This is how we draw objects in Mathematica.

Graphics[
{
{Thickness[0.01],Red,Line[Table[ {Cos[s],Sin[s]},{s,0,2 Pi,Pi/32}]]},
{PointSize[0.01],Table[Point[{Cos[s],Sin[s]}],{s,0,2 Pi,Pi/32}]},
{PointSize[0.04],Blue,Point[{Cos[0],Sin[0]}]}

},
PlotRange->{{-2,2},{-2,2}},Frame->True,ImageSize->250]

This is how we set objects in motion.

Manipulate[
Graphics[
{
{Thickness[0.01],Red,Line[Table[ {Cos[s],Sin[s]},{s,0,2 Pi,Pi/32}]]},
{PointSize[0.01],Table[Point[{Cos[s],Sin[s]}],{s,0,2 Pi,Pi/32}]},
{PointSize[0.04],Blue,Point[{Cos[t],Sin[t]}]}

},
PlotRange->{{-2,2},{-2,2}},Frame->True,ImageSize->250],
{t,0,4 Pi}]

Changes formulas produces various curves.

Manipulate[
Graphics[
{
{Thickness[0.01],Red,Line[Table[ {(s/Pi) Cos[s],(s/Pi) Sin[s]},{s,0,2 Pi,Pi/32}]]},
{PointSize[0.01],Table[Point[{(s/Pi) Cos[s],(s/Pi) Sin[s]}],{s,0,2 Pi,Pi/32}]},
{PointSize[0.04],Blue,Point[{(t/Pi) Cos[t],(t/Pi) Sin[t]}]}

},
PlotRange->{{-2,2},{-2,2}},Frame->True,ImageSize->250],
{t,0,2 Pi}]

Find the derivative of the above vector function.

D[{(t/Pi) Cos[t],(t/Pi) Sin[t]},t]

Illustrate the role of the derivative vector.

Manipulate[
Graphics[
{

{Thickness[0.01],Red,Line[Table[ {(s/Pi) Cos[s],(s/Pi) Sin[s]},{s,0,2 Pi,Pi/32}]]},
{PointSize[0.01],Table[Point[{(s/Pi) Cos[s],(s/Pi) Sin[s]}],{s,0,2 Pi,Pi/32}]},
{PointSize[0.04],Blue,Point[{(t/Pi) Cos[t],(t/Pi) Sin[t]}]},

{Thickness[0.01],Arrow[{{0,0}+{(t/Pi) Cos[t],(t/Pi) Sin[t]},{(t/Pi) Cos[t],(t/Pi) Sin[t]}+{((Cos[t])/π)-((t Sin[t])/π),((t Cos[t])/π)+((Sin[t])/π)}}]}

},
PlotRange->{{-2,3},{-2,2}},Frame->True,ImageSize->250],
{t,0,2 Pi}]

What we see in the above plot is that the blue point moves faster and faster.

Next we calculate the arc length of the above spiral. First find the length of the velocity vector, then integrate.

FullSimplify[Norm[{((Cos[t])/π)-((t Sin[t])/π),((t Cos[t])/π)+((Sin[t])/π)}],t>0]

Integrate[((Sqrt[1+(t^2)])/π),t]

Integrate[((Sqrt[1+(t^2)])/π),{t,0,2 Pi}]

N[%]

Next we draw curves in 3 dimensional space.

Manipulate[
Graphics3D[
{
{Thickness[0.01],Red,Line[Table[ {Cos[s],Sin[s],(s/(2Pi))},{s,-4 Pi,4 Pi,Pi/32}]]},
{PointSize[0.01],Table[Point[{Cos[s],Sin[s],(s/(2Pi))}],{s,-4 Pi,4 Pi,Pi/32}]},
{PointSize[0.04],Blue,Point[{Cos[t],Sin[t],(t/(2Pi))}]}

},
PlotRange->{{-2,2},{-2,2},{-2,2}},Boxed->True,Axes->True,ImageSize->250],
{t,-4 Pi,4 Pi}]

Next we use the formula for the unit sphere  x = Cos[θ] Sin[ϕ],  y = Sin[θ] Sin[ϕ],  z = Cos[ϕ]  to place “helix” on the unit sphere. The logic below is that we want the blue point to stay on the sphere as it moves upwards or downwards. Therefore we set θ = t and choose ϕ as a function of  t such that ϕ goes from -π/2 to 0 as t moves from  0 to 4π.  Thus we need

