In[]:=
GeometricLocusPointPerpendicularSegment::usage="
​References:https://e.math.cornell.edu>people;Personal.";
Development:
I like to visualize things in 3D, then:
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ζ[t_]:={t,
2
t
,0};​​N1[t_]:=
(ζ'[t]ζ''[t])ζ'[t]
Norm[ζ'[t]ζ''[t]]Norm[ζ'[t]]
;​​
Animation:
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​​Manipulateζ[t_]:={t,
2
t
,0};N1[t_]:=
(ζ'[t]ζ''[t])ζ'[t]
Norm[ζ'[t]ζ''[t]]Norm[ζ'[t]]
;​​Show[{ParametricPlot3D[ζ[t],{t,-2,2},PlotStyleBlack]/.Line[pts_,rest___]:>Tube[pts,0.015,rest],ParametricPlot3D[ζ[t]+N1[t],{t,-2,t1},PlotStyle->Blue]/.Line[pts_,rest___]:>Tube[pts,0.015,rest],Graphics3D[{Blue,Ball[ζ[t1]+N1[t1],0.05]}],Graphics3D[{Blue,Point[ζ[t1]],Arrowheads[0],Arrow[Tube[{ζ[t1],ζ[t1]+N1[t1]}]]}]},AxesLabel->(Style[#,15,Blue]&/@{"X","Y","Z"}),AxesOrigin{0,0,0},AxesTrue,BoxedFalse,BoxRatiosAutomatic,PlotRange{{-2,2},{-4,4},{0,1}},ViewPoint{1.3,-2.4,2.},ImageSize900],Style["Geometric locus of a point in a perpendicular line segment of unit length:",Bold,Large],{{t1,-2+0.00002,"Value (θ)"},-2+0.00001,2,0.00001},ControlPlacementTop
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​
Geometric locus of a point in a perpendicular line segment of unit length:
Value (θ)
Parametric equations of the geometric locus:
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ζ[t]+N1[t]
t-
2t
1+4
2
t
,
2
t
+
1
1+4
2
t
,0;