## How to iteration all the points on curves

I plot curves by

ContourPlot[x^2 y + x y^2 == 1, {x, -10, 10}, {y, -10, 10}] 

It shows three curves on the (x,y) plane, but now my question is how to iterate all the points on the curves, the iterative equation is complex (nonlinear), for example, x = x+y*x, and y = x*y+y^2.

I have an idea, solve x^2 y + x y^2 == 1 and get (x,y), then plot it by ParametricPlot one by one, but it will be tedious (as it is nonlinear, so solve it is difficult sometimes). Is there any better way?

## Intersection algorithim for finding the intersection points of two arbitrary mathematical curves

I’m looking for an intersection algorithm for finding the intersection points of two arbitrary mathematical curves. Websites like Desmos, GeoGebra, and WolframAlpha allow you to graph and/or input two(Or more but I’m only interested in two) curves to find their intersection points. I know all of the services I’ve listed have some form of API, my problems with those APIs are the technical and practical restrictions. Therefore, I’d like to just use the algorithms they use under the hood. Problem comes when I don’t even know the names of those algorithms or even whether they’re public domain. I’ve done some searching around but I can’t seem to find what I’m looking for. So I guess what I’m asking for are some basic design descriptions/requirements for these algorithms so I can implement one of them. I figured this problem wasn’t really a question for the mathematics StackExchange or StackOverflow So I decided to place it here. Let me know if that’s a problem.

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## Problems with using lissajous curves for sniper scope sway

I like to pick apart games in my spare time and recreate some mechanics from simple to complex which I think would be fun. The game I mainly pick apart is Call of Duty 4: Modern Warfare because of its large modding community and mod tools which make it easy to learn how the game works. The in-game developer console also makes it easy to debug the game on my own as well.

One of the mechanics I am currently trying to pick apart is the sniper scope. One thing I noticed right away is the sway pattern which is a lissajous pattern. I quickly implemented my own lissajous pattern which is the classic figure-8 used in most games and applied it to the camera. Here is the script:

using UnityEngine;  public class ScopeSway : MonoBehaviour {     public int a = 1,                b = 2;      private float size = 1f,                   speed = 0.5f,                   time = 0,                   posX = 0,                   posY = 0;      void Update()     {         time = Time.time * speed;         posX = size * Mathf.Sin((a * time) + (Mathf.PI / 2));         posY = size * Mathf.Sin(b * time);          Vector3 offsetX = transform.up * posX,                 offsetY = transform.right * posY,                 offsetZ = transform.forward * transform.position.z;          transform.localPosition = offsetX + offsetY + offsetZ;     } } 

One thing I want to get out of the way is that if you visualize the figure 8 the height and width are the same size, so instead of making a width and height I merged them into a single size variable which makes a perfect square. The a and b variables modify the frequency which allow you to make more curves instead of a figure-8. Using this source you will be able to plug in your own a and b values from the patterns you see. This is also the source I used when writing this script.. The script doesn’t have any problems, it does exactly what I wrote it to do. However, there is one micro detail and core mechanic I am failing to recreate.

For those that have played the game and have access to it there is one thing you probably have never noticed until you actually spot what is happening. In the game whenever you are looking through the sniper scope you will notice that the scope sways like normal. However, no matter how close or how far whatever it is that you are aiming at the size of the curve pattern is the same. In order to visualize this, take the script I pasted here and put it on a camera and place a cube 100 meters away, also set your cameras FoV to 15 (CoD 4 uses 15 FoV for sniper scopes). The scope will seem like it sways normally but not until you move the cube directly in front of the camera. Now you will notice that the camera sway is huge on targets up close but small for targets far away. In CoD, the sway amount is the same no matter how far or how close you are to what you are aiming at. What exactly is going on here?

After thinking about it I thought that maybe they are changing the size of the pattern based on the distance of where you are pointing. So I decided to add a raycast for up to 100 meters and change the size of the pattern based on the returned hit distance. It failed miserably. It created lots of stuttering and snapping motions which where very visible and hard on the eyes. After that I thought that maybe I can just interpolate between the two points but that won’t work because in CoD it does not look like the view is being interpolated, the sway does not even make a single twitch. If you take my script and aim at the corner of the object and let the sway point off of the object the sway amount goes all the way back up, which does not happen in the game.

