Computational methods in Mathematics

Transcription

1 Computational methods in Mathematics José Carlos Díaz Ramos Cristina Vidal Castiñeira June 13, Graphics Mathematica represents all graphics in terms of a collection of graphics primitives. The primitives are objects like Point, Line and Polygon, that represent elements of a graphical image, as well as directives such as RGBColor or Thickness. The appearance of these can be modified using options. Each complete piece of graphics in Mathematica is represented as a graphics object. There are several different kinds of graphics objects, corresponding to different types of graphics. Each kind of graphics object has a definite head which identifies its type. We will essentially use the following two: Graphics[list ] Graphics3D[list ] general two-dimensional graphics general three-dimensional graphics The most basic way of producing a graphic is writing a command like: graphic type[{{directives 1, primitives 1 }, {directives 2, primitives 2 },...}, options]. Here we have a list of the some basic primitives: Point[{x, y}] point at position {x, y} Line[{{x 1, y 1 }, {x 2, y 2 }, }] line through the points {x 1, y 1 }, {x 2, y 2 }, Polygon[{{x 1, y 1 }, {x 2, y 2 }, }] filled polygon with the specified list of corners Circle[{x, y}, r] circle with radius r centered at {x, y} Tube[{{x 1, y 1, z 1 },{x 2, y 2, z 2 },... }, r] Sphere[{x, y, z}, r] Some of the basic directives are: represents a tube of radius r represents a sphere of radius r centred at {x, y, z} RGBColor[r,g,b] color with specified red, green and blue components, each between 0 and 1 Hue[h] color with hue h between 0 and 1 PointSize[d] give all points a diameter d as a fraction of the width of the whole plot Thickness[w] give all lines a thickness w as a fraction of the width of the whole plot 1

2 In the latest versions of Mathematica one can use named colours (like Red, Blue, Green, etc.), sizes (like Large, Small, Tiny), or thickness (like Thin or Thick). After you have produced a graphics object of some kind, the next thing you have to do is to display it. This is accomplished by Show. Show[g 1, g 2,..., opt 1 val 1, opt 2 val 2,... ] display several graphics objects combined with certain options Here are a few examples of two dimensional graphics. In[1]:= pt = Table[Point[{Cos[ 2πn 2πn 4 ], Sin[ 4 ]}], {n, 0, 4}]; lin = Line[Table[{Cos[ 2πn 2πn 6 ], Sin[ 6 ]}, {n, 0, 6}]]; pol = Polygon[Table[{Cos[ 2πn 2πn 3 ], Sin[ 3 ]}, {n, 0, 3}]]; cir = Circle[{0, 0}, 1]; This produces the corresponding graphics object and displays it. In[2]:= Graphics[ {{PointSize[Large], Red, pt}, {Thick, Green, lin}, {Blue, pol}, cir}] Out[2]= To produce 3-dimensional graphics, the procedure is analogous. One has to replace Graphics by Graphics3D and use three coordinates instead of two. You should keep in mind that all the graphics produced with Mathematica follow this approach regardless of whether you see it or not. For example, this shows you how a two dimensional plot is saved internally by Mathematica. We skip the long output. In[3]:= InputForm[Plot[Sin[x],{x, 0, 2π}]] 1.1 2D graphics We start with commands for plotting two dimensional data. All the commands listed here will produce graphic objects of type Graphics. These are a few options that any of the functions described below will admit (as well as Show, when you combine two dimensional graphics). We will demonstrate them in what follows. 2

3 option name default value AspectRatio 1/GoldenRatio the height-to-width ratio for the plot; Automatic sets it from the absolute x and y coordinates Axes Automatic whether to include axes AxesLabel None labels to be put on the axes; ylabel specifies a label for the y axis, {xlabel, ylabel } for both axes AxesOrigin Automatic the point at which axes cross PlotLabel None an expression to be printed as a label for the plot PlotRange Automatic the range of coordinates to include in the plot; All includes all points; Ticks Automatic which tick marks to draw if there are axes; None gives no tick marks 1.2 Plot This commands plots a function f : [a, b] R. The syntax is Plot[f, {x, a, b}], where f is an expression depending on x that evaluates to a real number for all x [a, b]. To plot several functions at a time in the same interval use Plot[{f 1, f 2, }, {x, a, b}]. You can provide options that change the appearance of the graphic. The following table gives a few options available for Plot which are not present for Graphics objects in general. option name default value PlotStyle Automatic a list of lists of graphics primitives to use for each curve (as explained in the previous section) PlotPoints Automatic the minimum number of points at which to sample the function An example: In[4]:= Plot[{Tan[x], Sin[x], Cos[3x]}, {x,0,2 Pi}, PlotStyle {Red, Green, Blue}] Out[4]= The function Plot, like other Mathematica functions such as Table, evaluates the first of its arguments for each of the different nodes of the interval where the function is to be plotted. If the first argument is an expression that 3

