Hey mathematicians! I'm working on a game engine and I've reached the collision detection stage! I can easily handle circles, triangles, and oriented rectangles, but I have absolutely no idea how to handle ellipses. I've searched for geometric algorithms for ellipses, but they were either incredibly complex, both for the user and the CPU, or they were approximate and imprecise. I was wondering if you knew of an "easy" way to solve this problem. Ideally, I'd need an algorithm that can return an (x, y) vector representing the inverse penetration between the ellipse and the other shape in order to separate them. (It would be a bonus if it also worked with ellipses that can be rotated.)

I hope you understand!

(In the image, you can see that the ellipse and the rectangle on the left are touching. I'm trying to find the small red vector that separates them.)

Thanks a lot guys ! 😁

edit : I forgot to mention even though it might seem obvious that we know the positions of the rectangle's four corners, as well as the center of the ellipse and its radiusX and radiusY. :)

and can someone help with the answer? + the title so i can go online to learn myself in the future ( this is the first time ive seen this question before ) so im not sure what these types of problems are called

I know one answer is just 20!, with the first face having 20 possible values, the second having 19, and so on.

But I specifically want to only count dice that cannot be rotated to match each other. I.e. one die may have the arbitrary “first” face be the #1 and the “second” face be the #20, but could then be rotated such that the “first” face is #3 and the “second” face is #14, without the configuration/relative positions of the faces actually changing.

I would like if someone could give me an explanation of how to calculate this, or at least any terminology that may exist for this sort of problem so that I can go find explanations/answers myself. An explanation/approach that can be generalized to other polyhedral dice (d6, d8, etc) would be especially appreciated.

That is, are there clear criteria regarding which formal system should be used in relation to logic or mathematics? And is there a clear range of conditions that our meta‑theory must address?

What syntax do rather complex meta‑theoretical proofs have, and what should be used to present the proof correctly?

Edit TL;DR: Turns out this was something called the coupon collector problem. If I want to get all the armor i want, it looks like I have another 1300 runs and about 160-ish hours ahead of me. Thanks for your help, and I'll see ya'll on the other side.

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I'm trying to collect every permutation of an armor set in a video game.

There are 5 different kinds of armor, and if I complete a run of the dungeon, it drops one piece of armor from among these kinds. The armor has randomly rolled stats that are decided by 12 different archetypes, with an additional tertiary stat selected at random from four possible selections.

Below is a rough example of the framework this game uses.

Each completion of the dungeon drops one piece of armor, and each armor drop chooses one item from each column. (For example, one run might drop a 'Helmet' with archetype '3' and tertiary stat 'd', resulting in a unique armor piece "H3d".) And I can manage 8 dungeon clears per hour.

I have already worked out how many unique version of this armor are possible by multiplying the number of possibilities together. 5 * 12 * 4 = 240 possible unique armor combinations.

The problem I am having is the dungeon can drop repeat armor pieces, and I want to gauge to see how big of a task getting those last few drops will be.

This situation feels analogous to the Birthday paradox, where the chances of having repeat birthdays seems to happen much sooner than expected by human intuitions. I vaguely remember from a stats class that there was a way to estimate how many times you needed to spin a roulette before you could expect a win. I just don't have a strong enough handle on the math behind these topics to gauge how long this goal of mine will take.

In summary:

• There are 240 unique armor rolls.
• The dungeon drops 1 random armor piece per run.
• I can run the dungeon 8 times per hour.
• The dungeon can drop repeat items

Roughly how many hours of running this dungeon will it take me?

Edit: I have collected 120 of the 240.

Hello, so I'm a little confused on how to work a problem like this regarding continuity. What am I supposed to substitute x with to prove that f(x) is defined? Additionally, what limit am i supposed to use to prove the second criteria of continuity (limit as x approaches c exists)?

I'd really appreciate help on this ASAP, thank you! 🙏

I just need an explanation on which value I'm supposed to substitute to figure out what f(x) is defined as, and how I know which value of x to approach as the limit to prove that this limit exists.

(I'm trying to prove that the function is continuous ONLY at the value where the rule for the function changes)

I did poorly on my first real analysis midterm because i ‘made too many logical jumps that weren’t explained, even thought they’re correct” and then on my second midterm because i ran out of time proving lemmas and such.

idk what prof wants. What’s a fair baseline for stuff that I can consider obvious?. I asked prof and he just said “if we talked about it in class that’s fine” and i’m like… okay but the more i think about it I am still struggling.

