## `plot::Function3d` -- graphical primitive for a three-dimensional graph of a function

#### Introduction

`plot::Function3d``(f, x = a..b, y = c..d)` represents a plot of the function f(x,y) with (x,y) in [a,b] x [c,d].

#### Creating Elements

`plot::Function3d(f, x = a..b, y = c..d <, option1, option2...>)`

#### Parameters

`f` | - | arithmetical expression in `x` and `y`
`x, y` | - | identifiers
`a, b, c, d` | - | arithmetical expressions
`option1, option2, ...` | - | plot option(s) of the form `OptionName = value`

#### Related Domains

`plot::Curve2d`, `plot::Function2d`, `plot::Surface3d`, `RGB`

#### Related Functions

`plot`, `plot3d`, `plotfunc3d`, `plot::copy`

#### Details

- Objects generated by `plot::Function3d` represent graphical primitives for two-dimensional graphs of functions that can be displayed via `plot(...)`, or used with other graphical primitives of the `plot` library.
- Note that `plotfunc3d` is also used to plot three-dimensional graphs of functions. But in contrast to `plot::Function3d`, it does not return the graph in form of a graphical object but displays the graph immediatly after executing the command.

  The main advantage of using `plot::Function3d` is, that you get the representation of the graph as an object. It can be manipulated afterwards or combined easily with other graphical primitives, such as polygons, curves or surfaces, to a common graphical scene.

  See example 2 below.

- An object of `plot::Function3d` has type `"graphprim"`, i.e., if `o` is such an object, then the result of `type(o)` is the string `"graphprim"`.
- Options `option1`, `option2`... are specified by equations `OptionName = value`. The following table gives an overview of the available options:

  `OptionName` | admissible values | default value
  Color | `[Flat]`, `[Flat, [r,g,b]]`, `[Height]`, `[Height, [r,g,b], [R,G,B]]`, `[Function, f]` | `[Height]`
  Grid | `[`integer`]` | `[20,20]`
  LineStyle | SolidLines, DashedLines | SolidLines
  LineWidth | positive integers | `1`
  PointStyle | Circles, FilledCircles, FilledSquares, Squares | FilledSquares
  PointWidth | positive integers | `30`
  Smoothness | `[`integer`]` | `[0]`
  Style | `[Points]` | `[ColorPatches,`
  | `[WireFrame, Mesh]` | `AndMesh]`
  | `[WireFrame, ULine]` |
  | `[WireFrame, VLine]` |
  | `[HiddenLine, Mesh]` |
  | `[HiddenLine, ULine]` |
  | `[HiddenLine, VLine]` |
  | `[ColorPatches, Only]` |
  | `[ColorPatches, AndMesh]` |
  | `[ColorPatches, AndULine]` |
  | `[ColorPatches, AndVLine]` |
  | `[Transparent, Only]` |
  | `[Transparent, AndMesh]` |
  | `[Transparent, AndULine]` |
  | `[Transparent, AndVLine]` |
  Title | strings |
  TitlePosition | `[x, y]` |

  See `plot3d` for further details on each option.

- Scene options for the parameters `option1`, `option2`... are not allowed! One may pass scene options to the function `plot`, or use `plot::Scene` to create an object representing a graphical scene. Cf. example 1.

- A so-called attribute of a graphical primitive is a named entry of the object which can be accessed via the slot operator `::`.

  Each attribute has the property ``read'', i.e., the value of an attribute `attr` of a graphical primitive `o` can be read with `o::attr`. If the attribute also has the ``write'' property, then the value of the attribute can be changed with `o::attr := new_value`. See example 4.

