Working Material

Hypocycloids: Using Vector Functions (I)

Tomas Garza

Author

Tomas Garza

Title

Hypocycloids: Using Vector Functions (I)

Description

Generating various types of hypocycloids using vector parametrization

Hypocycloids: Using Vector Functions (I)

Tomas Garza

Description

In previous notebooks [*] we have shown how elementary vector algebra significantly simplifies the study of curves, such as cycloids and epicycloids. In particular, the parametrization is greatly simplified, and Mathematica resources allows for useful graphic presentations of geometric objects. We now extend the results to the case of hypocycloids.

[*] “Cycloids: Using Vector Functions” (https://notebookarchive.org/2023-02-0z50kpj)
“Epicycloids: Using Vector Functions” (https://notebookarchive.org/2023-02-47osxcf)

We then consider the case where a moving circle rolls along the inside of the circumference of a fixed circle (the moving, or rolling, circle is also called generating circle). In general, the curve that describes a point located at the end of a given radius of the rolling circle is called a hypocycloid. The particular shape of the curve will depend on the ratio of the radii of the fixed and the rolling circles.

Generating the image

The deltoid

As a first example, suppose the radius of the moving circle equals one third of the radius of the fixed circle. Using essentially the same methods as in the previous cases, the curve is obtained, called deltoid (because of its similarity to the Greek letter delta). Once again, we can see that the curve is obtained as the sum of two vector angle-dependent functions (in white and red, in the figure, if the constructive view is used). In the code, look at the definition of greenVector, which clearly displays this fact.

In[]:=

FramedColumnRow[{Spacer[40],Style[Grid[{{"Hypocycloid: the green circle rolls, without slipping,"},{"along the interior of the white circumference, and the radius"},{"of the green one is one-third of the radius of the white one."}}],"Label",14]}],Panel@ManipulateModule{cen={0,0},dist,r,fixedCircle,centersCircle,yellowPoint,rollingCircle,greenAngle,grayAngle,greenVector,greenPoint,whiteArrow,redArrow,greenArrow},dist=

;r=

(1-dist)

dist

(*radiusofrollingcircle*);fixedCircle={White,Circle[cen,1]};centersCircle={Opacity[0.8],Yellow,Circle[cen,1-dist]};yellowPoint={Yellow,PointSize[0.02],Point[(1-dist)cs[ang]]};rollingCircle={Green,Opacity[0.7],Circle[(1-dist)cs[ang],dist]};greenAngle={Opacity[0.5],Green,Disk[cen,dist,{0,ang}]};grayAngle={Opacity[0.7],White,Disk[(1-dist)cs[ang],dist,{0,-rang}]};greenVector=-(dist)cs[π-rang]+(1-dist)cs[ang];greenPoint={Green,PointSize[0.02],Point[greenVector]};whiteArrow={White,Arrowheads[0.03],Arrow[{cen,(-dist)cs[π-rang]}]};redArrow={Red,Arrowheads[0.03],Arrow[{cen,(1-dist)cs[ang]}]};greenArrow={Green,Arrowheads[0.03],Arrow[{cen,greenVector}]};ParametricPlot[-(dist)cs[π-rt]+(1-dist)cs[t],{t,0.01,ang},PlotStyle{Orange,Thickness[0.006]},Epilog->{Inset[Style[Grid[{{"the green point describes a deltoid"},{"(a hypocycloid with three cusps)"}}],White,13],{0,1.4}],Point[cen],greenPoint,yellowPoint,fixedCircle,rollingCircle,Translate[whiteArrow,(1-dist)cs[ang]],If[k>1,{greenAngle,grayAngle,centersCircle,whiteArrow,greenArrow,redArrow,{Red,PointSize[0.02],Point[cen]}}]},Axes->True,TicksNone,Background->Black,PlotRange1.7,ImageSize450],{{k,1,Grid[{{"1 simple view"},{"2 constructive view"}}]},{1,2}},{{ang,0,"ϕ"},0,2π},Initialization{cs[x_]:={Cos[x],Sin[x]}},RoundingRadius8

Out[]=

Hypocycloid: the green circle rolls, without slipping,

along the interior of the white circumference, and the radius

of the green one is one-third of the radius of the white one.

1 simple view

2 constructive view

The astroid

A rolling circle with a radius of one quarter of the radius of the fixed circle is considered below. The curve obtained in this way is called an astroid.

In[]:=

FramedColumnRow[{Spacer[40],Style[Grid[{{"Hypocycloid: the green circle rolls, without slipping,"},{"along the interior of the white circumference, and the radius"},{"of the green one is one-fourth of the radius of the white one."}}],"Label",14]}],Panel@ManipulateModule{cen={0,0},dist,r,fixedCircle,centersCircle,yellowPoint,rollingCircle,greenAngle,grayAngle,greenVector,greenPoint,whiteArrow,redArrow,greenArrow},dist=

