Mixing
Estimate coordinates of vertices
Let $hat{t}$ be the reference triangle with the vertices $hat{A_1} = (0,0)$, $hat{A_2} = (1,0)$, $hat{A_3} = (0,1)$ and let $t$ be the triangle with the vertices $A_1 = (0,0)$, $A_2 = (h_1, 0)$ and $A_3 = (0, h_2)$. Let $hat{u} := u circ F_t$, where $F_t: hat{t} to t$ denotes the affine map with $F_t(hat{A_i}) = A_i$ for i = 1, 2, 3.
What can we say about the constant $C$ in $$max {h_1^{-1}, h_2^{-1} } leq C h_t^{-1}$$
where $h_t$ is the diameter of the triangle $t$ ?
(I think the best would be to express $C$ in terms of the shape-regularity constant.)
What I've done so far:
The triangle $t$ is rectangular and we know that $h_t = sqrt{h_1^2 + h_2^2}$.
We want to find a $C$ such that $max{h_1^{-1}, h_2^{-1}} leq C h_t^{-1} = frac{C}{sqrt{h_1^2+h_2^2}}$
But...am I wrong or did I missunderstand something? Because such a $C$ can not exist, if, for example, $0 leq h_1 leq 1$ and $h_2 = 1$, then $max{h_1^{-1}, h_2^{-1}} = frac{1}{h_1} to infty$ if $h_1$ goes to $0$.
But the task is as described above, so I think that I'm wrong and did a mistake. But where?
Thanks for any help!
linear-algebra geometry numerical-methods triangulation
asked Nov 20 '18 at 13:04
StMan
1387
add a comment |
Let $hat{t}$ be the reference triangle with the vertices $hat{A_1} = (0,0)$, $hat{A_2} = (1,0)$, $hat{A_3} = (0,1)$ and let $t$ be the triangle with the vertices $A_1 = (0,0)$, $A_2 = (h_1, 0)$ and $A_3 = (0, h_2)$. Let $hat{u} := u circ F_t$, where $F_t: hat{t} to t$ denotes the affine map with $F_t(hat{A_i}) = A_i$ for i = 1, 2, 3.
What can we say about the constant $C$ in $$max {h_1^{-1}, h_2^{-1} } leq C h_t^{-1}$$
where $h_t$ is the diameter of the triangle $t$ ?
(I think the best would be to express $C$ in terms of the shape-regularity constant.)
What I've done so far:
The triangle $t$ is rectangular and we know that $h_t = sqrt{h_1^2 + h_2^2}$.
We want to find a $C$ such that $max{h_1^{-1}, h_2^{-1}} leq C h_t^{-1} = frac{C}{sqrt{h_1^2+h_2^2}}$
But...am I wrong or did I missunderstand something? Because such a $C$ can not exist, if, for example, $0 leq h_1 leq 1$ and $h_2 = 1$, then $max{h_1^{-1}, h_2^{-1}} = frac{1}{h_1} to infty$ if $h_1$ goes to $0$.
But the task is as described above, so I think that I'm wrong and did a mistake. But where?
Thanks for any help!
linear-algebra geometry numerical-methods triangulation
asked Nov 20 '18 at 13:04
StMan
1387
add a comment |
Let $hat{t}$ be the reference triangle with the vertices $hat{A_1} = (0,0)$, $hat{A_2} = (1,0)$, $hat{A_3} = (0,1)$ and let $t$ be the triangle with the vertices $A_1 = (0,0)$, $A_2 = (h_1, 0)$ and $A_3 = (0, h_2)$. Let $hat{u} := u circ F_t$, where $F_t: hat{t} to t$ denotes the affine map with $F_t(hat{A_i}) = A_i$ for i = 1, 2, 3.
What can we say about the constant $C$ in $$max {h_1^{-1}, h_2^{-1} } leq C h_t^{-1}$$
where $h_t$ is the diameter of the triangle $t$ ?
(I think the best would be to express $C$ in terms of the shape-regularity constant.)
What I've done so far:
The triangle $t$ is rectangular and we know that $h_t = sqrt{h_1^2 + h_2^2}$.
We want to find a $C$ such that $max{h_1^{-1}, h_2^{-1}} leq C h_t^{-1} = frac{C}{sqrt{h_1^2+h_2^2}}$
But...am I wrong or did I missunderstand something? Because such a $C$ can not exist, if, for example, $0 leq h_1 leq 1$ and $h_2 = 1$, then $max{h_1^{-1}, h_2^{-1}} = frac{1}{h_1} to infty$ if $h_1$ goes to $0$.
But the task is as described above, so I think that I'm wrong and did a mistake. But where?
Thanks for any help!
linear-algebra geometry numerical-methods triangulation
asked Nov 20 '18 at 13:04
StMan
1387
Let $hat{t}$ be the reference triangle with the vertices $hat{A_1} = (0,0)$, $hat{A_2} = (1,0)$, $hat{A_3} = (0,1)$ and let $t$ be the triangle with the vertices $A_1 = (0,0)$, $A_2 = (h_1, 0)$ and $A_3 = (0, h_2)$. Let $hat{u} := u circ F_t$, where $F_t: hat{t} to t$ denotes the affine map with $F_t(hat{A_i}) = A_i$ for i = 1, 2, 3.
What can we say about the constant $C$ in $$max {h_1^{-1}, h_2^{-1} } leq C h_t^{-1}$$
where $h_t$ is the diameter of the triangle $t$ ?
(I think the best would be to express $C$ in terms of the shape-regularity constant.)
What I've done so far:
The triangle $t$ is rectangular and we know that $h_t = sqrt{h_1^2 + h_2^2}$.
We want to find a $C$ such that $max{h_1^{-1}, h_2^{-1}} leq C h_t^{-1} = frac{C}{sqrt{h_1^2+h_2^2}}$
But...am I wrong or did I missunderstand something? Because such a $C$ can not exist, if, for example, $0 leq h_1 leq 1$ and $h_2 = 1$, then $max{h_1^{-1}, h_2^{-1}} = frac{1}{h_1} to infty$ if $h_1$ goes to $0$.
But the task is as described above, so I think that I'm wrong and did a mistake. But where?
Thanks for any help!
linear-algebra geometry numerical-methods triangulation
linear-algebra geometry numerical-methods triangulation
asked Nov 20 '18 at 13:04
StMan
1387
asked Nov 20 '18 at 13:04
StMan
1387
asked Nov 20 '18 at 13:04
StMan
1387
asked Nov 20 '18 at 13:04
StMan
1387
asked Nov 20 '18 at 13:04
StMan
1387
1387
add a comment |
add a comment |
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