  The following attributes are available for a function primitive:

  attribute | meaning | properties
  `options` | A table of plot options of the function primitive. Note that if you change the value of this attribute, the entries of the assigned table are not checked to be valid plot options for function primitives. Invalid entries lead to runtime errors. The initial value of this attribute is the table stored under the domain entry `"defaultOptions"`, where such options are replaced and added, respectively, which are given with the parameters `option1`, `option2`... of the creating call. | read/write
  `plotdata` | List of the plot data of the function primitive in a `plot3d` conforming syntax (see the method `"getPlotdata"` below). Note that the value of this attribute should only be used if the attribute `refreshPlotdata` has the value `FALSE` (see below). | read
  `range1` | The first variable of the function and its range in the form `ident1 = a..b`. The initial value is the parameter `x = a..b`. | read/write
  `range2` | The second variable of the function and its range in the form `ident2 = c..d`. The initial value is the parameter `y = c..d`. | read/write
  `refreshPlotdata` | A boolean value which signals whether the plot data of the function primitive must be (re-)build with the method `"getPlotdata"` (see below). If its value is `FALSE`, then the plot data of the function primitive is stored in the attribute `plotdata`. The initial value is `TRUE`. See the help page of `plot::Curve2d` for an example. | read/write
  `term` | The term of the function. The initial value is the parameter `f`. | read/write

#### Operands

An object of `plot::Function3d` consists of three operands. The first operand is the term `f` of the graph. The second operand is the first variable of the function and its range in the form `x = a..b`, and the third one is the second variable of `f` and its range in the form `y = c..d`.

#### Important Operations

Operands of a function primitive can be accessed either using the system function `op`, the index operator `[ ]`, or using the attributes described above. For example, if `function` is such an object, then the calls `op(function,1)`, `function[1]` and `function::term` return the expression `f`.

Via `function[1] := g` or `function::term := g`, the term of a function plot can be changed to the value `g`.

See the methods `"op"`, `"_index"`, `"set_index"` and `"slot"` below.

Use the slot operator `::` to get or set plot options of such objects afterwards, i.e., when they have been created. For example, if `function` is such an object, then `function::Color := RGB::Red` changes the color of the function primitive `function` to red.

#### Result of Evaluation

Evaluating an object of type `plot::Function3d` returns itself.

#### Function Call

Calling an object of `plot::Function3d` as a function yields the object itself, regardless of the arguments. The arguments are not evaluated.

#### Entries

defaultOptions

  is a table of plot options for function primitives and their default values. Each entry has the form `OptionName = default_value`.

  When an object of the domain `plot::Function3d` is created, then a copy of this table is stored under the attribute `options` (see the table of attributes above), where those options are added and replaced, respectively, which are given by the (optional) parameters `option1`, `option2`... of the creating call (see ``Creating Elements'' above).

  Plot options, which are not contained in the table stored under the attribute `options` will not be included in the plot data of the object created by the method `"getPlotdata"` (see below).

  For those options, the corresponding default value either is set by a graphical scene, if the option also exists as a scene option (such as the option PointWidth), or it is internally set by the function `plot3d` which is used to plot the object. See the table of plot options above, which gives a summary of the available plot options for function primitives and their default values. See example 3.

  To change the default value of some plot options, the option name and its default value may be added to the table `"defaultOptions"`, or replaced by a new value, respectively.

optionNames

  is a set of the available option names for plots of three-dimensional graphs of functions.

#### Method _index: indexed access to the operands of a function primitive

- `_index(dom function, positive integer i)`
- - Returns the `i`th operand of `function`. See ``Operands'' above for a description of the operands of `function`. If `i` is greater than 3, then `FAIL` is returned.
  - This method overloads the system function `_index`, i.e., one may use it in the form `function[i]`, or in functional notation `_index(function, i)`.

#### Method dimension: dimension of a function primitive

- `dimension(dom function)`
- - Returns the integer 3.

#### Method getPlotdata: the plot data of a function primitive

- `getPlotdata(dom function)`
- - Returns a list of an inner list, where the inner list is a plot description of `function` in a `plot3d` conforming syntax, i.e., it has the form `[Mode = Surface, [...], ...]`.

    For example, with `s := plot::Function3d::getPlotdata(function)` the call `plot3d(s[1])` gives a plot of `function`.

  - Only those plot options will be included in the plot data of the function, that are contained in the table stored under the attribute `options` (see the table of attributes above). For any other plot option not contained in this table, the corresponding default value set by the function `plot3d` for surfaces is used when plotting the object.
  - The result is stored as the value of the attribute `plotdata` of `function`.
  - A call of this method sets the value of the attribute `refreshPlotdata` of `function` to `FALSE`.
  - This method is called from `plot::Scene` to build the plot data of the graphical scene.

