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\deflang1033\pard\ri4\plain\f3\fs72\cf0 Chapter 2 Mathematical graphics with MuPAD\plain\f3\fs36\cf0 
\par 
\par 2.1 Secrets of the Virtual Camera
\par 
\par \pard\li300\ri5\fi-300{\*\pn\pnlvlblt\pnf1\pnindent300{\pntxtb\'b7}}\plain\f4\fs36\cf1 {\pntext\f1\'b7\tab}f := (x,y) ->  x^2 + y^2:
\par \pard\li600\ri1\fi-300\plain\f4\fs36\cf1 g := (x,y) -> -x^2 + y^2:
\par h := (x,y) ->  x^2 - y^2:
\par p := (x,y) -> -x^2 - y^2:
\par \pard\li300\ri5\fi-300{\*\pn\pnlvlblt\pnf1\pnindent300{\pntxtb\'b7}}\plain\f4\fs36\cf1 {\pntext\f1\'b7\tab}plotfunc3d(
\par \pard\li600\ri1\fi-300\plain\f4\fs36\cf1    f(x,y), g(x,y), h(x,y), p(x,y),
\par    x=-2..2, y=-2..2
\par )
\par \pard\ri4\plain\f4\fs36\cf1  \plain\f3\fs36\cf0    
\par  >>>>>>>>>> Time for Exercise 2.1  
\par  >>>>>>>>>> Time for Exercise 2.2  
\par 
\par 2.2 The structure of MuPAD graphs
\par 
\par 2.3 The gallery of mathematical plots
\par \pard\li300\ri5\fi-300{\*\pn\pnlvlblt\pnf1\pnindent300{\pntxtb\'b7}}\plain\f4\fs36\cf1 {\pntext\f1\'b7\tab}info(plot)
\par \pard\ri4\plain\f4\fs36\cf1 
\par \plain\f3\fs36\cf0 2.3.1 Function2d
\par \pard\li300\ri5\fi-300{\*\pn\pnlvlblt\pnf1\pnindent300{\pntxtb\'b7}}\plain\f4\fs36\cf1 {\pntext\f1\'b7\tab}G1 := plot::Function2d(sin(x), x=-5..5)
\par {\pntext\f1\'b7\tab}G1 := plot::Function2d(sin(x), x=-5..5,
\par \pard\li600\ri1\fi-300\plain\f4\fs36\cf1    LineColor=RGB::Blue,
\par    LineWidth=0.3*unit::mm
\par ):
\par \pard\li300\ri5\fi-300{\*\pn\pnlvlblt\pnf1\pnindent300{\pntxtb\'b7}}\plain\f4\fs36\cf1 {\pntext\f1\'b7\tab}G2 := plot::Function2d(sin(2*x), x=-5..5,
\par \pard\li600\ri1\fi-300\plain\f4\fs36\cf1    LineColor=RGB::Red,
\par    LineWidth=0.6*unit::mm
\par ):
\par \pard\li300\ri5\fi-300{\*\pn\pnlvlblt\pnf1\pnindent300{\pntxtb\'b7}}\plain\f4\fs36\cf1 {\pntext\f1\'b7\tab}G3 := plot::Function2d(sin(3*x), x=-5..5,
\par \pard\li600\ri1\fi-300\plain\f4\fs36\cf1    LineColor=RGB::Brown,
\par    LineWidth=0.8*unit::mm
\par ):
\par \pard\li300\ri5\fi-300{\*\pn\pnlvlblt\pnf1\pnindent300{\pntxtb\'b7}}\plain\f4\fs36\cf1 {\pntext\f1\'b7\tab}G4 := plot::Function2d(sin(4*x), x=-5..5,
\par \pard\li600\ri1\fi-300\plain\f4\fs36\cf1    LineColor=[0.5, 0.5, 0.5],
\par    LineWidth=1*unit::mm
\par ):
\par \pard\li300\ri5\fi-300{\*\pn\pnlvlblt\pnf1\pnindent300{\pntxtb\'b7}}\plain\f4\fs36\cf1 {\pntext\f1\'b7\tab} plot(G1,G2,G3,G4) // here we plot objects G1,..,G4
\par \pard\ri4\plain\f3\fs36\cf0 
\par >>>>>>>>>> Time for Exercise 2.3 
\par 
\par 2.3.2 Curve2d
\par 
\par \pard\li300\ri5\fi-300{\*\pn\pnlvlblt\pnf1\pnindent300{\pntxtb\'b7}}\plain\f4\fs36\cf1 {\pntext\f1\'b7\tab}C1 := plot::Curve2d(
\par \pard\li600\ri1\fi-300\plain\f4\fs36\cf1    [2*sin(3*t), 4*cos(5*t)], t=0..2*PI
\par ):
\par \pard\ri4\plain\f4\fs36\cf1 >\plain\f3\fs36\cf0 >>>>>>>>> Time for Exercise 2.4 
\par 
\par 2.3.3 Polar coordinates
\par 
\par \pard\li300\ri5\fi-300{\*\pn\pnlvlblt\pnf1\pnindent300{\pntxtb\'b7}}\plain\f4\fs36\cf1 {\pntext\f1\'b7\tab}P1 := plot::Polar([1, alpha], alpha=0..2*PI,
\par \pard\li600\ri1\fi-300\plain\f4\fs36\cf1    LineWidth=0.3
\par ):
\par P2 := plot::Polar(
