# Tagged Questions

140 views

### Is there any reason why $4-\pi$ is quite close to $\frac{\sqrt{3}}{2}$?

In this question obviously the error of our "approximation" is $4-\pi=0.858...$ . I tried to reconstruct the false argument with $\tau=2\pi$, and the error in that case would be $8-\tau=1.716...$, ...
27 views

### Find a maximum triangle that lies on a polyline (with constraints)

If there's a polyline (a GPS track, actually) with a lot of points (could be several thousand), that looks like this 1) How can I find such a triangle with the biggest possible perimeter, that its ...
70 views

### Why does $\arctan(\frac{\tan \theta}{2}) \approx \frac{1}{2 - \theta} - \frac{1}{2 + \theta}$ for small $\theta$?

In answering this question, I needed to show that $\arctan \left( \frac{\tan \theta}{2} \right) \approx \frac{\theta}{2}$ when $\theta$ was small. So, naturally, I computed the first few terms of the ...
495 views

### Geometric explanation of $\sqrt 2 + \sqrt 3 \approx \pi$

Just curious, is there a geometry picture explanation to show that $\sqrt 2 + \sqrt 3$ is close to $\pi$?
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### Approximation to $\sqrt{\cos(\theta)}$?

I have this formula, (it is just the law of cosines angle formula): $$d = \sqrt{a^2 + b^2 - 2ab \ cos(\theta)}$$ Here is my issue. I am wondering if there is a way to 'extract' the $cos$ term. My ...
64 views

### Name of the generalization of quadtree and octree?

What is the name of the equivalent of quadtrees and octrees in n-dimension ?
40 views

### Asymptotic behaviour of the area of a 2-dimensional flat subset of $\mathbb{R}^3$

I am interested by the area of the $2$-dimensional flat subset of $\mathbb{R}^3$ defined by the following equations with one parameter $t>1$: $x,y,z>0$ (positive octant) $x+y+z=t$ (hyperplane ...
26 views

### Approximate sector between two lines?

I need to approximate a red figure. I know coordinates of three points (little transparent circles). I also know a count of segments I need to divide this figure. The angle may be from 0 to Pi and ...
104 views

### probability involving matching of discrete shapes on a square grid

Figure F exists on a regular square grid. T transforms F by any combination of horizontal or vertical reflection as well as rotation by 90 or 180 degrees. A larger background grid of X by Y contains ...
597 views

### How to calculate the area of bizarre shapes

I'm looking for an algorithm to calculate the area of various shapes (created out of basic shapes such as circles, rectangles, etc...). There are various possibilities such as the area of 2 circles, 1 ...
322 views

### What is the correct value of $\pi$

I have seen that: $\pi = 22/7$ $\pi = 3.14\ldots$ $\pi = 17 - \sqrt{192}$. $22/7 \gt \pi$ $22/7 \lt\pi$ My brain storming doubt was, is A = B? Is B = C? Is C= A = B? how D and E are correct or ...
184 views

### calculate surface normal with random sampling of points

Given a surface in $R^3$ and a point P on the surface, I want to calculate the surface normal in this point, the vector that is perpendicular to the surface. However, I do not know the whole surface, ...
133 views

### How do I create a shape from a square corners' values?

I'm working on a 3D algorithm, so my problem applies to cubes, not squares. But for convenience, I'll stick to 2D. Each corner of a square can contain up to 100 units, depending of the values at each ...
1k views

### Generating control-point tangents for a Catmull-Rom spline

From Wikipedia, we have a few variations for calculating tangents when creating a spline based only on positions of control-points: Finite difference \mathbf m_k=\frac{\mathbf p_{k+1}-\mathbf ...
49 views

### Lower-Bounding angles in integer Lattices

Given an $n \times n$ integer grid I chose any two grid points $a,b$, draw a line $l$ through $a$ and $b$ and measure the angle between $l$ and a horizontal line. I can do this for any grid point pair ...