#### Method nops: number of operands of a function primitive

- `nops(dom function)`
- - Returns the integer 3.
  - This method overloads the system function `nops`, i.e., one may use it in the form `nops(function)`.

#### Method op: extract operands of a function primitive

- `op(dom function, positive integer i)`
- - Returns the `i`th operand of `function`. See ``Operands'' above for a description of the operands of `function`. If `i` is greater than 3, then `FAIL` is returned.
  - This method overloads the system function `op`, i.e., one may use it in the form `op(function, i)`.

#### Method set_index: set operands of a function primitive

- `set_index(dom function, positive integer i, any val)`
- - Replaces the `i`th operand of `function` by the value `val`. See ``Operands'' above for a description of the operands of `function`.
  - If `i` is greater than 3, or if `val` is not an admissible value for the `i`th operand, then a warning message is issued. In this case the call of this method has no effect on the object `function`.
  - A call of this method sets the value of the attribute `refreshPlotdata` of `function` to `TRUE`.

#### Method slot: read and write attributes and plot options

- `slot(dom function, string slotname)`
- - Reads the value of the slot `slotname` of `function`. `slotname` may either be the name of an attribute or the name of a plot option. See the tables of available plot options and attributes above.
  - If `slotname` is the name of a plot option, but the option is not contained in the table stored under the attribute `options`, then `FAIL` is returned.

    If `slotname` is an invalid attribute or option, then an error message is issued.

  - This method overloads the system function `slot`, i.e., one may use it in the form `function::slotname_id` (here, `slotname_id` must be the identifier corresponding to the string `slotname`), or in functional notation `slot(function, slotname)`.
- `slot(dom function, string slotname, any val)`
- - Changes the value of the attribute or option with the name `slotname` to the value `val`.
  - If there is no attribute or option with the name `slotname`, or if `val` is not an admissible value for `slotname`, then a warning message is issued. In this case, the value of `slotname` remains unchanged.
  - This method overloads the system function `slot`, i.e., one may use it in the form `function::slotname_id := val` (here, `slotname_id` must be the identifier corresponding to the string `slotname`), or in functional notation `slot(function, slotname, val)`.
  - The value of the attribute `refreshPlotdata` of `function` is set to `TRUE`.

#### Method checkOption: check a plot option

- `checkOption(equation OptionName = value)`
- - This method checks whether `OptionName` is a known plot option for function primitives (see the table of available plot options above), and `value` is an admissible value for this option.
  - If both is correct, then the list `[TRUE, OptionName, newValue]` is returned. Note that the value of the option could have been converted into an admissible format. Thus, `newValue` must be used as the value of the option `OptionName` instead of `value`.
  - Otherwise, the list `[FALSE, error_msg]` is returned. The string `error_msg` is a description of the located problem, which can be passed, for example, to the system function `error` to raise a user-specified exception.

#### Method copy: create a copy of a function primitive

- `copy(dom function)`
- - Returns a copy of the object `function`.
  - This method is called from the function `plot::copy`. See its help page for details.

#### Method modify: modify a copy of a function plot

- `modify(dom function, equation(s) Name1 = value1...)`
- - Creates a copy of the object `function` and changes the slots `Name1`... of this copy to the new values `value1`...
  - The identifiers `Name1`... must be names of attributes or plot options of the domain `plot::Function3d`. Otherwise a warning message is issued, and the slot remains unchanged. Also, if one of the values `value1`... is not an admissible value for the corresponding attribute or plot option, respectively, the change of the slot is ignored.

    See the tables of available options and attributes above.

  - A call of this method sets the value of the attribute `refreshPlotdata` of the copy of `function` to `TRUE`.
  - This method is called from the function `plot::modify`.

#### Method print: print a function primitive

- `print(dom function)`
- - This method returns an unevaluated expression of the form `plot::Function3d(f, x = a..b, y = c..d)`. It is used to print objects of `plot::Function3d` to the screen.
  - See the system function `print` for details.