\par    [sin(2*alpha), alpha], alpha=0..2*PI,
\par    LineWidth=0.5
\par ):
\par P3 := plot::Polar([exp(-1/u),u], u=0.1..4*PI,  
\par    LineWidth=1
\par ):
\par plot(P1, P2, P3, Scaling=Constrained)
\par \pard\ri4\plain\f3\fs36\cf0 
\par >>>>>>>>>> Time for Exercise 2.5 
\par 
\par 2.3.4 Implicit2d
\par 
\par \pard\li300\ri5\fi-300{\*\pn\pnlvlblt\pnf1\pnindent300{\pntxtb\'b7}}\plain\f4\fs36\cf1 {\pntext\f1\'b7\tab}I1 := plot::Implicit2d(x^2+y^2=1, x=-2..2, y=-2..2,
\par \pard\li600\ri1\fi-300\plain\f4\fs36\cf1    LineColor=RGB::Red
\par ):
\par I2 := plot::Implicit2d(x^2-y^2=1, x=-2..2,y=-2..2,
\par    LineColor=RGB::Blue,
\par    LineWidth=0.5
\par ):
\par I3 := plot::Implicit2d(x^4+y^4=1, x=-1..1, y=-1..1,
\par    LineColor=RGB::Green,
\par    LineWidth=0.7
\par ):
\par I4 := plot::Implicit2d(-x^2+y^2=1, x=-2..2,y=-2..2,
\par    LineColor=RGB::Brown,
\par    LineWidth=0.9
\par ):
\par plot(I1, I2, I3, I4, Scaling=Constrained)
\par \pard\ri4\plain\f3\fs36\cf0 
\par >>>>>>>>>> Time for Exercise 2.6
\par 
\par 2.3.5 Function3d
\par 
\par \pard\li300\ri5\fi-300{\*\pn\pnlvlblt\pnf1\pnindent300{\pntxtb\'b7}}\plain\f4\fs36\cf1 {\pntext\f1\'b7\tab}K := (x,y,z)->[abs(sin(x)), abs(sin(y)), abs(sin(z))]:
\par \pard\li600\ri1\fi-300\plain\f4\fs36\cf1 F1 := plot::Function3d(
\par    1.7*x*exp(-1/2*(x^2+y^2)),
\par    x=-4..4, y=-4..4,
\par    FillColorFunction=K
\par ):
\par F2 := plot::Function3d(-1, x=-4..4, y=-4..4):
\par plot(F1, F2)
\par \pard\ri4\plain\f4\fs36\cf1  \plain\f3\fs36\cf0    
\par \pard\li300\ri5\fi-300{\*\pn\pnlvlblt\pnf1\pnindent300{\pntxtb\'b7}}\plain\f4\fs36\cf1 {\pntext\f1\'b7\tab}K := (x,y,z)->  [abs(sin(x)),abs(sin(y)),abs(sin(z)),abs(sin(z))]:
\par \pard\li600\ri1\fi-300\plain\f4\fs36\cf1 F1 := plot::Function3d(
\par    1.7*x*exp(-1/2*(x^2+y^2)),
\par    x=-4..4, y=-4..4,
\par    XMesh=50,
\par    YMesh=50,
\par    FillColorFunction=K
\par ):
\par F2 := plot::Function3d(-1, x=-4..4, y=-4..4):
\par plot(F1, F2)
\par \pard\ri4\plain\f4\fs36\cf1  \plain\f3\fs36\cf0    
\par  >>>>>>>>>> Time for Exercise 2.7
\par 
\par 2.3.6 Surfaces
\par 
\par \pard\li300\ri5\fi-300{\*\pn\pnlvlblt\pnf1\pnindent300{\pntxtb\'b7}}\plain\f4\fs36\cf1 {\pntext\f1\'b7\tab}Cl := (u,v)->[u-trunc(u), v-trunc(v), 0]:
\par \pard\li600\ri1\fi-300\plain\f4\fs36\cf1 S1 := plot::Surface(
\par    [sin(u)*sin(v),
\par    sin(u)*cos(v),
\par    cos(u)*sin(v)],
\par    u=0..PI, v=0..PI,
\par    FillColorFunction=Cl
\par ):
\par plot(S1)
\par \pard\ri4\plain\f4\fs36\cf1  \plain\f3\fs36\cf0 
\par >>>>>>>>>> Time for Exercise 2.8
\par 
\par 2.3.7 Implicit3d
\par 
\par \pard\li300\ri5\fi-300{\*\pn\pnlvlblt\pnf1\pnindent300{\pntxtb\'b7}}\plain\f4\fs36\cf1 {\pntext\f1\'b7\tab}Steiner := plot::Implicit3d(
\par \pard\li600\ri1\fi-300\plain\f4\fs36\cf1    y^2*z^2 + z^2*x^2 + x^2*y^2 + 2*x*y*z = 0,
\par    x=-1..1, y=-1..1, z=-1..1,
\par    MeshVisible=TRUE,
\par    XMesh=25, YMesh=25, ZMesh=25
\par ):
\par plot(Steiner, Axes=None, Scaling=Constrained)
\par \pard\ri4\plain\f3\fs36\cf0 
\par \pard\li300\ri5\fi-300{\*\pn\pnlvlblt\pnf1\pnindent300{\pntxtb\'b7}}\plain\f4\fs36\cf1 {\pntext\f1\'b7\tab}Cl := (u,v,x,y,z)->[abs(x),abs(y),abs(z)]:
\par \pard\li600\ri1\fi-300\plain\f4\fs36\cf1 SteinerParam:=plot::Surface(    
\par    [sin(2*u)*(cos(v))^2, 
\par    sin(u)*sin(2*v), 
\par    cos(u)*sin(2*v)],
\par    u=-PI/2..PI/2, 
\par    v=-PI/2..PI/2,
\par    FillColorFunction=Cl,
\par    Scaling=Constrained,
\par    UMesh=50, 
\par    VMesh=50
\par ):
\par plot(SteinerParam, Axes=None)
\par \pard\ri4\plain\f4\fs36\cf1 
\par \plain\f3\fs36\cf0 
\par 2.3.8 Spherical coordinates
\par 
\par \pard\li300\ri5\fi-300{\*\pn\pnlvlblt\pnf1\pnindent300{\pntxtb\'b7}}\plain\f4\fs36\cf1 {\pntext\f1\'b7\tab}p := x -> abs(5*x - trunc(5*x)):
\par \pard\li600\ri1\fi-300\plain\f4\fs36\cf1 Cl := (u,v,x,y,z)->[p(x*y),p(y*z),p(z*x)]:
\par Shell := plot::Spherical(
\par    [u*v,u,v],u=0..2*PI,v=0..PI,
\par    FillColorFunction=Cl
\par ):
\par plot(
\par    Shell,
\par    Axes=Origin,
\par    AxesLineWidth=0.7,
\par    AxesTips=TRUE
\par )
\par \pard\ri4\plain\f4\fs36\cf1  \plain\f3\fs36\cf0    
\par >>>>>>>>>> Time for Exercise 2.8
\par 
\par 2.3.9 Cylindrical coordinates
\par 
\par \pard\li300\ri5\fi-300{\*\pn\pnlvlblt\pnf1\pnindent300{\pntxtb\'b7}}\plain\f4\fs36\cf1 {\pntext\f1\'b7\tab}Cl2 := (t,z)->[abs(1-z/PI),abs(1-t/PI),abs(t/PI)]:
\par \pard\li600\ri1\fi-300\plain\f4\fs36\cf1 Cy1 := plot::Cylindrical(
\par    [3*sin(2*z)+2*cos(3*t), t, z],
\par    t=0..PI, z=0..PI,
\par    Scaling=Unconstrained,
\par    FillColorFunction=Cl2
\par ):
\par plot(Cy1)
\par \pard\ri4\plain\f4\fs36\cf1  \plain\f3\fs36\cf0    
\par \pard\li300\ri5\fi-300{\*\pn\pnlvlblt\pnf1\pnindent300{\pntxtb\'b7}}\plain\f4\fs36\cf1 {\pntext\f1\'b7\tab}Cl2:= (u,z) -> [abs(1-z/PI) ,abs(1-u/PI),abs(u/PI)]:
\par \pard\li600\ri1\fi-300\plain\f4\fs36\cf1 Hat := plot::Cylindrical(
\par    [t*v, sin(t), sin(v)],  
\par    t=-2*PI..2*PI, v=0..2*PI,
\par    Scaling=Unconstrained,
\par    FillColorFunction=Cl2,
\par    UMesh=50,
\par    VMesh=50
\par ):
\par plot(Hat)
\par \pard\ri4\plain\f3\fs36\cf0 
\par >>>>>>>>>> Time for Exercise 2.9
\par 
\par 2.3.10 Curve3d and Tube
\par 
\par \pard\li300\ri5\fi-300{\*\pn\pnlvlblt\pnf1\pnindent300{\pntxtb\'b7}}\plain\f4\fs36\cf1 {\pntext\f1\'b7\tab}a := 7:
\par \pard\li600\ri1\fi-300\plain\f4\fs36\cf1 b := 3.5:
\par r := 5: 
\par x := t -> r*sin(a*t)*cos(b*t):
\par y := t -> r*sin(a*t)*sin(b*t):
\par z := t -> r*cos(a*t):
\par C3D := plot::Curve3d([x(t),y(t),z(t)], t=0..2*PI/b,
\par    LineWidth=0.8,
\par    LineColorType=Dichromatic
\par ):
\par plot(C3D)
\par \pard\ri4\plain\f3\fs36\cf0 
\par \pard\li300\ri5\fi-300{\*\pn\pnlvlblt\pnf1\pnindent300{\pntxtb\'b7}}\plain\f4\fs36\cf1 {\pntext\f1\'b7\tab}a := 7:
\par \pard\li600\ri1\fi-300\plain\f4\fs36\cf1 b := 3.5:
\par r := 5:
\par x := t -> r*sin(a*t)*cos(b*t):
\par y := t -> r*sin(a*t)*sin(b*t):
\par z := t -> r*cos(a*t):
\par T3D := plot::Tube([x(t),y(t),z(t)], 0.2, t=0..2*PI/b):
\par plot(T3D, Scaling=Unconstrained)
\par \pard\ri4\plain\f3\fs36\cf0 
\par \pard\li300\ri5\fi-300{\*\pn\pnlvlblt\pnf1\pnindent300{\pntxtb\'b7}}\plain\f4\fs36\cf1 {\pntext\f1\'b7\tab}a := 7:
\par \pard\li600\ri1\fi-300\plain\f4\fs36\cf1 b := 3.5:
\par r := 5:
\par x := t -> r*sin(a*t)*cos(b*t):