4 produces a list of graphics, it is important to force the evaluation of this argument before plotting the function to avoid this evaluation at each iteration of Plot. In this case we use the syntax Plot[Evaluate[f],{x, x min, x max }]. This makes a plot of the functions n cos x with n running from 1 to 6. The Evaluate command tells Mathematica first to make the table of functions, and only then to evaluate them for particular values of x. In[5]:= Plot[Evaluate[Table[n Cos[x], {n, 6}]],{x, 0, 10}] Out[5]= 1.3 3D graphics In this section we deal with 3-dimensional graphics which are essentially objects of type Graphics3D. Most of the comments given for 2-dimensional graphics go through now. We will present the basic plotting functions and forget about the more complicated low level commands. option name default value AspectRatio Automatic the height-to-width ratio for the plot Axes False whether to include axes AxesLabel None labels to be put on the axes; ylabel specifies a label for the y axis, {xlabel, ylabel } for both axes Boxed True whether to include a box for the graphic Lighting True whether to use ambient light to colour the graphic PlotLabel None an expression to be printed as a label for the plot PlotRange Automatic the range of coordinates to include in the plot; All includes all points; SphericalRegion False whether the final image should be scaled so that a sphere drawn around the threedimensional bounding box would fit in the display area specified ViewPoint {1.3,-2.4,2.} point in space from which the objects plotted are to be viewed This is an example of how to use graphics primitives and directives for 3- dimensional graphics. 4

5 In[6]:= Graphics3D[Green, Sphere[2,3,5], Red, Sphere[3,3,5,1/3], Brown, Sphere[2,2,5.5,1/4], Sphere[2,1.75,5.5,1/4], Sphere[2.3,1.75,5.5,1/4], Sphere[2,2,4.5,1/4], Sphere[2,1.75,4.5,1/4], Sphere[2.3,1.75,4.5,1/4], Blue, Sphere[3,3.5,5.5,1/4], Sphere[3,3.5,4.5,1/4], Boxed False] Out[6]= 1.4 ParametricPlot3D We use ParametricPlot3D for 3-dimensional parametric plots. Note that the syntax ParametricPlot3D[{f x, f y, f z }, {t, t min, t max }] is the direct analog in three dimensions of ParametricPlot[{f x, f y }, {t, t min, t max }] in two dimensions, which we have not discussed. This makes a parametric plot of a helical curve. In[7]:= ParametricPlot3D[{Cos[t],Sin[t],t/3}, {t,0,4π}] Out[7]= The function ParametricPlot3D can also be used to plot surfaces. It is your duty to make sure that the parametric plot you produce does not have self intersections or any strange behaviour. The syntax to plot a parametrised surface is ParametricPlot3D[{f x, f y, f z }, {u, u min, u max }, {v, v min, v max }]. In[8]:= ParametricPlot3D[{x, y, x 3-3 x y 2 }, {x,-1,1}, {y,-1,1}] 5

6 Out[8]= There are other possibilities to represent surfaces instead of ParametricPlot3D such as Plot3D, ContourPlot3D and others. 1.5 Manipulate and DynamicModule Sometimes it is useful to represent a graphic that depends on a parameter and dynamically manipulate this parameter. This is achieved in Mathematica using the function Manipulate. Manipulate[expr,{u, u min, u max }] generates a version of expr with controls added to allow interactive manipulation of the value of u. The following input allows us to manipulate a plot of two functions that depends on a parameter called a. In[9]:= Manipulate[Plot[{(5 a)sin[x], a Cos[x]},{x, 0, 10}], {a, 0, 5}] Out[9]= If the plot we want to manipulate is obtained as a procedure, as we will see later, then it is convenient to localize the variables used in this procedure. For dynamic manipulation purposes this is achieved using DynamicModule instead of Module. The syntax of these two functions is identical: DynamicModule[{x, y, },expr]. The symbols specified in a DynamicModule will by default have their values maintained even across Mathematica sessions. One can also specify initial values for x, y,..., as DynamicModule[{x = x 0, y = y 0, },expr] 6