;r=

(1-dist)

dist

(*radiusoftherollingcircle*);fixedCircle={White,Circle[cen,1]};centersCircle={Opacity[0.8],Yellow,Circle[cen,1-dist]};yellowPoint={Yellow,PointSize[0.02],Point[(1-dist)cs[ang]]};rollingCircle={Green,Opacity[0.7],Circle[(1-dist)cs[ang],dist]};greenAngle={Opacity[0.5],Green,Disk[cen,dist,{0,ang}]};grayAngle={Opacity[0.7],White,Disk[(1-dist)cs[ang],dist,{0,-rang}]};greenVector=-(dist)cs[π-rang]+(1-dist)cs[ang];greenPoint={Green,PointSize[0.02],Point[greenVector]};whiteArrow={White,Arrowheads[0.03],Arrow[{cen,(-dist)cs[π-rang]}]};redArrow={Red,Arrowheads[0.03],Arrow[{cen,(1-dist)cs[ang]}]};greenArrow={Green,Arrowheads[0.03],Arrow[{cen,greenVector}]};ParametricPlot[-(dist)cs[π-rt]+(1-dist)cs[t],{t,0.01,ang},PlotStyle{Orange,Thickness[0.006]},Epilog->{Inset[Style[Grid[{{"the green point describes an astroid"},{"(a hypocycloid with four cusps)"}}],14,White],{0,1.4}],Point[cen],greenPoint,yellowPoint,fixedCircle,rollingCircle,Translate[whiteArrow,(1-dist)cs[ang]],If[k>1,{greenAngle,grayAngle,centersCircle,whiteArrow,greenArrow,redArrow,{Red,PointSize[0.02],Point[cen]}}]},Axes->True,TicksNone,Background->Black,PlotRange1.7,ImageSize450],{{k,1,Grid[{{"1 simple view"},{"2 constructive view"}}]},{1,2}},{{ang,0,"ϕ"},0,2π},Initialization{cs[x_]:={Cos[x],Sin[x]}},RoundingRadius8

Out[]=

Hypocycloid: the green circle rolls, without slipping,

along the interior of the white circumference, and the radius

of the green one is one-fourth of the radius of the white one.

1 simple view

2 constructive view

Once again, we can see that the curve is obtained as the sum of two vector angle-dependent functions (in white and red, in the figure, if the constructive view is used).

A function for generating hypocycloids with any integral number of cusps

I now define a function to generate hypocycloids like the previous ones, where the radii of the rolling circles are of the form (1/n) , where n is a positive integer. This function is called stella:

In[]:=

Clear[stella];stella[n_?IntegerQ/;n≥3]:=FramedPanel@ManipulateModule{cen={0,0},dist,st,r,fixedCircle,centersCircle,yellowPoint,rollingCircle,greenAngle,grayAngle,greenVector,greenPoint,whiteArrow,redArrow,greenArrow},dist=

;r=

(1-dist)

dist

(*radiusoftherollingcircle*);st=ToString[n];fixedCircle={White,Circle[cen,1]};centersCircle={Opacity[0.8],Yellow,Circle[cen,1-dist]};yellowPoint={Yellow,PointSize[0.02],Point[(1-dist)cs[ang]]};rollingCircle={Green,Opacity[0.7],Circle[(1-dist)cs[ang],dist]};greenAngle={Opacity[0.5],Green,Disk[cen,dist,{0,ang}]};grayAngle={Opacity[0.7],White,Disk[(1-dist)cs[ang],dist,{0,-rang}]};greenVector=-(dist)cs[π-rang]+(1-dist)cs[ang];(*greenVectoristhepositionvectorofthegreenpoint,andgreenPointisthecorrespondingGraphicsprimitive*)greenPoint={Green,PointSize[0.02],Point[greenVector]};whiteArrow={White,Arrowheads[0.03],Arrow[{cen,(-dist)cs[π-rang]}]};redArrow={Red,Arrowheads[0.03],Arrow[{cen,(1-dist)cs[ang]}]};greenArrow={Green,Arrowheads[0.03],Arrow[{cen,greenVector}]};ParametricPlot[-(dist)cs[π-rt]+(1-dist)cs[t],{t,0.01,ang},PlotStyle{Orange,Thickness[0.006]},Epilog->{Inset[Style[Grid[{{"the green point describes a star with "<>st<>" peaks"},{"(i.e., a hypocycloid with "<>st<>" cusps)"}}],14,White],{0,1.4}],Point[cen],greenPoint,yellowPoint,fixedCircle,rollingCircle,Translate[whiteArrow,(1-dist)cs[ang]],If[k>1,{greenAngle,grayAngle,centersCircle,whiteArrow,greenArrow,redArrow,{Red,PointSize[0.02],Point[cen]}}]},Axes->True,TicksNone,Background->Black,PlotRange1.7,ImageSize450],{{k,1,Grid[{{"1 simple view"},{"2 constructive view"}}]},{1,2}},{{ang,0,"ϕ"},0,2π},Initialization{cs[x_]:={Cos[x],Sin[x]},SaveDefinitionsTrue},RoundingRadius->5

Once the previous cell is executed, the function stella may be used with any integer n ≥ 3. The constructive view will not show good detail for n > 10, and the simple view won’t show well for n>50. For example, if we take n=10:

In[]:=

stella[10]

Out[]=

1 simple view

2 constructive view

Cite this as: Tomas Garza, "Hypocycloids: Using Vector Functions (I)" from the Notebook Archive (2023), https://notebookarchive.org/2023-03-622hvfx