I do not know what I happening or how they are achieving this effect. I do know that the sway is programmatically done and is not using an animation because the sway has the same pattern as the gun sway when aiming from the hip. Whenever you scope in and scope out the scope does not start at the same place every time, it starts at a different place which means it is using lissajous curves based on a factor of time.

My question for you guys is, how are they achieving this and what am I doing wrong? I have spent five days now researching and picking apart some open source games but I have not found an solution or even a hint. It seems like this topic isn’t really well covered because it is a mechanic no one notices and we all take for granted, but it is the most important polish for full screen sniper scopes. How do I make the sway pattern the same size and speed no matter how far or how close it is that I am looking at.

A better explanation of what is going on is if you zoom in CoD right in front of a wall the scope will only sway in a box of lets say 20 pixels. And when you zoom to something far out it still only sways in a box of 20 pixels. My script will sway in a box of 20 pixels when aiming at things 100 meters away but when zooming in on things directly in front of the camera the camera will sway in a box of 100 or more pixels. I want my camera to sway in a box of 20 pixels or so no matter how far or how close the object you are aiming at is. It’s almost like they are shifting the cameras viewport and not the physical camera?

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## How to customize default enabled EC curves or the curves preference in Azul Zulu JDK

We use Zulu JDK and want to customize the default enabled EC curves. Oracle JRE provides this using jdk.tls.namedGroups system property in java.security.

https://www.oracle.com/technetwork/java/javase/8u121-relnotes-3315208.html

section “Improve the default strength of EC in JDK” describe this for Oracle JDK

Any suggestion?

Thanks

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I have a bunch of Differential Pulse Voltammetry data I am trying to import into Mathematica, and find the area between each curve and the blank. So far I have used the following commands:

"Data1 = Import[   "/Users/chelseaswank/Desktop/untitled folder/HSJ-6-26-19-7-#2 - \ Sheet1 4.47.09 PM.csv", "Table"] Data2 = Import[   "//Users/chelseaswank/Desktop/untitled folder/HSJ-6-26-19-13-#2 - \ Sheet1 4.47.09 PM.csv", "Table"] Data3 = Import[   "/Users/chelseaswank/Desktop/HSJ-6-26-19/HSJ-6-26-19-19-#2 - \ Sheet1.csv", "Table"] Data4 = Import[   "/Users/chelseaswank/Desktop/untitled folder/HSJ-6-26-19-25-#2 - \ Sheet1 4.47.09 PM.csv", "Table"] Data5 = Import[   "/Users/chelseaswank/Desktop/untitled folder/HSJ-6-26-19-31-#2 - \ Sheet1 4.47.09 PM.csv", "Table"] Data6 = Import[   "/Users/chelseaswank/Desktop/untitled folder/HSJ-6-26-19-37-#2 - \ Sheet1 4.47.09 PM.csv", "Table"] Data7 = Import[   "/Users/chelseaswank/Desktop/untitled folder/HSJ-6-26-19-43-#2 - \ Sheet1 4.47.09 PM.csv", "Table"] Data8 = Import[   "/Users/chelseaswank/Desktop/untitled folder/HSJ-6-26-19-49-#2 - \ Sheet1 4.47.09 PM.csv", "Table"] Data9 = Import[   "/Users/chelseaswank/Desktop/untitled folder/HSJ-6-26-19-55-#2 - \ Sheet1 4.47.09 PM.csv", "Table"]"                 and                                                   "f1 = ListLinePlot[    Import["/Users/chelseaswank/Desktop/HSJ-6-26-19/HSJ-6-26-19-7-#2 - \ Sheet1.csv"]]; f2 = ListLinePlot[    Import["/Users/chelseaswank/Desktop/untitled \ folder/HSJ-6-26-19-13-#2 - Sheet1.csv"]]; Plot[{f1[x], f2[x], f1[x] - f2[x]}, {x, 0, 1.1},   Filling -> {{1 -> {2}}, 3 -> Axis}, PlotLegends -> "Expressions"]" 

## Counting elliptic curves by discriminant

Enumerating elliptic curves $$E/\mathbb{Q}$$ sorted by (the absolute value of) their minimal discriminants is a difficult open problem, as is the (likely easier) problem of counting elliptic curves $$E/\mathbb{Q}$$ given by a minimal Weierstrass model $$E_{A,B} : y^2 = x^3 + Ax + B$$, ordered by the model discriminant $$\Delta(E_{A,B}) = -16(4A^3 + 27B^2)$$.

