## `plot::Surface3d` -- graphical primitive for a three-dimensional surface plot

#### Introduction

`plot::Surface3d``([x, y, z], u = a..b, v = c..d)` represents a plot of the surface defined by (u,v) -> (x(u,v); y(u,v); z(u,v)) with (u,v) in [a,b] x [c,d].

#### Creating Elements

`plot::Surface3d([x, y, z], u = a..b, v = c..d <, option1, option2...>)`

#### Parameters

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

#### Related Domains

`plot::Curve3d`, `plot::Function3d`, `RGB`

#### Related Functions

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

#### Details

- Objects generated by `plot::Surface3d` represent graphical primitives for three-dimensional surfaces that can be displayed via `plot(...)`, or used with other graphical primitives of the `plot` library.
- An object of `plot::Surface3d` 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 call of `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`.

  The following attributes are available for a surface primitive:

  attribute | values | properties
  `options` | A table of plot options of the surface 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 surface 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 surface 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 parameter of the surface and its range in the form `ident1 = a..b`. The initial value is the parameter `u = a..b`. | read/write
  `range2` | The second parameter of the surface and its range in the form `ident2 = c..d`. The initial value is the parameter `v = c..d`. | read/write
  `refreshPlotdata` | A boolean value which signals whether the plot data of the surface primitive must be (re-)build with the method `"getPlotdata"` (see below). If its value is `FALSE`, then the plot data of the surface 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 surface in form of a list of three arithmetical expressions. The initial value is the parameter `[x, y, z]`. | read/write

  See the examples of the help page of `plot::Function3d` about working with attributes.

#### Operands

An object of `plot::Surface3d` consists of three operands. The first operand is the list `[x, y, z]`. The second operand is the first parameter of the surface and its range in the form `u = a..b`, and the third one is the second parameter of the surface and its range in the form `v = c..d`.

#### Important Operations

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

Via `surface[1] := [x_new, y_new, z_new]` or `surface::term := [x_new, y_new, z_new`, the parametrization of a surface plot can be changed.

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 `surface` is such an object, then `surface::Color := RGB::Red` changes the color of the surface primitive `surface` to red.

#### Result of Evaluation

Evaluating an object of type `plot::Surface3d` returns itself.

#### Function Call

Calling an object of `plot::Surface3d` 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 surface primitives and their default values. Each entry has the form `OptionName = default_value`.

  When an object of the domain `plot::Surface3d` 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 the examples of the help page of `plot::Function3d`.

  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 surfaces.

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

- `_index(dom surface, positive integer i)`
- - Returns the `i`th operand of `surface`. See ``Operands'' above for a description of the operands of `surface`. 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 `surface[i]`, or in functional notation `_index(surface, i)`.

#### Method dimension: dimension of a surface primitive

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

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

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

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

  - Only those plot options will be included in the plot data of the surface, 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 `surface`.
  - A call of this method sets the value of the attribute `refreshPlotdata` of `surface` 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 surface primitive

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

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

- `op(dom surface, positive integer i)`
- - Returns the `i`th operand of `function`. See ``Operands'' above for a description of the operands of `surface`. 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(surface, i)`.

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

- `set_index(dom surface, positive integer i, any val)`
- - Replaces the `i`th operand of `surface` by the value `val`. See ``Operands'' above for a description of the operands of `surface`.
  - 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 `surface`.
  - A call of this method sets the value of the attribute `refreshPlotdata` of `surface` to `TRUE`.

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

- `slot(dom surface, string slotname)`
- - Reads the value of the slot `slotname` of `surface`. `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 `surface::slotname_id` (here, `slotname_id` must be the identifier corresponding to the string `slotname`), or in functional notation `slot(surface, slotname)`.
- `slot(dom surface, 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 `surface::slotname_id := val` (here, `slotname_id` must be the identifier corresponding to the string `slotname`), or in functional notation `slot(surface, slotname, val)`.
  - The value of the attribute `refreshPlotdata` of `surface` 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 surface 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 surface primitive

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

#### Method modify: modify a copy of a surface primitive

- `modify(dom surface, equation(s) Name1 = value1...)`
- - Creates a copy of the object `surface` 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::Surface3d`. 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 `surface` to `TRUE`.
  - This method is called from the function `plot::modify`.

#### Method print: print a surface primitive

- `print(dom surface)`
- - This method returns an unevaluated expression of the form `plot::Surface3d([x, y, z], u = a..b, v = c..d)`. It is used to print objects of `plot::Surface3d` to the screen.
  - See the system function `print` for details.

#### Example 1

The following call returns an object representing a plot of the surface defined by (u,v) -> (u, sin(v), cos(v)) with (u,v) in [0,2*PI] x [-1,1]:

    >> s1 := plot::Surface3d(
         [u, sin(v), cos(v)], u = 0..2*PI, v = -1..1
       )

           plot::Surface3d([u, sin(v), cos(v)], u = 0..2 PI, v = -1..1)

To plot this surface in a graphical scene, call:

    >> plot(s1)

Plot options of the surface can be given as additional parameters in the creating call, such as displaying the graph as an opaque object together with the parameter lines:

    >> s2 := plot::Surface3d(
         [u, sin(v), cos(v)], u = 0..2*PI, v = -1..1,
         Style = [HiddenLine, Mesh]
       )

           plot::Surface3d([u, sin(v), cos(v)], u = 0..2 PI, v = -1..1)

    >> plot(s2)

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(s2, Axes = Corner)

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

Please refer to the examples of the help page of `plot::Function3d` about working with options and attributes.

#### Changes

- new function

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