#### Example 1

The following call returns an object representing the graph of the function (x,y) -> sin(x*y) for x in [0, 2*PI] and y in [-PI, PI]:

    >> f1 := plot::Function3d(sin(x*y), x = 0..2*PI, y = -PI..PI)

               plot::Function3d(sin(x y), x = 0..2 PI, y = -PI..PI)

To plot this function in a graphical scene, call:

    >> plot(f1)

Plot options of the surface can be given as additional parameters in the creating call, such as plotting the graph in blue color:

    >> f2 := plot::Function3d(sin(x*y), x = 0..2*PI, y = -PI..PI,
         Color = [Flat, RGB::Blue])

               plot::Function3d(sin(x y), x = 0..2 PI, y = -PI..PI)

    >> plot(f2)

To change default values of some scene options, pass the scene options to the call of `plot` as additional arguments. For example, to change the style of the axes:

    >> plot(f1, Axes = Corner)

See the help page of `plot::Scene` for available scene options.

#### Example 2

We want to display a graph of the function (x,y) -> sin(x) and the curve defined by t -> (t,sin(t),sin(t)). We start by creating the two different graphical primitives as follows:

    >> f := plot::Function3d(sin(x), x = -PI..PI, y = -5..5);
       c := plot::Curve3d(
         [t, sin(t), sin(t)], t = -PI..PI,
         Color = RGB::Blue, LineWidth = 20
       )

                 plot::Function3d(sin(x), x = -PI..PI, y = -5..5)

                  plot::Curve3d([t, sin(t), sin(t)], t = -PI..PI)

To plot the scene of these objects, pass them as parameters to the function `plot`:

    >> plot(f, c)

#### Example 3

If a function primitive is created, values of some plot options of the created object can be read, or replaced by new values.

To illustrate this, we create the following function:

    >> f := plot::Function3d(sin(x)*cos(y), x = 0..2*PI, y = 0..2*PI)

             plot::Function3d(cos(y) sin(x), x = 0..2 PI, y = 0..2 PI)

and plot the graph:

    >> plot(f)

The graph is drawn in color patches, together with the parameter lines of the two parameters of the surface. To change the style of the graph, use the option Style. For example, to display the graph as an opaque object together with the parameter lines, call:

    >> f::Style:= [HiddenLine, Mesh]: plot(f)

and plot the graph:

    >> plot(f)

The options that are set for an object are stored under the attribute `options`. You can read the value of this attribute as follows:

    >> f::options

                           table(
                             Style = [HiddenLine, Mesh],
                             Grid = [20, 20]
                           )

If an option is not contained in this table, then its value is set either by `plot::Scene`, if the option also exists as a scene option, or internally set by the function `plot3d` when plotting the surface.

For example, the option Color is not contained in this table. If you try to read its value, you get the following result:

    >> f::Color

                                       FAIL

The default value of this option is the list `[Height]`, set by the function `plot3d` (see the table of options above). The default colors are taken from the preferences of the MuPAD's graphic tool VCam.

To override the default value of this option for the object `f`, enter:

    >> f::Color:= [Flat, RGB::Red]: plot(f)

If we now take a look at the table stored under the attribute `options`, we get:

    >> f::options

                        table(
                          Color = [Flat, [1.0, 0.0, 0.0]],
                          Style = [HiddenLine, Mesh],
                          Grid = [20, 20]
                        )

To delete some plot options set for a graphical primitive, call:

    >> delete f::options[Style]: f::options

                        table(
                          Color = [Flat, [1.0, 0.0, 0.0]],
                          Grid = [20, 20]
                        )

If we now plot the object `f`, the default value of the option Style is used, which is the list `[ColorPatches, AndMesh]`:

    >> plot(f)

#### Example 4

This example illustrates how to read and write attributes of function primitives (see the table of available attributes in ``Details'' above).

In the previous example we already introduced the attribute `options`. The attributes `range1` and `range2`, for example, hold the ranges for the variables of the function graph. We give an example:

    >> f := plot::Function3d(x^2 + y^2, x = -5..5,y = -5..5)

                                    2    2
                  plot::Function3d(x  + y , x = -5..5, y = -5..5)

We can read the ranges of the variables:

    >> f::range1, f::range2

                               x = -5..5, y = -5..5

Because these attributes have the ``write'' property, we can also change their values as follows:

    >> f::range1:= x = -10..10: f::range2:= y = -10..10:
       plot(f)

#### Changes

- new function

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