## `plot::Curve3d` -- graphical primitive for a three-dimensional curve

#### Introduction

`plot::Curve3d``([x, y, z], t = a..b)` represents a plot of the curve defined by t -> (x(t); y(t); z(t)) with t in [a,b].

#### Creating Elements

`plot::Curve3d([x, y, z], t = a..b <, option1, option2...>)`

#### Parameters

`x, y, z` | - | arithmetical expressions in `t`
`t` | - | identifier
`a, b` | - | arithmetical expressions
`option1, option2, ...` | - | plot option(s) of the form `OptionName = value`

#### Related Domains

`plot::Curve2d`, `plot::Function3d`, `plot::Surface3d`, `RGB`

#### Related Functions

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

#### Details

- Objects generated by `plot::Curve3d` represent graphical primitives for three-dimensional curves that can be displayed via `plot(...)`, or used with other graphical primitives of the `plot` library.
- An object of `plot::Curve3d` has the 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`]` | `[100]`
  LineStyle | SolidLines, DashedLines | SolidLines
  LineWidth | positive integers | `1`
  PointStyle | Circles, FilledCircles, FilledSquares, Squares | FilledSquares
  PointWidth | positive integers | `30`
  Smoothness | `[`integer`]` | `[0]`
  Style | `[Points]`, `[Lines]`, `[LinesPoints]`, `[Impulses]` | `[Lines]`
  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 curve primitive:

  attribute | meaning | properties
  `options` | A table of plot options of the curve 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 curve 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 curve 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
  `range` | The parameter of the curve and its range. The initial value is the parameter `t = a..b`. | read/write
  `refreshPlotdata` | A boolean value which signals whether the plot data of the curve primitive must be (re-)build with the method `"getPlotdata"` (see below). If its value is `FALSE`, then the plot data of the curve 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 curve. The initial value is the parameter `[x, y, z]`. | read/write

  See example 3.

#### Operands

An object of `plot::Curve3d` consists of two operands. The first operand is the term of the curve specified as the list `[x, y, z]`. The second one is the parameter of the curve and its range in the form `t = a..b`.

#### Important Operations

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

Via `curve[1] := [new_x, new_y, new_z]` or `curve::term := [new_x, new_y, new_z]`, the term of a curve primitive 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 `curve` is such an object, then `curve::Color := RGB::Red` changes the color of the curve primitive `curve` to red.

#### Result of Evaluation

Evaluating an object of type `plot::Curve3d` returns itself.

#### Function Call

Calling an object of `plot::Curve3d` 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 curve primitives and their default values. Each entry has the form `OptionName = default_value`.

  When an object of the domain `plot::Curve3d` 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 curve primitives and their default values. See example 2.

  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 curves.

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

- `_index(dom curve, positive integer i)`
- - Returns the `i`th operand of `curve`. See ``Operands'' above for a description of the operands of `curve`. If `i` is greater than 2, then `FAIL` is returned.
  - This method overloads the system function `_index`, i.e., one may use it in the form `curve[i]`, or in functional notation `_index(curve, i)`.

#### Method dimension: dimension of a curve primitive

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

#### Method getPlotdata: create the plot data of a curve primitive

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

    For example, with `s := plot::Curve3d::getPlotdata(curve)` the call `plot3d(s[1])` gives a plot of `curve`.

  - Only those plot options will be included in the plot data of the curve, 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 curves is used when plotting the object.
  - The result is stored as the value of the attribute `plotdata` of `curve`.
  - A call of this method sets the value of the attribute `refreshPlotdata` of `curve` 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 curve primitive

- `nops(dom curve)`
- - Returns the integer 2.
  - This method overloads the system function `nops`, i.e., one may use it in the form `nops(curve)`.

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

- `op(dom curve, positive integer i)`
- - Returns the `i`th operand of `curve`. See ``Operands'' above for a description of the operands of `curve`. If `i` is greater than 2, then `FAIL` is returned.
  - This method overloads the system function `op`, i.e., one may use it in the form `op(curve, i)`.

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

- `set_index(dom curve, positive integer i, any val)`
- - Replaces the `i`th operand of `curve` by the value `val`. See ``Operands'' above for a description of the operands of `curve`.
  - If `i` is greater than 2, 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 `curve`.
  - A call of this method sets the value of the attribute `refreshPlotdata` of `curve` to `TRUE`.