7 2 Curves In this section, we start using Mathematica for understanding several concepts of differential geometry. We start plotting curves of R 3. Latter, we will also investigate some other geometric properties such as curvature. We illustrate some of the basic geometric concepts associated with curves. We use some of the programming techniques described in previous sections. Many of the concepts defined here are well known and there is a vast bibliography to explore curves using software like Mathematica. We do not aim to be thorough: we just want to show a few examples. These will give us a clue of how to proceed in more general situations. In[1]:= helix[t ] = {Cos[t], Sin[t], t/3}; In[2]:= ParametricPlot3D[helix[t], {t, 0, 4π}] Out[2]= The basic geometric objects associated with regular curves in R 3 are curvature, torsion and the vectors of the Frenet-Serret frame: the tangent vector, the normal vector and the binormal vector. We quickly remind the basic formulas and definitions. Let α : t I α(t) R 3 be a regular curve, that is, α (t) 0 for all t. Such a regular curve can be parametrised by arc length s(t) = t t 0 α (t) dt. With respect to the arc length parameter (whose derivatives are denoted by a dot), we define the tangent vector as T (s) = α(s), the curvature as κ(s) = T (s), the normal vector as N(s) = T (s)/κ(s), the binormal vector as B(s) = T (s) N(s), and the torsion as τ(s) = Ḃ(s), N(s) whenever α(s) 0. It is clear that the arc length parameter always exists but it could be very difficult (or even impossible) to calculate. Hence, for practical purposes, it is sometimes better to work with the parameter t. Thus, one has to adapt the above formulas by doing the right change of variable: T (t) = α (t) α (t), N(t) = T (t) κ(t) α (t), B(t) = T (t) N(t), κ(t) = α (t) α (t) α (t) 3, τ(t) = det(α (t), α (t), α (t)) α (t) α (t) 2. 7

8 The following commands implement the above formulas. In[3]:= frenett[α ][t ] := α [t] α [t] α [t] ; frenett[α] In[4]:= frenetn[α ][t ] := [t] ; curvature[α][t] α [t] α [t] In[5]:= frenetb[α ][t ] := frenett[α][t] frenetn[α][t] In[6]:= curvature[α ][t] := In[7]:= torsion[α ][t ] := (α [t] α [t]) (α [t] α [t]) (α [t] α [t]) 3/2 Det[{α [t],α [t],α [t]}] (α [t] α [t]) (α [t] α [t]) As an example, we calculate these geometric objects for the helix above. Note the use of Simplify. In[8]:= Simplify[{curvature[helix][t], torsion[helix][t]}] Out[8]= { 9 10, 3 10 } In[9]:= Simplify[frenetT[helix][t]] Out[9]= { 3Sin[t] 10, 3Cos[t] 1 10, 10 } In[10]:= Simplify[frenetN[helix][t]] Out[10]= {-Cos[t], -Sin[t],0} In[11]:= Simplify[frenetB[helix][t]] Out[11]= { Sin[t] 10, Cos[t] 3 10, 10 } Sometimes, a plot helps more than the exact formulas we get using the symbolic capability of Mathematica. For example, we write here a function to plot the Frenet-Serret frame at a point of a curve. This uses the formulas for the Frenet-Serret frames above. We use the commands Arrow and Tube to plot the corresponding vectors of the Frenet-Serret frame. Once we have a vector, we wrap it using Graphics3D to produce the actual graphic. We plot normal and binormal vector modifying their lenght by multiplication by curvature and torsion. Finally, we plot the osculating circle. Later, we will use Show to render the result on screen. In[12]:= disk3d[center, r, t1, t2 ]:=Module[{s, t}, ParametricPlot3D[Evaluate[center + t Cos[s] t1 + t Sin[s] t2], {s, 0, 2π},{t, 0, r}, Mesh False, PerformanceGoal "Quality"]] In[13]:= plotosculating[curve ][t ] := Module[ {c = curve[t], v1 = frenett[curve][t], v2 = frenetn[curve][t], v3 = frenetb[curve][t], curv = curvature[curve][t]}, Show[ 8

