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Hướng Dẫn Vẽ Đồ Thị 3D trong MuPAD Của MatLab plot::Function3d – 3D function graphs plot::Function3d creates the 3D graph of a function in 2 variables. Calls: plot::Function3d(f, Options) plot::Function3d(f, x = , y = , <a = a min a max >, Options) Parameters: f: the function: an arithmetical expression or a piecewise object in the independent variables , and the animation parameter a. Alternatively, a MuPAD procedure that accepts 2 input parameter , or 3 input parameters , , and returns a numerical value when the input parameters are numerical. f is equivalent to the attribute Function. x: the first independent variable: an identifier or an indexed identifier. x is equivalent to the attribute XName. : the plot range in direction: , must be numerical real values or expressions of the animation parameter . If not specified, the default range x = -5 5 is used. is equivalent to the attributes XRange, XMin, XMax. y: the second independent variable: an identifier or an indexed identifier. y is equivalent to the attribute YName. : the plot range in direction: , must be numerical real values or expressions of the animation parameter . If not specified, the default range y = -5 5 is used. is equivalent to the attributes YRange, YMin, YMax. See Also: plot, plot::copy, plot::Function2d, plot::Surface, plotfunc2d, plotfunc3d Details: • The expression f(x, y) is evaluated at finitely many points , in the plot range. There may be singularities. Although a heuristics is used to find a reasonable range when singularities are present, it is highly recommended to specify a range via ViewingBoxZRange = with suitable numerical real values , . Cf. example 2. • Animations are triggered by specifying a range for a parameter a that is different from the indedependent variables x, y. Thus, in animations, the -range , the -range as well as the animation range must be specified. Cf. example 3. • The function f is evaluated on a regular equidistant mesh of sample points determined by the attributes XMesh and YMesh (or the shorthand-notation for both, Mesh). By default, the attribute AdaptiveMesh = 0 is set, i.e., no adaptive refinement of the equidistant mesh is used. If the standard mesh does not suffice to produce a sufficiently detailed plot, one may either increase the value of XMesh and YMesh or set AdaptiveMesh = n with some (small) positive integer n. This may result in up to times as many triangles as used with AdaptiveMesh = 0, potentially more when f has non-isolated singularities. Cf. example 4. • The “coordinate lines” (“parameter lines”) are curves on the function graph. The phrase “XLines” refers to the curves with the parameter running from to , while is some fixed value from the interval . The phrase “YLines” refers to the curves with the parameter running from ymin to ymax, while is some fixed value from the interval . By default, the parameter lines are visible. They may be “switched off” by specifying XLinesVisible = FALSE and YLinesVisible = FALSE, respectively. • The coordinate lines controlled by XLinesVisible = TRUE/FALSE and YLinesVisible = TRUE/FALSE indicate the equidistant regular mesh set via the Mesh attributes. If the mesh is refined by the Submesh attributes or by the adaptive mechanism controlled by AdaptiveMesh = n, no additional parameter lines are drawn. As far as the numerical approximation of the function graph is concerned, the settings , and , Submesh = [0, 0] are equivalent. However, in the first setting, nx parameter lines are visible in the direction, while in the latter setting parameter lines are visible. Cf. example 5. Attributes for plot::Function3d AdaptiveMesh = 0 adaptive sampling AffectViewingBox = TRUE influence of objects on the ViewingBox of a scene Color = RGB::Red the main color FillColor = RGB::Red color of areas and surfaces FillColor2 = RGB::CornflowerBlue second color of areas and surfaces for color blends FillColorDirection = [0, 0, 1] the direction of color transitions on surfaces FillColorDirectionX = 0 x-component of the direction of color transitions on surfaces FillColorDirectionY = 0 y-component of the direction of color transitions on surfaces FillColorDirectionZ = 1 z-component of the direction of color transitions on surfaces FillColorFunction functional area/surface coloring FillColorType = Dichromatic surface filling types Filled = TRUE filled or transparent areas and surfaces Frames = 50 the number of frames in an animation Function function expression or procedure Legend makes a legend entry LegendEntry = TRUE add this object to the legend? LegendText short explanatory text for legend LineColor = RGB::Black.