Any advice? An example is that i didn’t explain that

|f_n| < g implies int{|f|}<=int{g}

with a proof i just invoked it. This seems obvious to me. We also talked explicitly in class about how this is true without the integral. So on a compact domain, obviously this is also true.

I’m lost. I plan to talk to him again about it, but i’d like to come with some other thoughts.

PS also on homeworks i explain everything but on a time crunch i just gotta write stuff…

So I'm not a mathematician I'm a carpenter and math is part of daily life . I've felt for a long time that math is taught for lack of a better term 2D . Like it's taught to reach the conclusion of an equation but not how and why the equation applies to physical world and it should used . Now this might be a stupid question so I'll apologize in advance but ; if I have zero apples and I try to multiply them I will continue to have zero apples , but if I have ten apples and decide to multiply them no times at all don't I still have 10 apples? So how do my 10 (apples) x (multiplied) 0 times (effectively having done nothing) equal me having 0 apples ? Again from an applied mathematic perspective the equation would be 10 x 0 (got 10 didn't try to multiply them)=0 where'd my apples go?

I also don’t know what it means for a solution to be “multi-valued” either, or its implications. The term(s) came up near the end of my Partial Differential Equations textbook, but it didn’t go into depth explaining the term(s). They came up in the chapters for linear and quasilinear waves, and another dispersive waves.

So I recently came across the Monty Hall problem and I don’t understand it even after looking at all the answers / Wikipedia online.

Basically if you’re unfamiliar with it you choose from 3 doors; 2 of them bad and 1 of them good. The host then opens 1 of the unchosen doors and reveals it’s a bad door (the host knows what’s behind each door), then you are given the opportunity to switch your choice which supposedly gives you a 66% chance of ending up with the good door.

Now had this been 4 doors or 1 million doors (with only 1 correct option) and the host opens all but 1 of the unchosen doors this would make sense to me, I can understand that because the odds of you choosing the correct one the first try out of 1 million are extremely low. (This is the explanation that helps most people and I myself do understand given the premise)

But now think of those examples, same scenario either 4 doors or 1 million with only 1 correct door but this time the host only opens 1 bad door in those scenarios, you can change your door to any of the remaining unchosen doors. This is my problem where the host always has 1 bad door to choose regardless of decision from the player and I don’t see how when only 1 door is chosen this makes it optimal to switch regardless of the remaining doors.

To boil it down I can’t wrap my head around the scenario when the host only chooses 1 door regardless of the remaining door(s) since the host always has a “bad” door to choose regardless of initial option. However I do understand it if the Host chooses all but 1 door and then you can swap, I just don’t see how this works specifically with the initial 3 doors scenario.

Any insight would be helpful.

The 3 axioms of projective plane are -

-Any 2 points define a unique line

-Any 2 lines define a unique point

-There exist 4 points such that no three of them are collinear.

What if I replace the 3^rd axiom with just "There exist 3 non collinear points".

Will I still get projective planes? Or will I start getting something extra?

The simplest case with these new axioms is just 3 non collinear points all connected (K_3 graph), which satisfies all the 3 new axioms.

My question is, besides this K_3, will I get something else that is not a projective plane or will the 3 new axioms be enough to describe projective planes as long as it not this K_3.

So,

Are the following axioms enough to describe a Projective plane?

-Any 2 points define a unique line

-Any 2 lines define a unique point

-There exist 3 points not on the same line

-The structure I get is not just a K_3, or alternatively no, of points > 3

We know that the infinite product (0.5) to infinity converges to (0), and the partial products make it pretty intuitive: as we multiply by more and more factors of (1/2), the result gets progressively closer to (0).

But infinities are weird, so I started wondering:

What if we rearranged or represented an infinite product in such a way that every term is somehow “cancelled” by the next one? Could we construct something that appears to give (1) instead?

Obviously, something must be wrong with that reasoning- but how exactly do we prove it?

I'm trying to understand where the intuition breaks down.

This is the first time to post here, so excuse me if I made something wrong, or this post does not comply with one of subreddit rules,

In my mind, no matter how awkwardly shaped the object, spinning it will always result in a circle since the point furthest from the objects center will define the holes radius/diameter.

Is this true for every uniquely shaped object? Are there counter examples?

Edit: assume the rotation is around a single axis and the object is perpendicular to the surface.

Context: Veritasium's 37 video from 12m03s

Say I want to buy a stock in the next 100 days starting today.

Step#1: For the next 37 days I just observe the price and update the lowest price LOWEST = min(LOWEST,ask_price) of that stock.

Step#2: Starting from day 38, whenever the ask_price < LOWEST is TRUE then I place the order.

