How to plot dual cone of a given conical domain?

Question

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?

Ulrich Neumann · Answer

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...

tommsch · Answer

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}]

How to plot dual cone of a given conical domain?

2 Answers

Add your own answers!

Ask a Question