9 Graphics3D[ {{Blue, Arrow[Tube[{c, c+v1}]], Arrow[Tube[{c, c+v2}]], Arrow[Tube[{c, c + v3}]]}, {Red, Arrow[Tube[{c, c + curv v2},0.05]]}, {Green, Arrow[Tube[{c, c+torsion[curve][t] v3}, 0.05]]} {Yellow, Line[{c, c + 1/curv v2}]}}], disk3d[c + v2/curv, 1/curv, v1, v2]]]; In[14]:= curveobjects[curve, {t0, t1 }] := Module[{s}, DynamicModule[{t}, Manipulate[ Show[ ParametricPlot3D[curve[s], {s, t0, t1}], plotosculating[curve][t], Boxed False, Axes False, PlotRangePadding 1, SphericalRegion True], {t, t0, t1}, ControlPlacement Top]]]; Here is an example with the helix. In[15]:= curveobjects[helix, {0, 4 π}] Out[15]= 3 Surfaces We will consider parametrised surfaces and will not be interested in singularities or self-intersections. It will be the task of the user to check whether the parametrisation produces an embedded surface or not (that is, one has to check whether there are self intersections, singular points and so on). We will illustrate some of the geometric objects that can be constructed on a surface and give the code to calculate them. In order to try our code we start with a well-known example. Here it is the definition of a torus. 9

10 In[1]:= torus[r, r ][u, v ]= {(R + rcos[v])cos[u], (R + rcos[v])sin[u], rsin[v]}; The first obvious thing we can do with a surface is to plot it. In[2]:= ParametricPlot3D[torus[2, 1][u, v], {u, 0, 2π}, {v, 0, 2π}] Out[2]= The most basic object of a surface is the metric or first fundamental form. As we are given a surface by means of a parametrization, the reasonable thing to do is to calculate the metric with respect to the basis of tangent vectors. The code is straightforward. The only interesting point here is the way the derivatives are calculated: Derivate[i 1, i 2, ][f][x 1, x 2, ] represents i 1 +i 2 + f. For x i 1 1 x i 2 2 example 3 f x 2 y (x, y) is represented by Derivate[2, 1][f][x, y]. In[3]:= CoordinateVector[1][sup ][u, v ]:=Derivative[1,0][sup][u, v]; CoordinateVector[2][sup ][u, v ]:=Derivative[0,1][sup][u, v]; In[4]:= Metric[sup ][u, v ]:=Module[ {dx = {CoordinateVector[1][sup][u, v], CoordinateVector[2][sup][u, v]}}, {{dx[[1]] dx[[1]], dx[[1]] dx[[2]]}, {dx[[2]] dx [[1]], dx[[2]] dx[[2]]}}]; For example, this is the metric of the torus In[5]:= MatrixForm[Simplify[Metric[torus[R, r]][u, v]]] Out[5]//MatrixForm= ( r (R + rcos[u]) 2 ) Most of the geometry of a surface in R 3 is encoded in the Gauss map. This is defined by x 3 = (x 1 x 2 )/ x 1 x 2 where x i = x/ u i and x(u 1, u 2 ) is the parametrization. In[6]:= GaussMap[surf ace ][u, v ]:= Module[{x3 = Cross[CoordinateVector[1][surf ace][u, v], 10

