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plot::Implicit2d

Contour lines of a function from R^2 to R

Use only in the MuPAD Notebook Interface.

This functionality does not run in MATLAB.

Syntax

plot::Implicit2d(f, x = xmin .. xmax, y = ymin .. ymax, <a = amin .. amax>, options)

Description

plot::Implicit2d(f(x, y), x = `x_{min}`..`x_{max}` , y = `y_{min}`..`y_{max}` ) plots the curves where the smooth function f is zero.

plot::Implicit2d(f, x = `x_{min}`..`x_{max}` , y = `y_{min}`..`y_{max}` ) plots the zeroes of f in the given range, i.e., the set .

plot::Implicit2d assumes that f is regular almost everywhere on this curve, which means that f must be differentiable and at least one of its partial derivatives must be nonzero.

To plot other contours than zeroes, use the option Contours.

Attributes

AttributePurposeDefault Value
AffectViewingBoxinfluence of objects on the ViewingBox of a sceneTRUE
AntiAliasedantialiased lines and points?TRUE
Colorthe main colorRGB::Blue
Contoursthe contours of an implicit function[0]
Framesthe number of frames in an animation50
Functionfunction expression or procedure 
Legendmakes a legend entry 
LegendTextshort explanatory text for legend 
LegendEntryadd this object to the legend?TRUE
LineColorcolor of linesRGB::Blue
LineWidthwidth of lines0.35
LineColor2color of linesRGB::DeepPink
LineStylesolid, dashed or dotted lines?Solid
LinesVisiblevisibility of linesTRUE
LineColorTypeline coloring typesFlat
LineColorFunctionfunctional line coloring 
LineColorDirectionthe direction of color transitions on lines[0, 1]
LineColorDirectionXx-component of the direction of color transitions on lines0
LineColorDirectionYy-component of the direction of color transitions on lines1
Meshnumber of sample points[11, 11]
Namethe name of a plot object (for browser and legend) 
ParameterEndend value of the animation parameter 
ParameterNamename of the animation parameter 
ParameterBegininitial value of the animation parameter 
ParameterRangerange of the animation parameter 
TimeEndend time of the animation10.0
TimeBeginstart time of the animation0.0
TimeRangethe real time span of an animation0.0 .. 10.0
Titleobject title 
TitleFontfont of object titles[" sans-serif ", 11]
TitlePositionposition of object titles 
TitleAlignmenthorizontal alignment of titles w.r.t. their coordinatesCenter
TitlePositionXposition of object titles, x component 
TitlePositionYposition of object titles, y component 
VisiblevisibilityTRUE
VisibleAfterobject visible after this time value 
VisibleBeforeobject visible until this time value 
VisibleFromToobject visible during this time range 
VisibleAfterEndobject visible after its animation time ended?TRUE
VisibleBeforeBeginobject visible before its animation time starts?TRUE
XMaxfinal value of parameter "x" 
XMeshnumber of sample points for parameter "x"11
XMininitial value of parameter "x" 
XNamename of parameter "x" 
XRangerange of parameter "x" 
YMaxfinal value of parameter "y" 
YMeshnumber of sample points for parameter "y"11
YMininitial value of parameter "y" 
YNamename of parameter "y" 
YRangerange of parameter "y" 

Examples

Example 1

It is well-known that a circle can be described as :

plot(plot::Implicit2d(x^2+y^2-1, x = -1..1, y = -1..1))

Note that plot::Implicit2d uses the given range completely, even if there is nothing to plot at a border:

plot(plot::Implicit2d(x^2+y^2-1, x = -2..2, y = -2..2))

Example 2

plot::Implicit2d handles functions which are not regular at isolated points on the contours:

plot(plot::Implicit2d((x-y)*(x+y), x = -1..1, y = -1..1))

However, it fails if the function is singular on more than isolated points:

plot(plot::Implicit2d(0, x = -1..1, y = -1..1))

Example 3

We plot some of the elliptic curves y2 = x3 + 4 x + c:

plot(plot::Implicit2d(y^2 - x^3 + 4*x, x = -3..3, y = -4..4, 
                      Contours = [c $ c = -3..6]))

Example 4

Like most graphical objects, plot::Implicit2d can be animated easily:

plot(plot::Implicit2d(x^2 - y^2 = (x - a*y)*(x^2 + y^2),
                      x = -2..2, y = -2..2, a = -2..2))

Parameters

f

A real-valued expression or an equation in x, y, and possibly the animation parameter.

f is equivalent to the attribute Function.

x, y

identifiers.

x, y are equivalent to the attributes XName, YName.

xmin .. xmax, ymin .. ymax

Real-valued expressions, possibly in the animation parameter. The image is plotted with x in the range xminxxmax and yminyymax.

xmin .. xmax, ymin .. ymax are equivalent to the attributes XRange, XMin, XMax, YRange, YMin, YMax.

a

Animation parameter, specified as a = amin..amax, where amin is the initial parameter value, and amax is the final parameter value.

Algorithms

plot::Implicit2d uses a curve tracking method: It first generates starting points on the curve and then uses a predictor-corrector method to follow the curve thus found in both directions, using the implicit function theorem.

See Also

MuPAD Functions

MuPAD Graphical Primitives

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