For the question, it is not too hard to show that the number of curves $$E$$ with $$|\Delta(E_{A,B})| \leq X$$ is $$O(X)$$. Counting elliptic curves by their minimal Weierstrass models is essentially equivalent to counting monogenic cubic rings by discriminant, and the number of monic cubic rings of discriminant bounded by $$X$$ is surely less than the number of cubic rings having discriminant bounded by $$X$$, and the number of cubic rings having discriminant bounded by $$X$$ is $$O(X)$$ by the Davenport-Heilbronn theorem.

I am asking about whether there is any improvement over this bound: is it known that

$$\displaystyle N(X) = \#\{E_{A,B} : 16|4A^3 + 27B^2| \leq X\} = o(X)?$$

## Constructing elliptic curves defined over $\mathbb{Q}$ with fixed complex multiplication

I have the following problem: we know that for a field $$\kappa$$ of characteristic $$0$$ usually an elliptic curve $$E$$ defined over $$\kappa$$ is such that $$End(E)\cong \mathbb{Z}$$. This means that one cannot hope to find an elliptic curve with complex multiplication choosing it “randomly”.

Suppose that i want to produce and elliptic curve over $$\mathbb{Q}$$ whose $$End(E)$$ is and order in $$\mathbb{Q}(\sqrt{-D})$$: what are the methods currently known to do it? Is it possible to write explicitly the isogeny corresponding to $$\sqrt{-D}$$? (If it is in $$End(E)$$ and $$D$$ is not so large).

## The underlying curve of a family of genus zero $n$ punctured curves

Let $$X$$ be a curve of genus zero over an algebraically closed field $$k$$ so that $$X \cong \mathbb{P}_k^1$$. Let $$(C, s_1, \cdots, s_n)$$ a $$n$$ punctured genus zero curve over $$k$$ where $$s_i: k \to C$$ are sections. Sine $$C$$ is a genus zero over $$k$$, we have that $$C \cong \mathbb{P}_k^1$$.

Now let $$B$$ be an affine scheme over $$k$$ and suppose consider the family of $$n$$-punctured genus zero curves $$(C \to B, s_1, \cdots, s_n)$$ where $$s_i:B \to C$$. Let us always assume that $$n \ge 4$$.

At this point, I tend to come across statements inferring that since the fibers of the family $$(C \to B, s_i)$$ have non non-trivial automorphism $$C \cong \mathbb{P}_B^1$$. Here I by no means am claiming that the family $$(C \to B, s_i)$$ of $$n$$-punctured curves is trivial but rather that the (total space) of the family of genus zero curves over $$B$$ “underlying” the family $$(C \to B, s_i)$$ is trivial.

Why does the automorphism group of the fibers of a family $$(C \to B, s_i)$$ being trivial imply that the underlying family of genus zero curves over $$B$$ is trivial? Again I mean this in the sense that the total space $$C \cong \mathbb{P}_B^1$$.

## Singularity of Brill-Noether sub varieties of Picard varieties of smooth curves

Suppose $$C$$ is a smooth projective curve over complex numbers. The singularities of the theta divisor $$\Theta$$ in $$Pic^{g-1}(C)$$ is described in the literature. It is $$W^{1}_{g-1}=\{l\in Pic^{g-1}(C): h^0(l)\geq 2\}$$. I could find in the literature the expected dimension results. Are the singularities of $$W^1_{g-1}$$ known.

Question: What is the (exp)dimension of $$Sing(W^1_{g-1})$$, for a generic curve ?

## Do negative indecomposable bundles on curves have sections?

Let $$X$$ be a smooth projective curve, and $$E$$ an indecomposable vector bundle on $$X$$ with $$\mathrm{deg} E<0$$. Is it true that $$H^0(X,E)=0$$?

This is true if $$E$$ is a line bundle, which means it is also true whenever $$X$$ is $$\mathbb{P}^1$$, since all vector bundles split here.

It is also true by results of Atiyah if $$X$$ is an elliptic curve. What about for curves of higher genus?

The assumption that $$E$$ is indecomposable is of course necessary.