ParametricEquationsRotatedParabola::usage="
​​​References:e.math.cornell.edu,DifferentialGeometry,JimBelk,CornellUniversity,Exam2PracticeProblems,ParameterizingSpaceCurves,Prob.-5;Personal.";
Development:
Whenrotating30ºthevector{1,0,0}nowit's
3
2
,0,-
1
2
,sowithrespecttothestraightlinex=1andsincetheCos[30º]=
3
2
,thecoordinatexofthevertex​​itisnow1-
3
2
,andsincetheSin[30º]=
1
2
,thecoordinatezitisnow
1
2
.
Parametric equations of the parabola:
1-
3
2
,0,
1
2
+t
3
2
,0,-
1
2
+
2
t
{0,1,0};
In[]:=
ManipulateClearAll;​​ξ[t_]:=1-
3
2
,0,
1
2
+t
3
2
,0,-
1
2
+
2
t
{0,1,0};​​ShowParametricPlot3D[ξ[t],{t,-2,t1},PlotStylePurple]/.Line[pts_,rest___]:>Tube[pts,0.04,rest],ParametricPlot3D[{1,0,0}+t{0,1,0},{t,-5,5},PlotStyleGreen]/.Line[pts_,rest___]:>Tube[pts,0.04,rest],ParametricPlot3D[{t,
2
t
,0},{t,-2,2},PlotStyleOrange]/.Line[pts_,rest___]:>Tube[pts,0.04,rest],Graphics3DRed,Thick,Ball1-
3
2
,0,
1
2
,0.09,Graphics3D[{Black,Thick,Ball[ξ[t1],0.09]}],Graphics3D[{Black,Thick,Ball[{0,0,0},0.09]}],​​ContourPlot3D[z==0,{x,-5,5},{y,-5,5},{z,-5,5},ContourStyleDirective[Gray,Opacity[0.2],Specularity[White,30]],MeshFalse,PerformanceGoal"Quality"],ParametricPlot3D{1,0,0}+t
3
2
,0,-
1
2
,{t,-4,3},PlotStyle->{Black,Dashed}​​,AxesLabel->(Style[#,15,Blue]&/@{"X","Y","Z"}),AxesOrigin{0,0,0},AxesTrue,BoxedFalse,BoxRatiosAutomatic,PlotRange5,ImageSize900,ViewPoint{1.3,-3,1},Style["Rotated parabola 30º with respect to the straight line x = 1:",Bold,Large],{{t1,-1.9999,"Value (t1)"},-2+0.00001,2,0.00001},ControlPlacementTop
Out[]=
​
Rotated parabola 30º with respect to the straight line x = 1:
Value (t1)
Parabola on the XY plane in orange.The parabola turned 30º with respect to the straight line x = 1 (green colored), it appears in purple. The new vertex appears in red. The X-axis has rotated 30º (dotted line ).