11 CoordinateVector[2][surf ace][u, v]]}, x3/sqrt[x3 x3]]; This calculates the Gauss map of the torus. In this example we use the function PowerExpand to cancel square roots and squares. Mathematica assumes that variables are complex numbers, so this simplification cannot be performed by default. In[7]:= PowerExpand[Simplify[GaussMap[torus[R, r]][u, v]]] Out[7]= {-Cos[u] Cos[v], -Cos[u] Sin[v], -Sin[u]} The differential of the Gauss map contains very important geometric information of a surface. It essentially encodes the second fundamental form. Its trace is the mean curvature and its determinant the Gaussian curvature. In[8]:= GaussianCurvature[surf ace ][u, v ] := Module[ {x11 = Derivative[2, 0][surf ace][u, v], x12 = Derivative[1, 1][surf ace][u, v], x22 = Derivative[0, 2][surf ace][u, v], x3 = GaussMap[surf ace][u, v]}, Det[{{x11 x3, x12 x3}, {x12 x3, x22 x3}}]/det[metric[surf ace][u, v]]]; In[9]:= MeanCurvature[surf ace ][u, v ] := Module[ {x11 = Derivative[2, 0][surf ace][u, v], x12 = Derivative[1, 1][surf ace][u, v], x22 = Derivative[0, 2][surf ace][u, v], x3 = GaussMap[surf ace][u, v], g = Metric[surface][u, v]}, (x11 x3 g[[2, 2]] 2x12 x3 g[[1, 2]]+x22 x3 g[[1, 1]])/Det[g]/2]; For example, these are the Gaussian and mean curvature of the torus. In[10]:= Simplify[GaussianCurvature[torus[R, r]][u, v]] Out[10]= Cos[u] rr+r 2 Cos[u] In[11]:= PowerExpand[Simplify[MeanCurvature[torus[R, r]][u, v]]] Out[11]= R+2rCos[u] 2r(R+rCos[u]) The Christoffel symbols are used later for the geodesic equation. Here, they are programmed in a more sophisticated way. In[12]:= ChristoffelSymbols[surf ace ][u, v ] :=Module[{g, dg}, g = Evaluate[Metric[surface][#1, #2]] &; dg = {Derivative[1, 0][g][u, v],derivative[0,1][g][u, v]}; ((Transpose[dg, {2, 1, 3}]+dg Transpose[dg, {3, 2, 1}]) Inverse[g[u, v]])/2]; In[13]:= Simplify[ChristoffelSymbols[torus[R, r]][u, v]] 11

12 Out[13]= {{{0,0}, {0, rsin[u] rsin[u] (R+rCos[u])Sin[u] R+rCos[u] }}, {{0, R+rCos[u] }, { r,0}}} The way the geodesic equation is written here is even more complicated. The purpose of this code is to introduce you to functional programming. It is very easy to modify this code so that it works for objects of arbitrary dimensions. Check the Mathematica help for further details. In[14]:= GeodesicEquation[surf ace ][u ][t ]:= Module[{i, j, k}, Table[{u}[[k]] [t]+sum[ (ChristoffelSymbols[surface]@@(#[t]&/@{u}))[[i, j, k]] {u}[[i]] [t] {u}[[j]] [t], {i, 2}, {j, 2}] == 0, {k, 2}]]; In[15]:= Simplify[GeodesicEquation[torus[R, r]][u, v][t]] Out[15]= { (R+rCos[u[t]])Sin[u[t]]v [t] 2 +ru [t] r = 0, 2rSin[u[t]]u [t]v [t]+(r+rcos[u[t]])v [t] R+rCos[u[t]] = 0} 3.1 Ploting surfaces The following code generates a random surface that is the graph of a function defined on the rectangle [ 1, 1] [ 1, 1]. In[16]:= surface[x, y ]= Module[ {f =ListInterpolation[Array[RandomReal[{0, 1}]&,{6, 6}], {{ 1.2, 1.2}, { 1.2, 1.2}}]}, {x, y, f[x, y]}]; In[17]:= ParametricPlot3D[surf ace[x, y], {x, 1, 1}, {y, 1, 1}] Out[17]= The following function takes a surface and its domain and plots it with its normal vector and tangent plane. It basically uses CoordinateVector to draw the tangent plane and GaussMap to calculate the normal vector. These values are saved so that they do not have to be calculated everytime the point is moved. The dynamic interactivity is performed with the function Manipulate. In[18]:= PlotTangentNormal[surf ace, {u0, u1 }, {v0, v1 }]:=Module[ {u, v, g =ParametricPlot3D[surf ace[u, v], {u, u0, u1},{v, v0, v1}, 12

