OrthogonallyProjectedParabola::usage="
​References:Belk,Jim.e.math.cornell.edu,DifferentialGeometry.CornellUniversity:Homework5(ParametricCurvesinThreeDimensions),Prob.-3;Personal.";
Development:
Normal line to the plane x + y + z = 1 which passes through the points of parabola on the plane XY:
ℒ
N
:{t,
2
t
,0}+v{1,1,1};
Intersection of the straight line
ℒ
N
with the plane x + y + z = 1:
t+v+
2
t
+v+v==1;​​v=
1
3
(1-t-
2
t
);​​
1
3
{2t-
2
t
+1,2
2
t
-t+1,1-t-
2
t
};
In[]:=
ManipulateClearAll;c[t_]:={t,
2
t
,0};​​ξ[t_]:=
1
3
{2t-
2
t
+1,2
2
t
-t+1,1-t-
2
t
};​​Show[{ParametricPlot3D[c[t],{t,-3,t1},PlotStyleBlue]/.Line[pts_,rest___]:>Tube[pts,0.08,rest],ParametricPlot3D[ξ[t],{t,-3,t1},PlotStyleRed]/.Line[pts_,rest___]:>Tube[pts,0.08,rest],Graphics3D[{Red,Thick,Ball[ξ[t1],0.15]}],​​Graphics3D[{Blue,Thick,Ball[c[t1],0.15]}],ContourPlot3D[z==0,{x,-10,10},{y,-10,10},{z,-10,10},ContourStyleDirective[Yellow,Opacity[0.5],Specularity[White,30]],MeshFalse,PerformanceGoal"Quality"],​​ContourPlot3D[x+y+z==1,{x,-10,10},{y,-10,10},{z,-5,5},ContourStyleDirective[Gray,Opacity[0.2],Specularity[White,30]],MeshFalse,PerformanceGoal"Quality"],Graphics3D[{Black,Dashed,Thick,Line[{c[t1],ξ[t1]}]}]​​},AxesLabel->(Style[#,15,Blue]&/@{"X","Y","Z"}),AxesOrigin{0,0,0},AxesTrue,BoxedFalse,BoxRatiosAutomatic,PlotRange8,ImageSize900,ViewPoint{6,-2.4,2}],Style["Orthogonally projected parabola on the plane x + y + z = 1:",Bold,Large],{{t1,-2.9999,"Value (t1)"},-3+0.00001,3,0.00001},ControlPlacementTop
Out[]=
​
Orthogonally projected parabola on the plane x + y + z = 1:
Value (t1)
Parabola in the XY plane in blue, parabola projected on the plane x + y + z = 1 in red.
The XY plane in yellow and the plane x + y + z = 1 in gray.