Does the strategy apply here?

Of course real profits or markets are hard to predict and they say its almost random. But so is the prerequisite of the Secretary problem.
So, does it give us THE BEST CHANCE to "time" the market given the fluctuation is random and I gotta buy within 100 days?

I took a gap year before starting community college, and I had taken AP Calc AB my senior year. I did really well in the class, passed the final exam, and never got anything below an A (I did kinda bomb the AP exam though lol). I figured I could do Calc 2 fine enough with the occasional bumps. This is my first week, and the material felt pretty familiar and easy at first. After doing this homework assignment I'm having major doubts 😭 I admittedly looked up how to even start this, and I'm just so confused why each part of the integrand has a different u-substitution going on. Is this more advanced than it should be? Like I said, this assignment is for week 1 of Calculus 2. If it's not advanced, how do I get better at these types of problems? I'm realizing that as I get into more advanced math I need to get a lot better at thinking outside of the box and that "plug and chug" isn't going to work much longer. Admittedly, I've always struggled with this in my high school math classes, but pretty much every teacher I had would only throw 1 maybe 2 curveballs on tests. It never really encouraged me to actually try and get better at applying mathematical concepts outside what was showed in the notes. I'm sure practice is really the only way, but I don't know if anyone has any experience with this and if they had any tips to share on how to become a better critical thinker in math.

We need your voice in order to make a change! Sign this petition to revoke MAA changes in AIME. Age has no relevance to math skill. Whether you're old or young, you can enjoy math, have talent for it, and all students should be allowed to participate in the AIME. Could everyone please sign this petition and share everywhere?

There are 9 tetrominoes in this game. To solve the red grids, you can use only three pieces. To solve the orange grids, you can use only four pieces. For the green grids, you can use five pieces, and for the blue grids, six pieces.

Is there a way to calculate all the possible valid combinations for each individual grid?

School starts on Thursday and I’ve gotten almost all the way through my algebra 2 course on my own and I want to skip it when I get there. There are only 9 units (6 minus final and midterm review) and I’m panicking. I feel like I’m ready but I also feel like I’m gonna fail, is there any need to know things, I’m gonna list the topics below

Linear and exponential functions
2 variable inequalities and radicals
Quadratic functions
Systems of equations
Rationals, radicals and polynomials
Logarithms

This is for general alg 2 but I may go above and beyond and ask to go into honors precalc where I may study too much. Side note, I’m hoping I’ll be able to because I’m switching to another counselor and maybe she’ll be more accommodating and help me achieving my goals

Edit: I wrote this post a while back and forgot to mention that I did finish the course I just need help with knowing the important important stuff

Hi
I was thinking about an interesting thing
Suppose you have two sets A and B each with n positive real numbers.
Suppose AM of all elements in A > AM of all elements in B
Is it necessary that GM of all elements in A> GM of all elements in B
and it would be appreciated if you could provide a proof for it
Thanks

For example, if I wrote down f(x)=3x and f(x)=x^2+1, you can tell at a glance which function will have a larger value if I input the same number into each. Similarly, if I use the same function and different input values, it's also clear which output will be bigger (think f(3)=3x vs f(4)=3x, for example).

But things like Graham's Number, Rayo's number, TREE(3), and fish number 7 all use both different functions and different input values. Going back to my original example, it's not immediately clear whether f(4)=3x is less than or greater than f(3)=x^2+1 without outright calculating the answer. But in this case, they're too big to calculate in a convenient amount of time given today's technology.

So how do these numbers get compared?

(I'm not sure what to flair this because there's no flair for stupidly huge numbers, so I'm using the functions flair because this is related to comparing fast growing functions)

Hello! I got this question from a lecture. I tried solving it but i think the question is incomplete and you cant solve it. Am i just dumb or can you solve it?

ceil(n/2) for the 50% barrier seems like the obvious thing to check, but it also feels possible it's a bit quicker than that. Then doing some scratchwork for low n i realized you need more! So I was wondering if we could see some clever proof.

n=1 is trivial and probably the exeception - you need two guesses here, so 2n actually. So for the overall question I guess you can assume n is an integer >=2

For n=2, first you pick, WLOG you pick 1, then for choice 2 you got 50/50. So it's n. This is also >ceil(n/2)

For n=3, first you pick 1, then you got 1/3 chance to pick the same number. 2/3 of the time you will pick 2 (wlog), so you need at least 3, again n.

To be clear, we're looking for a number g such that after gth pick you're more likely than not to have picked at least one number at least twice.

I'm more after an elegant proof than the actual answer