\par y := t -> r*sin(a*t)*sin(b*t):
\par z := t -> r*cos(a*t):
\par T3D := plot::Tube(
\par    [x(t),y(t),z(t)], sin(t*b)/2, t=0..2*PI/b
\par ):
\par plot(T3D, Scaling=Unconstrained)
\par \pard\ri4\plain\f3\fs36\cf0 
\par >>>>>>>>>> Time for Exercise 2.10 
\par 
\par 2.4 Plotting geometry models
\par 
\par 2.4.1 Points, lines and polygons
\par 
\par \pard\li300\ri5\fi-300{\*\pn\pnlvlblt\pnf1\pnindent300{\pntxtb\'b7}}\plain\f4\fs36\cf1 {\pntext\f1\'b7\tab}// point coordinates
\par \pard\li600\ri1\fi-300\plain\f4\fs36\cf1 x1 := -1: x2 :=  1: x3 := 0:
\par y1 := -1: y2 := -1: y3 := 1:
\par \pard\li300\ri5\fi-300{\*\pn\pnlvlblt\pnf1\pnindent300{\pntxtb\'b7}}\plain\f4\fs36\cf1 {\pntext\f1\'b7\tab} // declarations of 2D points
\par \pard\li600\ri1\fi-300\plain\f4\fs36\cf1 point1 := plot::Point2d([x1,y1]):
\par point2 := plot::Point2d([x2,y2]):
\par point3 := plot::Point2d([x3,y3]):
\par points := plot::Group2d(
\par    point1, point2, point3,
\par    PointColor=[0,0.5,1],
\par    PointSize = 3*unit::mm
\par ):
\par \pard\li300\ri5\fi-300{\*\pn\pnlvlblt\pnf1\pnindent300{\pntxtb\'b7}}\plain\f4\fs36\cf1 {\pntext\f1\'b7\tab}// sides of the triangle
\par \pard\li600\ri1\fi-300\plain\f4\fs36\cf1 side1 := plot::Line2d([x1,y1],[x2,y2]):
\par side2 := plot::Line2d([x2,y2],[x3,y3]):
\par side3 := plot::Line2d([x3,y3],[x1,y1]):
\par sides := plot::Group2d(
\par    side1, side2, side3, LineWidth=1
\par ):
\par \pard\li300\ri5\fi-300{\*\pn\pnlvlblt\pnf1\pnindent300{\pntxtb\'b7}}\plain\f4\fs36\cf1 {\pntext\f1\'b7\tab}// midpoints of sides
\par \pard\li600\ri1\fi-300\plain\f4\fs36\cf1 midpoint1 := plot::Point2d([(x1+x2)/2,(y1+y2)/2]):
\par midpoint2 := plot::Point2d([(x2+x3)/2,(y2+y3)/2]):
\par midpoint3 := plot::Point2d([(x1+x3)/2,(y1+y3)/2]):
\par midpoints := plot::Group2d(
\par    midpoint1, midpoint2, midpoint3,
\par    PointColor=[1,0.5,0],
\par    PointSize = 3*unit::mm
\par ):
\par \pard\li300\ri5\fi-300{\*\pn\pnlvlblt\pnf1\pnindent300{\pntxtb\'b7}}\plain\f4\fs36\cf1 {\pntext\f1\'b7\tab}// medians
\par \pard\li600\ri1\fi-300\plain\f4\fs36\cf1 median1 := plot::Line2d([x1,y1],[(x2+x3)/2,(y2+y3)/2]):
\par median2 := plot::Line2d([x2,y2],[(x1+x3)/2,(y1+y3)/2]):
\par median3 := plot::Line2d([x3,y3],[(x1+x2)/2,(y1+y2)/2]):
\par medians := plot::Group2d(
\par    median1, median2, median3,
\par    LineColor=[0.8,0,0],
\par    LineWidth=0.5
\par ):
\par \pard\li300\ri5\fi-300{\*\pn\pnlvlblt\pnf1\pnindent300{\pntxtb\'b7}}\plain\f4\fs36\cf1 {\pntext\f1\'b7\tab}plot(sides, medians, midpoints,points,
\par \pard\li600\ri1\fi-300\plain\f4\fs36\cf1    Scaling=Constrained,
\par    Axes=None
\par )
\par \pard\ri4\plain\f4\fs36\cf1  \plain\f3\fs36\cf0   
\par  >>>>>>>>>> Technical comment:    
\par As we have already mentioned, MuPAD graphics is based on OpenGL. 
\par Therefore most of OpenGL limitations have some impact on our constructions. 
\par One of them is that in 3D we cannot use filled polygons with more than three 
\par vertices. Therefore, in most of 3D constructions any filled flat polygon with 
\par more than three vertices should be developed as a collection of filled triangles.