13 SphericalRegion True, Axes False], x1 =Evaluate[CoordinateVector[1][surf ace][#1, #2]]&, x2 =Evaluate[CoordinateVector[2][surf ace][#1, #2]]&, gm =Evaluate[GaussMap[surface][#1, #2]] &}, Manipulate[ Show[g, Graphics3D[ + x1@@p}]], Arrow[Tube[{surface@@p, surface@@p + x2@@p}]], Red, Arrow[Tube[{N[surf ace@@p], N[surf ace@@p+gm@@p]}]], EdgeForm[], Green, Polygon[{ N[surface@@p + (x1@@p) + (x2@@p)], N[surface@@p + (x1@@p) (x2@@p)], N[surface@@p (x1@@p) (x2@@p)], N[surface@@p (x1@@p) + (x2@@p)], N[surface@@p + (x1@@p) + (x2@@p)]}]}], PlotRangePadding 1], {{p, {u0, (v0 + v1)/2}, "Point"}, {u0, v0}, {u1, v1}}, ControlPlacement {Left}]]; In[19]:= PlotTangentNormal[surf ace, { 1, 1}, { 1, 1}] Out[19]= In[20]:= PlotTangentNormal[torus[2, 1], {0, 2π}, {0, 2π}] 13

14 Out[20]= The following function plots a surface painting it according to curvature. It takes the surface and the domain as parameters. Then it calculates the Gaussian curvature and saves the result in a variable. It uses a "TemparatureMap" function to colour: hot colours correspond to positive curvature and cold colours to negative curvature. The curvature is rescaled so that it fits in the interval [0,1]. Several other options are given. The function uses the command Quiet to prevent error messages from appearing when there are singular points. In[21]:= PlotGaussianCurvature[surf ace, {u0, u1 }, {v0, v1 }]:= Module[ {sc =Evaluate[GaussianCurvature[surf ace][#1, #2]]&,u, v}, Quiet[ParametricPlot3D[surf ace[u, v], {u, u0, u1}, {v, v0, v1}, ColorFunction (Glow[ColorData["TemperatureMap"] [0.5 + ArcTan[2.0sc[#4, #5]]/π]]]&), ColorFunctionScaling False, SphericalRegion True, PerformanceGoal "Quality",Axes False]]]; In[22]:= PlotGaussianCurvature[surf ace, { 1, 1}, { 1, 1}] Out[22]= In[23]:= PlotGaussianCurvature[torus[2, 1], {0, 2π}, {0, 2π}] 14

15 Out[23]= 3.2 Plotting geodesics Geodesics are very important objects of surfaces, but they are very difficult (usually impossible) to calculate in general. Here we show a piece of code that solves the geodesic equation numerically and plots a geodesic on a surface. In[24]:= PlotGeodesic[sup, {u0, u1 }, {v0, v1 }, opacity : 1]:= Module[ {u, v, x, sol, geq, max = 1.566, g = Evaluate[Metric[sup][#1, #2]]&, graphsup =ParametricPlot3D[sup[u, v], {u, u0, u1}, {v, v0, v1}, SphericalRegion True, Axes False, Mesh False, PerformanceGoal "Quality", PlotStyle Opacity[opacity]]}, geq =GeodesicEquation[sup][u[1], u[2]][v]; x =Evaluate[ {{1, 0}, { g[#1, #2][[1, 2]], g[#1, #2][[1, 1]]}/ Sqrt[g[#1, #2][[1, 1]]g[#1, #2][[2, 2]] g[#1, #2][[1, 2]] 2 ]}/ Sqrt[g[#1, #2][[1, 1]]]]&; Manipulate[Quiet[ sol =({u[1], u[2]}/.ndsolve[ Join[gEq, Thread[{u[1][0], u[2][0]} == pt], Thread[{u[1] [0], u[2] [0]} == (x@@pt).{cos[dir], Sin[dir]}]], {u[1], u[2]}, {v, 0.0, Tan[r]}][[1]]); Show[ graphsup, Graphics3D[{Sphere[N[sup@@pt], 0.05]}], ParametricPlot3D[ Evaluate[sup@@(#[u]&/@sol)], {u, 0, T an[r]}, PlotStyle Thick, PlotPoints 500], PlotRangePadding 0.1]], {{pt, {(u0+u1)/2, (v0+v1)/2}, "Point" }, {u0, v0}, {u1, v1}}, {{dir, 0, "Direction"}, 0, 2π}, {{r, Pi /4, "Length"}, 0.01, max}, Style[Dynamic["Real length: " <> ToString[N[Tan[r]]]]], 15

16 ControlPlacement Left, TrackedSymbols:>{pt, dir, r}]]; In[25]:= PlotGeodesic[surf ace, { 1, 1}, { 1, 1}] Out[25]= In[26]:= PlotGeodesic[torus[2, 1], {0, 2π}, {0, 2π}, 0.5] Out[26]= 16