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Published: 09/09/2022 (4 weeks ago)
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support phone chat time recording video recording of message audiuonimnarubo · Mr. Wong’s review of Tantalize 5 is outstanding.Q:

Finding the closest point on the gradient

Given the 2d-transformed point $T(x,y)$, I’m trying to find the point $(x_0,y_0)$ on the gradient $
abla(w)$ that is the closest to the transformed point (or vice-versa) using iterations. What I know from the gradient
$$
abla(w) = frac{partial w}{partial x}frac{partial x}{partial x_0} + frac{partial w}{partial y}frac{partial y}{partial x_0} + frac{partial w}{partial x}frac{partial x}{partial y_0} + frac{partial w}{partial y}frac{partial y}{partial y_0}$$
Is that if we perform this transformation $T(x_0,y_0)$ where $(x_0,y_0)$ is on the gradient, then we can use any linear mapping and the result will be the closest point on the gradient $(x_0,y_0)$. So, for example, if $T(x,y) = [x_0 + ax, y_0+by]$, I can choose
$$
begin{cases}
x_0 = 0 \
x = x_0 \
a = frac{1}{x_0} \
a=0
end{cases}
$$
and get $(x,y) = (0,b)$ as the closest point on the gradient. However, this would be tedious to apply to every other $T(x,y)$ I may be applying. Is there a better way to find $(x_0,y_0)$ that is on the gradient $
abla(w)$ and that also works for any arbitrary $T(x,y)$?
Thanks in advance.

A:

I presume the gradient in question is the one for the problem $u=f(T(x,y))$ given by $f(t)=frac{partial u}{partial x} frac{partial T}{

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