\par 
\par \pard\li300\ri5\fi-300{\*\pn\pnlvlblt\pnf1\pnindent300{\pntxtb\'b7}}\plain\f4\fs36\cf1 {\pntext\f1\'b7\tab}// point coordinates
\par \pard\li600\ri1\fi-300\plain\f4\fs36\cf1 x1 := -1: x2 :=  1: x3 := 0:
\par y1 := -1: y2 := -1: y3 := 1:
\par zd := -1: zg :=  1:
\par \pard\li300\ri5\fi-300{\*\pn\pnlvlblt\pnf1\pnindent300{\pntxtb\'b7}}\plain\f4\fs36\cf1 {\pntext\f1\'b7\tab} // bases declaration
\par \pard\li600\ri1\fi-300\plain\f4\fs36\cf1 top := plot::Polygon3d(
\par    [[x1,y1,zg], [x2,y2,zg],[x3,y3,zg]],
\par    Closed=TRUE,
\par    Filled=TRUE
\par ):
\par \pard\li300\ri5\fi-300{\*\pn\pnlvlblt\pnf1\pnindent300{\pntxtb\'b7}}\plain\f4\fs36\cf1 {\pntext\f1\'b7\tab}bottom := plot::Polygon3d(
\par \pard\li600\ri1\fi-300\plain\f4\fs36\cf1    [[x1,y1,zd], [x2,y2,zd],[x3,y3,zd]],
\par    Closed=TRUE,
\par    Filled=TRUE
\par ):
\par \pard\li300\ri5\fi-300{\*\pn\pnlvlblt\pnf1\pnindent300{\pntxtb\'b7}}\plain\f4\fs36\cf1 {\pntext\f1\'b7\tab} // one side declaration
\par \pard\li600\ri1\fi-300\plain\f4\fs36\cf1 tr1 := plot::Polygon3d(
\par    [[x1,y1,zd], [x2,y2,zd],[x2,y2,zg]],
\par    Closed=TRUE,
\par    Filled=TRUE,
\par    LinesVisible=FALSE
\par ):
\par \pard\li300\ri5\fi-300{\*\pn\pnlvlblt\pnf1\pnindent300{\pntxtb\'b7}}\plain\f4\fs36\cf1 {\pntext\f1\'b7\tab}tr2 := plot::Polygon3d(
\par \pard\li600\ri1\fi-300\plain\f4\fs36\cf1    [[x1,y1,zg], [x2,y2,zg],[x1,y1,zd]],
\par    Closed=TRUE,
\par    Filled=TRUE,
\par    LinesVisible=FALSE
\par ):
\par \pard\li300\ri5\fi-300{\*\pn\pnlvlblt\pnf1\pnindent300{\pntxtb\'b7}}\plain\f4\fs36\cf1 {\pntext\f1\'b7\tab} // solid declaration
\par \pard\li600\ri1\fi-300\plain\f4\fs36\cf1 prism := plot::Group3d(top, bottom, tr1, tr2):
\par \pard\li300\ri5\fi-300{\*\pn\pnlvlblt\pnf1\pnindent300{\pntxtb\'b7}}\plain\f4\fs36\cf1 {\pntext\f1\'b7\tab} // final plot  
\par \pard\li600\ri1\fi-300\plain\f4\fs36\cf1 plot( prism)
\par \pard\ri4\plain\f3\fs36\cf0 
\par \pard\li300\ri5\fi-300{\*\pn\pnlvlblt\pnf1\pnindent300{\pntxtb\'b7}}\plain\f4\fs36\cf1 {\pntext\f1\'b7\tab}// here we declare our sequence
\par \pard\li600\ri1\fi-300\plain\f4\fs36\cf1 data := [[n,1/(-2)^n] $ n=20..30]:
\par \pard\li300\ri5\fi-300{\*\pn\pnlvlblt\pnf1\pnindent300{\pntxtb\'b7}}\plain\f4\fs36\cf1 {\pntext\f1\'b7\tab} // now we declare plot object for the sequence
\par \pard\li600\ri1\fi-300\plain\f4\fs36\cf1 sequence := plot::PointList2d(data,   
\par    PointSize=2*unit::mm,
\par    PointColor=RGB::Red
\par ):
\par  plot(sequence)
\par \pard\ri4\plain\f3\fs36\cf0 
\par \pard\li300\ri5\fi-300{\*\pn\pnlvlblt\pnf1\pnindent300{\pntxtb\'b7}}\plain\f4\fs36\cf1 {\pntext\f1\'b7\tab}sequence2 := plot::Polygon2d(data,
\par \pard\li600\ri1\fi-300\plain\f4\fs36\cf1   PointSize=2,
\par   PointsVisible=TRUE
\par ):
\par  plot(sequence2)
\par \pard\ri4\plain\f3\fs36\cf0 
\par >>>>>>>>>> Time for Exercise 2.11   
\par 
\par 2.4.2 Solids in MuPAD
\par 
\par \pard\li300\ri5\fi-300{\*\pn\pnlvlblt\pnf1\pnindent300{\pntxtb\'b7}}\plain\f4\fs36\cf1 {\pntext\f1\'b7\tab}// declarations of four solids
\par \pard\li600\ri1\fi-300\plain\f4\fs36\cf1 RedSphere := plot::Sphere(1,[0,0,0],
\par    FillColor=[1,0,0]
\par ):
\par  GreenCylinder := plot::Cylinder(1,[0,0,0],[0,1,1],
\par    FillColor=[0,0.5,0]
\par ):
\par  PinkBox := plot::Box([0,0,0], [1,1,1],
\par    FillColor=[0.6, 0,0.6]
\par ):
\par  BlueCone := plot::Cone(1,[0,0,0], 0.2,[1,1,1]):
\par \pard\li300\ri5\fi-300{\*\pn\pnlvlblt\pnf1\pnindent300{\pntxtb\'b7}}\plain\f4\fs36\cf1 {\pntext\f1\'b7\tab}plot(RedSphere, GreenCylinder, BlueCone, PinkBox)
