I want to create the following animation with Mathematica.
What is the simplest way to do so?
My first effort:
GenerateLines[a_, b_, r_, n_] := Table[r {Cos[\[Theta]], Sin[\[Theta]], 0}, {\[Theta], a, b, (b - a)/ n}]; basePts = GenerateLines[\[Pi]/6, 2 \[Pi] - \[Pi]/6, 3, 200]; peakPt = {{0, 0, 3}}; baseLine = Line@basePts; slantedLines = Line /@ Tuples[{basePts, peakPt}]; Graphics3D[{baseLine, slantedLines}] 
ClearAll[cone, radius] radius[h_] = r /. First@ Solve[{Pi r Sqrt[r^2 + h^2] (1 + h )/2 == Pi /2, h > 0}, r, Reals]; radius[0] = 1; cone[h_] := ParametricPlot3D[{Cos[θ] (1 - z) radius[h], Sin[θ] (1 - z) radius[h], h z}, {θ, 0, Pi + h Pi}, {z, 0, 1}, PlotStyle -> None, MeshFunctions -> {#4&}, Boxed -> False, PlotRange -> 1.5 {{-1, 1}, {-1, 1}, {0, 1}}, BoundaryStyle -> GrayLevel[.2], PerformanceGoal -> "Quality", Axes -> False]; axes = Graphics3D[{Red, Arrowheads[Medium], MapThread[{Arrow[{{0, 0, 0}, 1.4 #2}], Text[#, 1.5 #2]} &, {{"X", "Y", "Z"}, IdentityMatrix[3]}]}]; Manipulate[Show[cone @ h, axes], {{h, 0, "height"}, 0, 1}] 
Animation above produced using:
frames = Table[Show[cone @ h, axes], {h, 0, 1, .01}]; Export["animatecone.gif", frames, AnimationRepetitions -> ∞, DisplayAllSteps -> True] 
ClearAll[draw]; draw[deg_, segmentsNumber_Integer, z_] := With[{singleSegment = Polygon@{{0, 0, z}, {1, 0, 0}, {Cos[deg Degree], Sin[deg Degree], 0}}}, NestList[ GeometricTransformation[#, RotationTransform[deg Degree, {0, 0, 1}]] &, singleSegment, segmentsNumber]] Manipulate[ Graphics3D[{Red, Arrow[{{0, 0, 0}, {1.5, 0, 0}}], Arrow[{{0, 0, 0}, {0, 1.5, 0}}], Arrow[{{0, 0, 0}, {0, 0, 1.5}}], Text["x", {1.6, 0, 0}], Text["y", {0, 1.6, 0}], Text["z", {0, 0, 1.6}], Transparent, draw[10 + i*8, 19, i]}, Boxed -> False, PlotRange -> {{-2, 2}, {-2, 2}, {-2, 2}}], {i, 0, 1}] For removing axes moves, use PlotRange in Graphics3D:
PlotRange -> {{-2, 2}, {-2, 2}, {-2, 2}} 
Animate[ r = Sqrt[1 - h^2]; phi = Sqrt[2] Pi /r; pts = Table[r {Cos[p], Sin[p], 0}, {p, 0, phi, phi/10}]; ptsc = Table[r {Cos[p], Sin[p], 0}, {p, 0, phi, phi/100}]; Graphics3D[{Line[ptsc], Line[{#, {0, 0, h}}] & /@ pts, Red, Arrow[{{0, 0, 0}, {1, 0, 0}}], Arrow[{{0, 0, 0}, {0, 1, 0}}], Arrow[{{0, 0, 0}, {0, 0, 1}}], Text["X", {1.1, 0, 0}], Text["Y", {0, 1.1, 0}], Text["Z", {0, 0, 1.1}]}, Boxed -> False, PlotRange -> {{-1, 1}, {-1, 1}, {0, 1}}] , {h, 0, 1/Sqrt[2]}, TrackedSymbols -> h] 
Just a start:
Manipulate[ Graphics3D[Cone[{{0, 0, h}, {0, 0, 1}}]], {h, 0, 0.9} ] 
And a bit more...
Manipulate[ Graphics3D[{ {Opacity[0.5], Cone[{{0, 0, h}, {0, 0, 1}}]}, Line[{{1.25, 0, 0}, {0, 0, 0}}], Line[{{0, 1.25, 0}, {0, 0, 0}}], Line[{{0, 0, 1.25}, {0, 0, 0}}] }, Boxed -> False], {h, 0, 0.9} ] 
At first we shrinking the boundary of sector to circle, keeping the arc length. That is l*α==R*θ
α = 0.8*2 π; l = 4; Manipulate[ ParametricPlot3D[R*{Cos[s], Sin[s], 0} /.R->(l*α)/θ, {s, 0, θ}, PlotRange -> 4, PerformanceGoal -> "Quality"], {θ, α, 2 π}] 
Then we draw line from the shrinking curve to the vertex of the cone {0,0,Sqrt[l^2 - R^2]} so we keep the length of generatrix say l.
α = 0.8*2 π; l = 4; Manipulate[ ParametricPlot3D[(1 - t)*(R*{Cos[s], Sin[s], 0}) + t*{0, 0, Sqrt[l^2 - R^2]} /. R -> (l*α)/θ, {s, 0, θ}, {t, 0, 1}, PlotRange -> 4, PerformanceGoal -> "Quality", MeshFunctions -> {#4 &}, Mesh -> 20, Boxed -> False], {θ, α, 2 π, .2}] 
