# How do I establish if a factor is indoor to an elliptical exerciser cone?

Take into consideration an approved elliptical exerciser cone $C$ with its vertex at the beginning, with elevation $h,$ and also with a base offered by : \begin{equation*} \left(\frac{x}{a}\right)^2+\left(\frac{y}{b}\right)^2=1;~z=h \end{equation*} for $a, b$ not equivalent to $0$. Offered a factor $p=(p_x,p_y,p_z)$, just how would certainly you establish if $p$ is indoor to $C$? I know I can first examine to see if $p$ is beyond the elliptical exerciser right cyndrical tube where one base the base of the elliptical exerciser cone and also the various other gets on the $XY$ aircraft, so for smiles, think that check has actually currently been made, and also $p$ remains in reality inside that cyndrical tube. One of the most noticeable remedy I see is to linearly scale $a$ and also $b$ by $\frac{p_z}{h}$ and also see if $p$ is inside the ellipse : \begin{equation*} \left(\frac{h}{p_z}\right)^2\left[\left(\frac{x}{a}\right)^2+\left(\frac{y}{b}\right)^2\right]=1;~z=p_z. \end{equation*} Exist various other strategies? Many thanks.

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2019-05-13 03:26:36
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Write down the formula for the cone: it is $$(x/a)^2 + (y/b)^2 - (z/h)^2 = 0.$$ Currently the inside of the cone will certainly be specified by the inequalities $$(x/a)^2 + (y/b)^2 - (z/h)^2 < 0 \text{ and } 0 < z < h.$$ So offered a factor $(x,y,z)$, thinking that $0 < z < h$ (to make sure that at the very least the $z$ - coordinate works with hing on the inside of the cone), you review $(x/a)^2 + (y/b)^2 - (z/h)^2$ and also take into consideration whether it is adverse.
NB: I am creating $(x,y,z)$ as opposed to $(p_x,p_y,p_z)$ simply to make the symbols less complex.