\par \pard\ri4\plain\f4\fs36\cf1  \plain\f3\fs36\cf0   
\par \pard\li300\ri5\fi-300{\*\pn\pnlvlblt\pnf1\pnindent300{\pntxtb\'b7}}\plain\f4\fs36\cf1 {\pntext\f1\'b7\tab}// transformations
\par \pard\li600\ri1\fi-300\plain\f4\fs36\cf1 A:=plot::Translate3d([1,1,1],
\par    plot::Scale3d([0.5,0.5,1],RedSphere)
\par ):
\par B:=plot::Translate3d([-1,1,0],
\par    plot::Scale3d([0.5,1,1],GreenCylinder)
\par ):
\par C:=plot::Translate3d([-1,-1,-1],PinkBox):
\par S:=plot::Translate3d([1,-1,0],BlueCone):
\par \pard\li300\ri5\fi-300{\*\pn\pnlvlblt\pnf1\pnindent300{\pntxtb\'b7}}\plain\f4\fs36\cf1 {\pntext\f1\'b7\tab} // finally we plot translated solids
\par \pard\li600\ri1\fi-300\plain\f4\fs36\cf1 plot(A,B,C,S)
\par \pard\ri4\plain\f3\fs36\cf0 
\par \pard\li300\ri5\fi-300{\*\pn\pnlvlblt\pnf1\pnindent300{\pntxtb\'b7}}\plain\f4\fs36\cf1 {\pntext\f1\'b7\tab}Cube := plot::Box([0,0,0],[4,4,4],
\par \pard\li600\ri1\fi-300\plain\f4\fs36\cf1    FillColor=[0.7,0.7,1,0.6]
\par ):
\par He := plot::Hexahedron (
\par    Center = [2.9,2.9,2.9],
\par    Radius = 1
\par ):
\par Te := plot::Tetrahedron (
\par    Center = [1,1,3],
\par    Radius = 1
\par ):
\par Oc := plot::Octahedron (
\par    Center = [1,1,1],
\par    Radius = 1
\par ):
\par Ic := plot::Icosahedron (
\par    Center = [1, 3, 1],
\par    Radius = 1
\par ):
\par Do := plot::Dodecahedron(
\par    Center = [3, 1, 1],
\par    Radius = 1
\par ):
\par Cam := plot::Camera(
\par    [28,-23,20],[1.5,1.5,1.5], PI/18
\par ):
\par \pard\li300\ri5\fi-300{\*\pn\pnlvlblt\pnf1\pnindent300{\pntxtb\'b7}}\plain\f4\fs36\cf1 {\pntext\f1\'b7\tab}plot(Cube, He, Te, Oc, Ic, Do, Cam, Axes=None)
\par \pard\ri4\plain\f3\fs36\cf0 
\par >>>>>>>>>> Time for Exercise 2.12 
\par 
\par 2.5 Turtle graphics and L-systems
\par 
\par 2.5.1 A brief introduction to turtle graphics
\par 
\par Method 1\tab          \tab Method2               \tab what hat it does
\par Left(angle)\tab \tab \tab left(angle)\tab \tab \tab turn left (angle in radians)
\par Right(angle)\tab \tab \tab right(angle)\tab \tab \tab turn right (angle in radians)
\par Forward(dist)\tab \tab forward(dist)\tab \tab \tab draw a line given length dist
\par Up\tab \tab \tab \tab \tab penUp()\tab \tab \tab \tab pen up and stop drawing
\par Down\tab \tab \tab \tab penDown()\tab \tab \tab pen down and start drawing 
\par Push\tab \tab \tab \tab push()\tab \tab \tab \tab save the current position
\par Pop\tab \tab \tab \tab \tab pop()\tab \tab \tab \tab go back to last saved position
\par LineColor(color)\tab \tab setLineColor(color)\tab change the color of the pen
\par 
\par 
\par \pard\li300\ri5\fi-300{\*\pn\pnlvlblt\pnf1\pnindent300{\pntxtb\'b7}}\plain\f4\fs36\cf1 {\pntext\f1\'b7\tab}T:=plot::Turtle():
\par \pard\li600\ri1\fi-300\plain\f4\fs36\cf1 T::forward(100): T::right(PI/2):
\par T::forward(100): T::right(PI/2):
\par T::forward(50):  T::right(PI/2):
\par T::forward(50):  T::right(PI/2):
\par T::forward(100): T::right(PI/2):
\par T::forward(25):  T::right(PI/2):
\par T::forward(25):  T::right(PI/2):
\par T::forward(50):
\par plot(T)
\par \pard\ri4\plain\f4\fs36\cf1  \plain\f3\fs36\cf0    
\par Technical comment:
\par =================   
\par Turtle graphics does not use coordinates explicitly. However, we can 
\par trace any location of the turtle by knowing that a new turtle always 
\par starts from the center of coordinate system and it is facing towards 
\par the positive direction of the y-axis.