Plot[-(Pi/2)+(t/8),{t,0,4 Pi},PlotRange->{-Pi/2,Pi/2},AspectRatio-> Automatic,Ticks->{Range[0,4 Pi,Pi],{-Pi/2,0,Pi/2}}]

Look what is happening to Cos (Red) and Sin (Blue) as the function of  -(Pi/2)+(t/8)

Plot[{Cos[-(Pi/2)+(t/8)],Sin[-(Pi/2)+(t/8)]},{t,0,4 Pi},PlotStyle->{Red,Blue},PlotRange->{-Pi/2,Pi/2},AspectRatio-> Automatic,Ticks->{Range[0,4 Pi,Pi],{-Pi/2,0,Pi/2}}]

Manipulate[
Graphics3D[
{
{Thickness[0.01],Red,Line[Table[ {Cos[s] Sin[-(Pi/2)+(s/8)],Sin[s]Sin[-(Pi/2)+(s/8)],Cos[-(Pi/2)+(s/8)]},{s,0,4 Pi,Pi/32}]]},
{PointSize[0.01],Table[Point[{Cos[s]Sin[-(Pi/2)+(s/8)],Sin[s]Sin[-(Pi/2)+(s/8)],Cos[-(Pi/2)+(s/8)]}],{s,0,4 Pi,Pi/32}]},
{PointSize[0.04],Blue,Point[{Cos[t]Sin[-(Pi/2)+(t/8)],Sin[t]Sin[-(Pi/2)+(t/8)],Cos[-(Pi/2)+(t/8)]}]}

},
PlotRange->{{-1.2,1.2},{-1.2,1.2},{-1.2,1.2}},Boxed->True,Axes->True,ImageSize->350],
{t,0,4 Pi}]

Now extend the domain for t and look what is happening to Cos (Red) and Sin (Blue) as the function of  -(Pi/2)+(t/8)

Plot[{Cos[-(Pi/2)+(t/8)],Sin[-(Pi/2)+(t/8)]},{t,-4 Pi,4 Pi},PlotStyle->{Red,Blue},PlotRange->{-Pi/2,Pi/2},AspectRatio-> Automatic,Ticks->{Range[-4 Pi,4 Pi,Pi],{-Pi/2,0,Pi/2}}]

Manipulate[
Graphics3D[
{
{Thickness[0.01],Red,Line[Table[ {Cos[s] Sin[-(Pi/2)+(s/8)],Sin[s]Sin[-(Pi/2)+(s/8)],Cos[-(Pi/2)+(s/8)]},{s,-4 Pi,4 Pi,Pi/32}]]},
{PointSize[0.01],Table[Point[{Cos[s]Sin[-(Pi/2)+(s/8)],Sin[s]Sin[-(Pi/2)+(s/8)],Cos[-(Pi/2)+(s/8)]}],{s,-4 Pi,4 Pi,Pi/32}]},
{PointSize[0.04],Blue,Point[{Cos[t]Sin[-(Pi/2)+(t/8)],Sin[t]Sin[-(Pi/2)+(t/8)],Cos[-(Pi/2)+(t/8)]}]}

},
PlotRange->{{-1.2,1.2},{-1.2,1.2},{-1.2,1.2}},Boxed->True,Axes->True,ImageSize->350],
{{t,0},-4 Pi,4 Pi}]

Now increase the number of turns of the spherical helix:

Manipulate[
Graphics3D[
{
{Thickness[0.01],Red,Line[Table[ {Cos[s] Sin[-(Pi/2)+(s/12)],Sin[s]Sin[-(Pi/2)+(s/12)],Cos[-(Pi/2)+(s/12)]},{s,-6 Pi,6 Pi,Pi/32}]]},
{PointSize[0.01],Table[Point[{Cos[s]Sin[-(Pi/2)+(s/12)],Sin[s]Sin[-(Pi/2)+(s/12)],Cos[-(Pi/2)+(s/12)]}],{s,-6 Pi,6 Pi,Pi/32}]},
{PointSize[0.04],Blue,Point[{Cos[t]Sin[-(Pi/2)+(t/12)],Sin[t]Sin[-(Pi/2)+(t/12)],Cos[-(Pi/2)+(t/12)]}]}

},
PlotRange->{{-1.2,1.2},{-1.2,1.2},{-1.2,1.2}},Boxed->True,Axes->True,ImageSize->350],
{{t,0},-6 Pi,6 Pi}]