Multiple integral equation with Barycentric coordinate
$begingroup$
$Delta(i, j, k)$ is a triangle with vertex $i(x_i, y_i), j(x_j, y_j), k(x_k, y_k)$ in counterclockwise, $P(x, y)$ is a point in $Delta(i, j, k)$, $S, S_i, S_j, S_k$ is the area of $Delta(i, j, k), Delta(j, k, P), Delta(k, i, P), Delta(i, j, P)$
we have $$left{begin{align}2S &= begin{vmatrix}1 & x_i & y_i \1 & x_j & y_j \1 & x_k & y_kend{vmatrix}, 2S_i = begin{vmatrix}1 & x & y \1 & x_j & y_j \1 & x_k & y_kend{vmatrix} \2S_j &= begin{vmatrix}1 & x_i & y_i \1 & x & y \1 & x_k & y_kend{vmatrix}, 2S_k = begin{vmatrix}1 & x_i & y_i \1 & x_j & y_j \1 & x & y end{vmatrix} end{align}right.$$
$P$'s barycentric coordinate is $(L_i, L_j, L_k)$
$$left{begin{align}L_i &= dfrac{S_i}{S} = dfrac{1}{2S}[(x_j y_k - x_k y_j) + (y_j - y_k)x + (x_k - x_j)y] \ L_j &= dfrac{S_j}{S} = dfrac{1}{2S}[(x_k y_i - x_i y_k) + (y_k - y_i)x + (x_i - x_k)y] \ L_k &= dfrac{S_k}{S} = dfrac{1}{2S}[(x_i y_j - x_j y_i) + (y_i - y_j)x + (x_j - x_i)y]end{align} right.$$
and
$$left{begin{align}x &= x_i L_i + x_j L_j + x_k L_k \ y &= y_i L_i + y_j L_j + y_k L_kend{align}right.$$
The problem is how to proof
$$begin{equation}iintlimits_{Delta(i, j, k)}L^p_i L^q_j L^r_k dxdy = 2Sdfrac{p!q!r!}{p + q + r + 2}end{equation}$$
I have replace $dxdy$ with $(x_i dL_i + x_j dL_j + x_k dL_k)(y_i dL_i + y_j dL_j + y_k dL_k)$ but still do not know how to continue.
integration barycentric-coordinates
asked Jan 3 at 9:00
BluedropsBluedrops
136
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add a comment |
$begingroup$
$Delta(i, j, k)$ is a triangle with vertex $i(x_i, y_i), j(x_j, y_j), k(x_k, y_k)$ in counterclockwise, $P(x, y)$ is a point in $Delta(i, j, k)$, $S, S_i, S_j, S_k$ is the area of $Delta(i, j, k), Delta(j, k, P), Delta(k, i, P), Delta(i, j, P)$
we have $$left{begin{align}2S &= begin{vmatrix}1 & x_i & y_i \1 & x_j & y_j \1 & x_k & y_kend{vmatrix}, 2S_i = begin{vmatrix}1 & x & y \1 & x_j & y_j \1 & x_k & y_kend{vmatrix} \2S_j &= begin{vmatrix}1 & x_i & y_i \1 & x & y \1 & x_k & y_kend{vmatrix}, 2S_k = begin{vmatrix}1 & x_i & y_i \1 & x_j & y_j \1 & x & y end{vmatrix} end{align}right.$$
$P$'s barycentric coordinate is $(L_i, L_j, L_k)$
$$left{begin{align}L_i &= dfrac{S_i}{S} = dfrac{1}{2S}[(x_j y_k - x_k y_j) + (y_j - y_k)x + (x_k - x_j)y] \ L_j &= dfrac{S_j}{S} = dfrac{1}{2S}[(x_k y_i - x_i y_k) + (y_k - y_i)x + (x_i - x_k)y] \ L_k &= dfrac{S_k}{S} = dfrac{1}{2S}[(x_i y_j - x_j y_i) + (y_i - y_j)x + (x_j - x_i)y]end{align} right.$$
and
$$left{begin{align}x &= x_i L_i + x_j L_j + x_k L_k \ y &= y_i L_i + y_j L_j + y_k L_kend{align}right.$$
The problem is how to proof
$$begin{equation}iintlimits_{Delta(i, j, k)}L^p_i L^q_j L^r_k dxdy = 2Sdfrac{p!q!r!}{p + q + r + 2}end{equation}$$
I have replace $dxdy$ with $(x_i dL_i + x_j dL_j + x_k dL_k)(y_i dL_i + y_j dL_j + y_k dL_k)$ but still do not know how to continue.
