just to limit how many times we can move the light forward.
 
 An algorithm doing this would look a bit like this (in pseudocode, inspired by Python):
 
 ![Refer to "Algorithm" section of the Appendix](./img/algorithm.png)
 
-as a reminder, here is the definiton of <code>F<sup>&rarr;</sup></code>:
+as a reminder, here is the definition of <code>F<sup>&rarr;</sup></code>:
 
 ![](./img/antonelli-formula-expanded.png)
 
+<br>
+
+That's it!
+We have all of our theoretical background, formulae and algorithm to start work on our actual program.
+
+To Be Continued!
+
 ## Appendix
 
-##### LaTeX source for equations
+### LaTeX source for equations
 
 The images for my custom LaTeX equations were created on [latexeditor.lagrida.com](https://latexeditor.lagrida.com/), by using Firefox's built-in ability to screenshot HTML nodes, in particular the [MathJax](https://www.mathjax.org/) node with the equation.
 
-###### Antonelli's formula - explained
+#### Antonelli's formula - explained
 
 ```latex
 \begin{align*}
@@ -340,7 +370,7 @@ The images for my custom LaTeX equations were created on [latexeditor.lagrida.co
 \end{align*}
 ```
 
-###### Antonelli's formula - simplified
+#### Antonelli's formula - simplified
 
 ```latex
 \begin{align*}
@@ -352,7 +382,7 @@ The images for my custom LaTeX equations were created on [latexeditor.lagrida.co
 \end{align*}
 ```
 
-###### Algorithm
+#### Algorithm
 
 ```latex
 \begin{align*}
@@ -382,9 +412,3 @@ The images for my custom LaTeX equations were created on [latexeditor.lagrida.co
 \end{align*}
 ```
 
-[^matter-together]: Technically, it's more accurate to say "matter apart", since after the collapse matter is as close together as physically possible, and the nuclear forces keep everything in order but also separated. However, considering the intuitive approach I'm striving for, "matter together" is simpler to understand.
-[^isco]: It's not really true that gravity stopps affecting spacetime after the ISCO, even though things can orbit Earth that doesn't mean that Earth doesn't bend spacetime. However, for the purposes of a simple shader, this assumption produces perfectly useable results.
-[^hitting-trains]: Of course, I'm asuming the trains are strong enough they won't break upon the collision.
-[^angular-momentum]: Angular momentum is actually [defined using the cross product](https://en.wikipedia.org/wiki/Specific_angular_momentum#Definition)
-[^light-has-no-mass]: Actually, light has no mass, so this version of the formula wouldn't work. You might think that makes no sense, since we cannot divide by zero, so the specific angular momentum shouldn't exist, but then with the cross product we find a value for it. You would be correct: Antonelli does some funky approximations, combining Newtonian physics with General theory of relativity, to make the formula that simple. Someone in physics will probably vomit looking at it, but for our purposes it's more than perfect.
-[^schwm-assumptions]: He also assumed the body didn't have an electric charge and the [cosmological constant](TODO) is zero. These are way outside of my understanding and our purposes, so I skiped them for simplicity.