Mathematica Asked on September 26, 2021
If $K$ is a given cone in $mathbb{R}^3$ which passes through origin, the dual cone $K^*$ is defined as $K^*={mathbf{x}inmathbb{R}^3 big| ,, mathbf{x}cdotmathbf{y}ge 0 ,, forall yin K}$.
Given a cone, e.g., $K={(x,y,z)inmathbb{R}^3,,big|,, xy-z^2ge 0}$, how do I plot its dual cone $K^*$, using RegionPlot3d or ContourPlot3d?
The cone K can be plotted as
RegionPlot3D[x y - z^2 >= 0, {x, -1, 1}, {y, -1 , 1}, {z, -1, 1},MaxRecursion -> 5, AxesLabel -> {x, y, z} ]
The dual cone might be plotted in the same we using the additional conditions...
Answered by Ulrich Neumann on September 26, 2021
You are looking for ForAll
.
As pointed out by Henrik Schumacher, for your particular example $K^*={0}$ since $K=-K$. Thus, I will show how to plot the dual cone one the freely chosen set
$xgeq0$, $ygeq0$ and $x+yleq5$.
The function ForAll
when given three parameters,
ForAll[{x1,...xn},cond,expr]
states that expr
is true for all xi
satisfying the condition cond
.
Resolve
or Reduce
then solves such systems. Thus, a MWE looks like
cone = x>=0 && y>=0 && x+y<=5
dual = Resolve[ForAll[{x, y}, cone, px x + py y <= 0], {px, py}]
RegionPlot[cone, {x, -10, 10}, {y, -10, 10}]
RegionPlot[dual, {px, -10, 10}, {py, -10, 10}]
Answered by tommsch on September 26, 2021
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