Hướng dẫn vẽ đồ thị 3d trong Matlab

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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 = amin amax>, 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 attribute XName identifier or an indexed identifier   x is equivalent to the

:

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 attribute YName identifier or an indexed identifier   y is equivalent to the

:

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

MeshVisible = FALSE visibility of irregular mesh lines in 3D

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)

XMesh = 25 number of sample points for 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)

YMesh = 25 number of sample points for 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]))

When the mesh is refined via the Submesh attributes, the numerical approximation of the surface becomes smoother However, the number of parameter lines is not increased:

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))

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))),

CameraDirection=[-1,0,0.5])