# Projection of curve 3D (x,y,z) on the x-y plane

I have a following systeme :

$$\left\{ \begin{array}{rcr} (\beta+\frac{1}{2}\delta^{2})\nu_{1}(u)-\delta\nu_{2}(u)+\nu_{3}(u)& = &0\\ (\beta+\frac{1}{2}\delta^{2})\upsilon_{1}(u)-\delta\upsilon_{2}(u)+\upsilon_{3}(u)& < &0\\ \end{array} \right.$$

where

\begin{align*} &\nu_{1}(u)= q(u)f[q(u)],\\ &\nu_{2}(u)= f[q(u)]^2 + F(x)q(u)f[q(u)],\\ &\nu_{3}(u)= \left(u (f[q(u)])^2 + \frac{1}{2}u^2q(u)f[q(u)]\right),\\ &q[u] := Quantile[NormalDistribution[0, 1], u], \ (the \ quantale \ at \ u)\\ &f[q(u)]=PDF[NormalDistribution[0, 1], q(u)] \ (the \ density \ at \ q(u)), \end{align*}

and for the inequality, $\upsilon_{i}(u)=\nu_{i}^\prime(u)$ with respect $q(u)$ for $i\in \{1,2,3\}$ with $(\beta,\delta,u)\in [0.1]\times[0.1]\times[0.1]$.

I would like to look at the projection of this system of equation onto the plane $(\beta,\delta)$ and $(u,\delta)$ in Mathematica or in r.

My code sets up the full 3-d view of the 3D plot, which is not what I want. And I think there are some thing wrong in my picture. I'd like to see two of the views $(\beta,\delta)$ and $(u,\delta)$ on different figure.

q[u_] := Quantile[NormalDistribution[0, 1], u]
f[x_] := PDF[NormalDistribution[0, 1], x]
h1[u_, a_, e_] := ((((a^2)/2 + e)*(-q[u])*(f[q[u]])) -
a*(-q[u]*f[q[u]]*u  +   f[q[u]]^2) +
u*f[q[u]]^2 - (u^2)/2*q[u]*f[q[u]])/(1/6 - a/2 + (a^2)/2 + e)
h2[u_, a_, e_] := (((a^2)/2 + e)*((q[u]^2) - 1)*f[q[u]]) -
a*(u*(q[u]^2 - 1)*f[q[u]] - 3* q[u]*  f[q[u]]^2) + f[q[u]]^3 -
2*u*q[u]*f[q[u]]^2 -
u*q[u]*f[q[u]]^2 + ((u^2)/2)*(q[u]^2 - 1)*f[q[u]]
ContourPlot3D[ h1[u, a, e] == 0,
{a, 0, 1}, {u, 0.1, 0.9}, {e, 0, 1.5},
RegionFunction -> Function[{u, a, e}, h2[u, a, e] > 0]]


The graph 3D :

I want to try do verify if my plot 3D is true and if it is possible to have the projection on both plan $(\beta,\delta)$ and $(u,\delta)$.

Any thoughts on the best way to do this?

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As I understood, projections are solutions of an initial system with one of coordinates is equal to zero. So, (Beta, Delta) view appears as when u=0. And it could be drawn in usual ContourPlot Additionally, you can see projections at your 3D graph just using appropriate ViewPoint –  Rom38 May 19 '14 at 22:34
No the projection is all the image of the curve on the surface (Beta, Delta) , for example if we consider a sphere x^2+y^2+z^2=0 this is the equation of the sphere, and the projection on (y,z) is a circle. I want to do the same here. –  Lea May 19 '14 at 22:51
Aha :) So, ViewPoint helps you to see it –  Rom38 May 19 '14 at 23:09
Your expression contains a bracket mismatch. Could you please check? –  Yves Klett May 20 '14 at 8:48
This takes ages to evaluate so it would be handy if you included a simple example. I think this, this and this are heavily related if not duplicates –  gpap May 20 '14 at 11:09

This may serve as a starting point - extract all Lines from the 3D graphics and get rid of the third component in all coordinate triples (an old but useful trick):

plot = ContourPlot3D[
h1[u, a, e] == 0, {a, 0, 1}, {u, 0.1, 0.9}, {e, 0, 1.5},
RegionFunction -> Function[{u, a, e}, h2[u, a, e] > 0]];

Cases[plot[[1]] // Normal, _Line,
Infinity] /. {x_?NumericQ, y_?NumericQ, z_?NumericQ} :> {x,y} // Graphics


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Damn you and your quick typing! –  gpap May 20 '14 at 11:13
@gpap meet me at noon here :D –  Yves Klett May 20 '14 at 12:14

I guess, the best way to draw projections is select the resulting 3D-graphics and to choose viewpoint when styling menu appeared. For orthogonal projections it should be  ViewPoint -> {0, \[Infinity], 0} ViewPoint -> {\[Infinity], 0, 0} ViewPoint -> {0, 0, \[Infinity]}

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