\par 
\par \pard\li300\ri5\fi-300{\*\pn\pnlvlblt\pnf1\pnindent300{\pntxtb\'b7}}\plain\f4\fs36\cf1 {\pntext\f1\'b7\tab}T2 := plot::Rotate2d(PI/2,[0,0],T):
\par \pard\li600\ri1\fi-300\plain\f4\fs36\cf1 T3 := plot::Rotate2d(PI,[0,0],T):
\par T4 := plot::Rotate2d(3*PI/2,[0,0],T):
\par plot(T,T2,T3,T4)
\par \pard\ri4\plain\f4\fs36\cf1  \plain\f3\fs36\cf0    
\par \pard\li300\ri5\fi-300{\*\pn\pnlvlblt\pnf1\pnindent300{\pntxtb\'b7}}\plain\f4\fs36\cf1 {\pntext\f1\'b7\tab}A1 := plot::Translate2d([50,-75], T):
\par \pard\li600\ri1\fi-300\plain\f4\fs36\cf1 A2 := plot::Rotate2d(  PI/4,[0,0],A1):
\par A3 := plot::Rotate2d(2*PI/4,[0,0],A1):
\par A4 := plot::Rotate2d(3*PI/4,[0,0],A1):
\par A5 := plot::Rotate2d(4*PI/4,[0,0],A1):
\par A6 := plot::Rotate2d(5*PI/4,[0,0],A1):
\par A7 := plot::Rotate2d(6*PI/4,[0,0],A1):
\par A8 := plot::Rotate2d(7*PI/4,[0,0],A1):
\par plot(A1,A2,A3,A4,A5,A6,A7,A8,
\par    LineColor=RGB::Red,
\par    LineWidth=0.5
\par )
\par \pard\ri4\plain\f4\fs36\cf1  \plain\f3\fs36\cf0    
\par \pard\li300\ri5\fi-300{\*\pn\pnlvlblt\pnf1\pnindent300{\pntxtb\'b7}}\plain\f4\fs36\cf1 {\pntext\f1\'b7\tab}A1 := plot::modify(T, LineColor=RGB::Red):
\par \pard\li600\ri1\fi-300\plain\f4\fs36\cf1 A1 := plot::Translate2d([50,-75], A1):
\par A3 := plot::modify(T, LineColor=RGB::Blue, LineWidth=2):
\par A3 := plot::Translate2d([50,-75], A3):
\par A3 := plot::Rotate2d(2*PI/4,[0,0],A3):
\par A5 := plot::modify(T,
\par    LineColor=RGB::Green,
\par    LineStyle=Dashed,
\par    LineWidth=0.5
\par ):
\par A5 := plot::Translate2d([50,-75], A5):
\par A5 := plot::Rotate2d(4*PI/4,[0,0],A5):
\par A7 := plot::modify(T, LineColor=RGB::Brown):
\par A7 := plot::Translate2d([50,-75], A7):
\par A7 := plot::Rotate2d(6*PI/4,[0,0],A7):
\par plot(A1,A3,A5,A7, LineWidth=0.75)
\par \pard\ri4\plain\f4\fs36\cf1  \plain\f3\fs36\cf0   
\par Method 1
\par \pard\li300\ri5\fi-300{\*\pn\pnlvlblt\pnf1\pnindent300{\pntxtb\'b7}}\plain\f4\fs36\cf1 {\pntext\f1\'b7\tab}A := plot::Turtle(
\par \pard\li600\ri1\fi-300\plain\f4\fs36\cf1    Forward(1), Right(PI/2), Forward(1)
\par ):
\par plot(A)
\par \pard\ri4\plain\f4\fs36\cf0 
\par M\plain\f3\fs36\cf0 ethod 2
\par \pard\li300\ri5\fi-300{\*\pn\pnlvlblt\pnf1\pnindent300{\pntxtb\'b7}}\plain\f4\fs36\cf1 {\pntext\f1\'b7\tab}A := plot::Turtle():
\par \pard\li600\ri1\fi-300\plain\f4\fs36\cf1 A::forward(1):  A::right(PI/2): A::forward(1):
\par plot(A)
\par \pard\ri4\plain\f3\fs36\cf0 
\par >>>>>>>>>> Time for Exercise 2.13
\par 
\par 2.5.2 Lindenmayer systems
\par 
\par \pard\li300\ri5\fi-300{\*\pn\pnlvlblt\pnf1\pnindent300{\pntxtb\'b7}}\plain\f4\fs36\cf1 {\pntext\f1\'b7\tab}KochCurve := plot::Lsys( // start a new L-system
\par \pard\li600\ri1\fi-300\plain\f4\fs36\cf1    PI/3,             // turtle must always turn PI/3
\par    "F++F++F",        // this is the seed  
\par    "F"="F-F++F-F",   // iteration rule
\par    Generations=4     // number of generations
\par ):
\par plot(KochCurve)      // now plot the turtle path
\par \pard\ri4\plain\f3\fs36\cf0 
\par \pard\li300\ri5\fi-300{\*\pn\pnlvlblt\pnf1\pnindent300{\pntxtb\'b7}}\plain\f4\fs36\cf1 {\pntext\f1\'b7\tab}Koch1 := plot::Lsys(PI/2, "F-F-F-F",
