Linked Questions

13
votes
5answers
38k views

Is there any equation for triangle?

Like there's an equation of a circle, is there any equation of a triangle? I've been trying to build one and the closest thing I've managed to do is to create an equation of 2 lines and use the $x$ ...
20
votes
4answers
905 views

How prove this inequality $\frac{2}{(a+b)(4-ab)}+\frac{2}{(b+c)(4-bc)}+\frac{2}{(a+c)(4-ac)}\ge 1$

let $a,b,c>0$,and such $a+b+c=3$, show that $$\dfrac{2}{(a+b)(4-ab)}+\dfrac{2}{(b+c)(4-bc)}+\dfrac{2}{(a+c)(4-ac)}\ge 1$$ I think this inequality use this $$ab\le\dfrac{(a+b)^2}{4}$$
21
votes
2answers
1k views

How to prove this inequality $(a^2+bc^4)(b^2+ca^4)(c^2+ab^4) \leq 64$?

Question: If $a,b,c$ are nonnegative real numbers such that $a+b+c=3,$ then $$(a^2+bc^4)(b^2+ca^4)(c^2+ab^4) \leq 64$$ My try: I found the equality holds only if $(a,b,c)=(2,0,1)$ or all of its ...
3
votes
3answers
2k views

Quadrilateral Interpolation

The simplest finite element shape in two dimensions is a triangle. In a finite element context, any geometrical shape is endowed with an interpolation, which is linear for triangles (most of the time),...
8
votes
2answers
1k views

Barycentric coordinates in a triangle - proof

I want to prove that the barycentric coordinates of a point $P$ inside the triangle with vertices in $(1,0,0), (0,1,0), (0,0,1)$ are distances from $P$ to the sides of the triangle. Let's denote the ...
3
votes
1answer
612 views

Example of a Problem Made Easier with Skew Coordinates

Skew or oblique coordinate systems are coordinate systems where the angle between the axes is not 90 degrees. The second answer to this question has formulas to convert between these systems with an ...
5
votes
2answers
432 views

Is there an easy way to determine a point from its barycentric coordinates geometrically?

For example if I have a point $(\alpha_1,\alpha_2,\alpha_3,\alpha_4)$ and a polygon with vertices $v_1,v_2,v_3,v_4$, can I determine the actual location without calculating the result with the formula ...
0
votes
3answers
232 views

Computing a three-dimensional Lebesgue measure of a bounded set

How can I compute the three-dimensional Lebesgue-measure of the set $A$ which is bounded by the areas $x+y+z =6$, $x=0$, $z=0$ and $x+2y=4$? A hint on how I solve problems like this in general would ...
0
votes
1answer
714 views

Jacobian determinant for bi-linear Quadrilaterals

Mapping from a square $\left[-\frac{1}{2},\frac{1}{2}\right]\times\left[-\frac{1}{2},\frac{1}{2}\right]$ with local coordinate system $\,(\xi,\eta)\,$ to an arbitrary quadrilateral with global ...
1
vote
1answer
360 views

Linear isoparametrics with dual finite elements

The subject presented here is some content of the Wikipedia page about Platonic solids combined with my own experience on Finite Elements.To start with the latter, there is a standard piece of Finite ...
1
vote
2answers
204 views

finite elements local vs global basisfunction

I always stumble across the term "local" and "global" basis functions for finite elements. But could not find an explanation what the difference is. What is the difference and where do they occur?
6
votes
1answer
160 views

Finite Element on surfaces: evaluate solution

While working with a finite element for a PDEs solver on Riemannian Surfaces embedded in $\mathbb R^3$, I got stuck when needing to evaluate the solution $u$ at a given point $(x_0,y_0,z_0)$ The ...
3
votes
1answer
96 views

Do Finite Elements really have to be disjoint?

When looking up the tag description of (finite-element-method), we saw the following : [ .. ] It consists of a method of discretization splitting the domain into disjoint subdomains [ .. ] Take a ...
1
vote
2answers
67 views

Finite element heat equation on a single simplex?

I am currently trying to learn the finite element method. Ultimately, I want to solve the heat equation in arbitrary dimensions. For the purpose of this question, however, assume that I am interested ...
4
votes
1answer
45 views

Computation of piece-wise linear hat functions

I have a discretized 3D surface for which I want to compute piece-wise linear hat functions. I assumed these functions are of the following form: $$\phi = ax + by + cz + d$$ with the property of $\...

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