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

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

#### Method checkOption: check a plot option

- `checkOption(equation OptionName = value)`
- - This method checks whether `OptionName` is an available plot option for curve 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 curve primitive

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

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

- `modify(dom curve, equation(s) Name1 = value1...)`
- - Creates a copy of the object `curve` 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::Curve3d`. 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 `curve` to `TRUE`.
  - This method is called from the function `plot::modify`.

#### Method print: print a curve primitive

- `print(dom curve)`
- - This method returns an unevaluated expression of the form `plot::Curve3d([x, y, z], t = a..b)`. It is used to print objects of `plot::Curve3d` to the screen.
  - See the system function `print` for details.

#### Example 1

The following call returns an object representing the graph of a spiral:

    >> c1 := plot::Curve3d([sin(t), cos(t), t], t = 0..2*PI)

                  plot::Curve3d([sin(t), cos(t), t], t = 0..2 PI)

To plot this curve in a graphical scene, call `plot`:

    >> plot(c1)

Plot options of the curve can be given as additional parameters in the creating call, such as increasing the width of the lines of a graph and plotting the graph in green color:

    >> c2 := plot::Curve3d([sin(t), cos(t), t], t = 0..2*PI,
         Color = RGB::Green, LineWidth = 50
       )

                  plot::Curve3d([sin(t), cos(t), t], t = 0..2 PI)

    >> plot(c2)

To change default values of some scene options, pass the scene options to the function `plot` as additional arguments. For example, change the scaling of the plot and change the number of ticks on the axes:

    >> plot(c2, Scaling = Constrained, Ticks = [3, 3, 10])

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

#### Example 2

If a curve 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 curve:

    >> c1 := plot::Curve3d([t*sin(t), t*cos(t), t], t = 0..4*PI)

                plot::Curve3d([t sin(t), t cos(t), t], t = 0..4 PI)

We create a copy of this curve, change some plot options of the copied object, and plot both objects in a graphical scene:

    >> c2 := plot::copy(c1):
       c2::Style := [Points]: c2::Grid := [30]:
       plot(c1, c2)

Plot options, which are explicitely set for a curve primitive, are stored under the attribute `options` and can be read with the slot operator `::`. The plot options for the first created object are:

    >> c1::options

                                  table(
                                    Grid = [100]
                                  )

These are default values of some plot options of two-dimensional curve primitives, defined by the entry `"defaultOptions"` of the domain `plot::Curve3d`:

    >> plot::Curve3d::defaultOptions

                                  table(
                                    Grid = [100]
                                  )

When the plot data of a curve primitive is created (calling the method `"getPlotdata"`), only those plot options are used that are contained in the table `c1::options`. Here these are:

    >> plot::Curve3d::getPlotdata(c1)

          [[Mode = Curve, [t sin(t), t cos(t), t],

             t = [0.0, 12.56637061], Grid = [100]]]

This means that for any other available plot option not contained in the table `c1::options`, the default value is either set by `plot::Scene`, if the option also exists as a scene option, or internally set by the function `plot2d` when plotting the object.

The curve primitive `c2` contains the following options:

    >> c2::options

                                table(
                                  Style = [Points],
                                  Grid = [30]
                                )

As you see, the option Style was added to this table, and the default value of the option Grid was replaced by the new value `[30]`. Use `delete` to remove plot options:

    >> delete c2::options[Style]: c2::options

                                   table(
                                     Grid = [30]
                                   )

#### Example 3

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

In the last example, we already used the attribute `options`, which stores plot options defined individually for a curve primitive and which overrides the corresponding default values set by MuPAD.

The attribute `term` holds the term of a curve primitive. For example, if `curve` is an object of `plot::Curve3d` such as:

    >> c := plot::Curve3d([cos(t)*sin(t), t, t], t = -5..5, Color = RGB::Blue)

                  plot::Curve3d([cos(t) sin(t), t, t], t = -5..5)

then `curve::term` returns the list:

    >> c::term

                               [cos(t) sin(t), t, t]

We plot this curve:

    >> plot(c)

Because the attribute `term` has the ``write'' property, you can change the value of this attribute as follows:

    >> c::term:= [t, t, sin(t)*cos(t)]: plot(c)

The value of `term` must be a list of three arithmetical expressions, otherwise an error message is issued.

#### Changes

- new function

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