[0.25] color of lines LineColor2 = RGB::DeepPink color of lines LineColorDirection = [0, 0, 1] the direction of color transitions on lines LineColorDirectionX = 0 x-component of the direction of color transitions on lines LineColorDirectionY = 0 y-component of the direction of color transitions on lines LineColorDirectionZ = 1 z-component of the direction of color transitions on lines LineColorFunction functional line coloring LineColorType = Flat line coloring types LineStyle = Solid solid, dashed or dotted lines? LinesVisible = TRUE visibility of lines LineWidth = 0.35 width of lines Mesh = [25, 25] number of sample points MeshVisible = FALSE visibility of irregular mesh lines in 3D Name the name of a plot object (for browser and legend) ParameterBegin initial value of the animation parameter ParameterEnd end value of the animation parameter ParameterName name of the animation parameter ParameterRange range of the animation parameter PointSize = 1.5 the size of points PointStyle = FilledCircles the presentation style of points PointsVisible = FALSE visibility of mesh points Shading = Smooth smooth color blend of surfaces Submesh = [0, 0] density of submesh (additional sample points) TimeBegin = 0.0 start time of the animation TimeEnd = 10.0 end time of the animation TimeRange = 0.0 10.0 the real time span of an animation Title object title TitleAlignment = Center horizontal alignment of titles w.r.t. their coordinates TitleFont = ["sans-serif", 11] font of object titles TitlePosition position of object titles TitlePositionX position of object titles, x component TitlePositionY position of object titles, y component TitlePositionZ position of object titles, z component Visible = TRUE visibility VisibleAfter object visible after this time value VisibleAfterEnd = TRUE object visible after its animation time ended? VisibleBefore object visible until this time value VisibleBeforeBegin = TRUE object visible before its animation time starts? VisibleFromTo object visible during this time range XLinesVisible = TRUE visibility of parameter lines (x lines) XMax = 5 final value of parameter “x” XMesh = 25 number of sample points for parameter “x” XMin = -5 initial value of parameter “x” XName name of parameter “x” XRange = -5 5 range of parameter “x” XSubmesh = 0 density of additional sample points for parameter “x” YLinesVisible = TRUE visibility of parameter lines (y lines) YMax = 5 final value of parameter “y” YMesh = 25 number of sample points for parameter “y” YMin = -5 initial value of parameter “y” YName name of parameter “y” YRange = -5 5 range of parameter “y” YSubmesh = 0 density of additional sample points for parameter “y” ZContours contour lines at constant z values Example 1 The following call returns an object representing the graph of the function over the region , : g := plot::Function3d(sin(x^2 + y^2), x = -2 2, y = -2 2) Call plot to plot the graph: plot(g) Functions can also be specified by piecewise objects or procedures: f := piecewise([x < y, 0], [x >= y, (x - y)^2]): plot(plot::Function3d(f, x = -2 4, y = -1 3)) f := proc(x, y) begin if x + y^2 + 2*y < 0 then 0 else x + y^2 + 2*y end_if: end_proc: plot(plot::Function3d(f, x = -3 2, y = -2 2)) delete g, f Example 2 We plot a function with singularities: f := plot::Function3d(x/y + y/x, x = -1 1, y = - 1 1): plot(f) We specify an explicit viewing range for the direction: plot(f, ViewingBoxZRange = -20 20) delete f Example 3 We generate an animation of a parametrized function: plot(plot::Function3d(sin((x - a)^2 + y^2), x = -2 2, y = -2 2, a = 0 5)) Example 4 The standard mesh for the numerical evaluation of a function graph does not suffice to generate a satisfying graphics in the following case: plot(plot::Function3d(besselJ(0, sqrt(x^2 + y^2)), x = -20 20, y = -20 20)) We increase the number of mesh points. Here, we use XSubmesh and YSubmesh to place 2 additional points in each direction between each pair of neighboring points of the default mesh. This increases the runtime by a factor of : plot(plot::Function3d(besselJ(0, sqrt(x^2 + y^2)), x = -20 20, y = -20 20, Submesh = [2, 2])) Alternatively, we enable adaptive sampling by setting the value of AdaptiveMesh to some positive value: plot(plot::Function3d(besselJ(0, sqrt(x^2 + y^2)), x = -20 20, y = -20 20, AdaptiveMesh = 2)) Example 5 By default, the parameter lines of a function graph are “switched on”: plot(plot::Function3d(x^2 + y^2, x = 0 1, y = 0 1)) The parameter lines are “switched off” by setting XLinesVisible, YLinesVisible: plot(plot::Function3d(x^2 + y^2, x = 0 1, y = 0 1, XLinesVisible = FALSE, YLinesVisible = FALSE)) The number of parameter lines are determined by the Mesh attributes: plot(plot::Function3d(x^2 + y^2, x = 0 1, y = 0 1, Mesh = [5, 12])) [...]... plot(plot::Function3d(x^2 + y^2, x = 0 1, y = 0 1, Mesh = [5, 12], XSubmesh = 1, YSubmesh = 2)) Example 6 Functions need not be defined over the whole parameter range: plot(plot::Function3d(sqrt(1-x^2-y^2), x=-1 1, y=-1 1)) plot(plot::Function3d(sqrt(sin(x)+cos(y)))) This makes for an easy way of plotting a function over a non-rectangular area: chi := piecewise([x^2 < abs(y), 1]) plot(plot::Function3d(chi*sin(x+cos(y))), . Hướng Dẫn Vẽ Đồ Thị 3D trong MuPAD Của MatLab plot::Function3d – 3D function graphs plot::Function3d creates the 3D graph of a function in 2 variables. Calls: plot::Function3d(f, Options) plot::Function3d(f,. y)^2]): plot(plot::Function3d(f, x = -2 4, y = -1 3)) f := proc(x, y) begin if x + y^2 + 2*y < 0 then 0 else x + y^2 + 2*y end_if: end_proc: plot(plot::Function3d(f, x = -3 2, y = -2 2)) delete. value: plot(plot::Function3d(besselJ(0, sqrt(x^2 + y^2)), x = -20 20, y = -20 20, AdaptiveMesh = 2)) Example 5 By default, the parameter lines of a function graph are “switched on”: plot(plot::Function3d(x^2