\par \pard\li600\ri1\fi-300\plain\f4\fs36\cf1   "F"="FF-F--F-F",
\par   Generations=4
\par ):
\par plot(Koch1)
\par \pard\ri4\plain\f4\fs36\cf1 
\par \pard\li300\ri5\fi-300{\*\pn\pnlvlblt\pnf1\pnindent300{\pntxtb\'b7}}\plain\f4\fs36\cf1 {\pntext\f1\'b7\tab}Koch2 := plot::Lsys(PI/2, "F+F+F+F",
\par \pard\li600\ri1\fi-300\plain\f4\fs36\cf1   "F"="FF-F-F+F+FF",
\par    Generations=4
\par ):
\par plot(Koch2)
\par \pard\ri4\plain\f4\fs36\cf1  \plain\f3\fs36\cf0   
\par \pard\li300\ri5\fi-300{\*\pn\pnlvlblt\pnf1\pnindent300{\pntxtb\'b7}}\plain\f4\fs36\cf1 {\pntext\f1\'b7\tab}weed := plot::Lsys(PI/5, "F",
\par \pard\li600\ri1\fi-300\plain\f4\fs36\cf1    "F"="BR[+FHF][-FHF]FHF",
\par    "R"="BRR",
\par    "R"=Line,
\par    "B"=RGB::Brown,
\par    "H"=RGB::ForestGreen,
\par    Generations=5
\par ):
\par plot(weed)
\par \pard\ri4\plain\f4\fs36\cf1  \plain\f3\fs36\cf0    
\par >>>>>>>>>> Time for Exercise 2.13 
\par 
\par 2.6 Animation step-by-step
\par 
\par \pard\li300\ri5\fi-300{\*\pn\pnlvlblt\pnf1\pnindent300{\pntxtb\'b7}}\plain\f4\fs36\cf1 {\pntext\f1\'b7\tab}reset():
\par \pard\li600\ri1\fi-300\plain\f4\fs36\cf1 Curve := plot::Curve2d([t^2-4, t/2], t=-3..3):
\par Particle := plot::Point2d([a^2-4, a/2], a=-3..3):
\par plot(Curve, Particle)
\par \pard\ri4\plain\f4\fs36\cf1  \plain\f3\fs36\cf0    
\par \pard\li300\ri5\fi-300{\*\pn\pnlvlblt\pnf1\pnindent300{\pntxtb\'b7}}\plain\f4\fs36\cf1 {\pntext\f1\'b7\tab}Curve := plot::Curve2d([t^2-4, t/2], t=-3..a, a=-3..3):
\par \pard\li600\ri1\fi-300\plain\f4\fs36\cf1 Particle := plot::Point2d([a^2-4, a/2], a=-3..3,
\par    PointSize=2,
\par    PointColor=RGB::Red
\par ):
\par plot(Curve, Particle)
\par \pard\ri4\plain\f4\fs36\cf1  \plain\f3\fs36\cf0    
\par 
\par \pard\li300\ri5\fi-300{\*\pn\pnlvlblt\pnf1\pnindent300{\pntxtb\'b7}}\plain\f4\fs36\cf1 {\pntext\f1\'b7\tab}h := 1:
\par \pard\li600\ri1\fi-300\plain\f4\fs36\cf1 k := 1:
\par Parabola := plot::Function2d(
\par    a*(x-h)^2+k, x=-1..3, a=-2..2
\par ):
\par plot(Parabola)
\par \pard\li300\ri5\fi-300{\*\pn\pnlvlblt\pnf1\pnindent300{\pntxtb\'b7}}\plain\f4\fs36\cf1 {\pntext\f1\'b7\tab}// radii of both circles
\par \pard\li600\ri1\fi-300\plain\f4\fs36\cf1 r := 0.5:
\par R := 2:
\par  
\par // coordinates of the particle
\par coordinates :=
\par    [(R-r)*cos(phi)+r*cos(((R-r)/r)*phi),
\par    (R-r)*sin(phi)-r*sin(((R-r)/r)*phi)]:
\par  
\par // static objects
\par LargeCircle := plot::Circle2d(R,
\par    LineColor=RGB::Black
\par ):
\par  
\par // objects to animate
\par SmallCircle:= plot::Circle2d(
\par    r, [(R-r)*cos(a), (R-r)*sin(a)],
\par    a=0..2*PI,
\par    Color = [0.8,0,0],
\par    LineColor=RGB::Black,
\par    LineWidth=0.1,
\par    Filled=TRUE,
\par    FillPattern=Solid
\par ):
\par  
\par Particle:=plot::Point2d(
\par    coordinates,
\par    phi = 0..2*PI,
\par    PointSize = 2,
\par    PointColor = [0,0,1]
\par ):
\par  
\par Curve := plot::Curve2d(
\par    coordinates,
\par    phi =0..a, a=0..2*PI
\par ):
\par // now we plot all together
\par  
\par plot(LargeCircle, SmallCircle, Particle, Curve,
\par    Frames=80,
\par    Scaling=Constrained,
\par    AxesInFront=TRUE
\par )
\par \pard\ri4\plain\f3\fs36\cf0 
\par >>>>>>>>>> Time for Exercise 2.14
\par }