integration barycentric-coordinates
asked Jan 3 at 9:00
BluedropsBluedrops
136
$endgroup$
add a comment |
$begingroup$
$Delta(i, j, k)$ is a triangle with vertex $i(x_i, y_i), j(x_j, y_j), k(x_k, y_k)$ in counterclockwise, $P(x, y)$ is a point in $Delta(i, j, k)$, $S, S_i, S_j, S_k$ is the area of $Delta(i, j, k), Delta(j, k, P), Delta(k, i, P), Delta(i, j, P)$
we have $$left{begin{align}2S &= begin{vmatrix}1 & x_i & y_i \1 & x_j & y_j \1 & x_k & y_kend{vmatrix}, 2S_i = begin{vmatrix}1 & x & y \1 & x_j & y_j \1 & x_k & y_kend{vmatrix} \2S_j &= begin{vmatrix}1 & x_i & y_i \1 & x & y \1 & x_k & y_kend{vmatrix}, 2S_k = begin{vmatrix}1 & x_i & y_i \1 & x_j & y_j \1 & x & y end{vmatrix} end{align}right.$$
$P$'s barycentric coordinate is $(L_i, L_j, L_k)$
$$left{begin{align}L_i &= dfrac{S_i}{S} = dfrac{1}{2S}[(x_j y_k - x_k y_j) + (y_j - y_k)x + (x_k - x_j)y] \ L_j &= dfrac{S_j}{S} = dfrac{1}{2S}[(x_k y_i - x_i y_k) + (y_k - y_i)x + (x_i - x_k)y] \ L_k &= dfrac{S_k}{S} = dfrac{1}{2S}[(x_i y_j - x_j y_i) + (y_i - y_j)x + (x_j - x_i)y]end{align} right.$$
and
$$left{begin{align}x &= x_i L_i + x_j L_j + x_k L_k \ y &= y_i L_i + y_j L_j + y_k L_kend{align}right.$$
The problem is how to proof
$$begin{equation}iintlimits_{Delta(i, j, k)}L^p_i L^q_j L^r_k dxdy = 2Sdfrac{p!q!r!}{p + q + r + 2}end{equation}$$
I have replace $dxdy$ with $(x_i dL_i + x_j dL_j + x_k dL_k)(y_i dL_i + y_j dL_j + y_k dL_k)$ but still do not know how to continue.
integration barycentric-coordinates
asked Jan 3 at 9:00
BluedropsBluedrops
136
$endgroup$
$Delta(i, j, k)$ is a triangle with vertex $i(x_i, y_i), j(x_j, y_j), k(x_k, y_k)$ in counterclockwise, $P(x, y)$ is a point in $Delta(i, j, k)$, $S, S_i, S_j, S_k$ is the area of $Delta(i, j, k), Delta(j, k, P), Delta(k, i, P), Delta(i, j, P)$
we have $$left{begin{align}2S &= begin{vmatrix}1 & x_i & y_i \1 & x_j & y_j \1 & x_k & y_kend{vmatrix}, 2S_i = begin{vmatrix}1 & x & y \1 & x_j & y_j \1 & x_k & y_kend{vmatrix} \2S_j &= begin{vmatrix}1 & x_i & y_i \1 & x & y \1 & x_k & y_kend{vmatrix}, 2S_k = begin{vmatrix}1 & x_i & y_i \1 & x_j & y_j \1 & x & y end{vmatrix} end{align}right.$$
$P$'s barycentric coordinate is $(L_i, L_j, L_k)$
$$left{begin{align}L_i &= dfrac{S_i}{S} = dfrac{1}{2S}[(x_j y_k - x_k y_j) + (y_j - y_k)x + (x_k - x_j)y] \ L_j &= dfrac{S_j}{S} = dfrac{1}{2S}[(x_k y_i - x_i y_k) + (y_k - y_i)x + (x_i - x_k)y] \ L_k &= dfrac{S_k}{S} = dfrac{1}{2S}[(x_i y_j - x_j y_i) + (y_i - y_j)x + (x_j - x_i)y]end{align} right.$$
and
$$left{begin{align}x &= x_i L_i + x_j L_j + x_k L_k \ y &= y_i L_i + y_j L_j + y_k L_kend{align}right.$$
The problem is how to proof
$$begin{equation}iintlimits_{Delta(i, j, k)}L^p_i L^q_j L^r_k dxdy = 2Sdfrac{p!q!r!}{p + q + r + 2}end{equation}$$
I have replace $dxdy$ with $(x_i dL_i + x_j dL_j + x_k dL_k)(y_i dL_i + y_j dL_j + y_k dL_k)$ but still do not know how to continue.
integration barycentric-coordinates
integration barycentric-coordinates
asked Jan 3 at 9:00
BluedropsBluedrops
136
asked Jan 3 at 9:00
BluedropsBluedrops
136
asked Jan 3 at 9:00
BluedropsBluedrops
136
asked Jan 3 at 9:00
BluedropsBluedrops
136
asked Jan 3 at 9:00
BluedropsBluedrops
136
136
add a comment |
add a comment |
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