Cycloids: Using Vector Functions

Tomas Garza

Description

There are a variety of geometric curves related to the circle. These curves arise from examining the behavior of a given point of a circle when it moves, for example, along a line, or along the circumference of another circle, either from its internal or external part. In simple cases, they are described by functions, but the conditions that define them often lead to more complicated configurations that are difficult to specify analytically. The use of parametric equations is essential for studying them, although direct parametrization might be relatively cumbersome, especially if one does not have a good command of trigonometry.

The introduction of elementary vector algebra significantly simplifies the problem. In this notebook a well-known curve is studied, where parametrization is obtained easily through the use of vectors. Mathematica resources are also used for geometric transformations and the graphic representation of geometric objects. The code used in solving the problem is given, allowing Mathematica to be seen in action and illustrating how to use many of its functions in this topic.

The cycloid

The cycloid is a simple example of the type of curves I refer to. Consider a circle of radius 1 that rolls along a fixed tangent line. Somewhere on a given radius of that circle a point is placed, which describes a curve when the circle moves. This curve is called a cycloid. The initial radius returns to its original position when the circle has traveled a distance 2π, that is, the length of its circumference.

The following figure shows the situation: the green dot may be on the circumference, at the end of the radius which is initially in vertical position, touching the straight line. But it may also be in a different position on that radius, either inside or outside the circle. This can be set by the user by choosing any of these three positions: two-thirds of the length of the radius; on the circunference, or 3/2 of the length of the radius.

Once the position of the green dot is chosen, the circle is made to roll with the slider labeled as t. You can then see, in orange, the curve described by the green point. The position of the green dot can be changed at any time, so as to present different types of cycloid.

The idea here is to isolate each of the moving elements (in this problem, the motion of the point on the circunference, and the motion of the center of the rolling circle), and assign a vector (dependent on t, i.e., a vector function) to each, so that the combined effect is obtained as the sum of the two vectors, exhibiting the powerful concepts and techniques of vector algebra. The parametric equations are obtained in a straightforward way, as can be seen in the code.

With this in mind, an interesting feature in this code is the constructive view. The angle on the circunference described by the green point is shown as the white vector, and the position of the centre of the moving circle as it rolls is shown as the red vector. The position of the green point appears clearly as the sum of the white and red vectors, as a green vector. The cycloidal path appears in orange. The ordinary view and the constructive view are switched with a button control.

In[]:=

FramedPanel@ColumnRow[{Spacer[15],Style["Cycloid: The green circle rolls without slipping along the white line","Label",15]}],ManipulateModule{cen={0,0},angGreen,vGreen,pointGreen,vWhite,vRed,radius,arrowWhite,arrowRed,arrowGreen,lineBase,circleFixed,lineCenter,movingCircle,marca2π,marca4π},angGreen={Opacity[0.8],Green,Disk[cen,1,{3π/2,3π/2-If[t<2π,t,t-2π]}]};vGreen=hcs[3π/2-t];(*seetheinitializationsectionatthebottomofthecodeforthedefinitionofcs*)pointGreen={Green,PointSize[0.015],Point[vGreen]};vWhite=hcs[3π/2-t];vRed={t,0};(*invectorterms:vGreenequalsvWhite+vRed;*)radius={Green,Line[{vWhite,vWhite+vRed}]};arrowWhite={White,Arrowheads[0.02],Arrow[{cen,cs[3π/2-t]}]};arrowRed={Red,Arrowheads[0.02],Arrow[{cen,{t,0}}]};arrowGreen={Green,Arrowheads[0.02],Arrow[{cen,vWhite+vRed}]};lineBase={White,Line[{{0,-1},{4π+0.5,-1}}]};circleFixed={White,Circle[cen,1]};lineCenter={Opacity[0.1],Line[{cen,{2π,0}}]};movingCircle={Green,Opacity[0.7],Circle[{t,0},1]};marca2π={Red,Line[{{2π,-0.9},{2π,-1.1}}]};marca4π={Red,Line[{{4π,-0.9},{4π,-1.1}}]};ParametricPlot(*{x-hSin[x],-hCos[x]}*)hcs[3π/2-x]+{x,0},{x,0.1,t},PlotStyle{Orange,Thickness[0.006]},Prolog->Ift>π16,InsetStyleWhichh

,"the green point describes a curtate cycloid",h1,"the green point describes an ordinary cycloid",True,"the green point describes a prolate cycloid",14,White,{2π,1.9},marca2π,marca4π,radius,{White,PointSize[0.015],Point[{t,0}]},lineBase,movingCircle,circleFixed,Style[Text["2π",{2π,-1},{0,2}],Red],Style[Text["4π",{4π,-1},{0,2}],Red],{If[k>1,{{Red,PointSize[0.015],Point[cen]},circleFixed,movingCircle,arrowGreen,angGreen,arrowWhite,arrowRed,lineCenter}],GeometricTransformation[pointGreen,gt]},AxesNone,PlotRange{{-1.1,4π+1.1},{-1.95,2.5}},BackgroundBlack,ImageSize500,{{k,1,Grid[{{"1 simple view"},{"2 constructive view"}}]},{1,2}},{{t,0},0,4π},{h,1,Grid[{{"position of the green"},{"point on the radius"}}]},

,1,

,Initialization{cs[x_]:={Cos[x],Sin[x]},gt:=TranslationTransform[{t,0}]},SaveDefinitions->True,RoundingRadius13

Out[]=

Cycloid: The green circle rolls without slipping along the white line

1 simple view

2 constructive view

position of the green

point on the radius

Notice that the parametric equations in the ParametricPlot are written as the sum of the two vectors, white and red. The conventional form of the equations appear inside a comment (* *); of course, the two forms are equivalent.

Cite this as: Tomas Garza, "Cycloids: Using Vector Functions" from the Notebook Archive (2023), https://notebookarchive.org/2023-02-0z50kpj