Create a parametric curve feature python object in FreeCAD
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Install using the Addon Manager in the Tools menu from the Macros tab.
This macro is based on the FreeCAD macro 3D Parametric Curve by Lucio Gomez and Laurent Despeyroux. https://wiki.freecadweb.org/Macro_3D_Parametric_Curve The difference is this macro creates a feature python object that also has spreadsheet and JSON integration capabilities.
Run the Make_Parametric_Curve.py file as a macro to create the feature python object. Then modify the properties to setup the formula to use. It comes with a default formula to create a curve pictured above
You can also, in addition to modifying the properties directly, elect to use a spreadsheet to hold the formulas or read them in from a previously created text file in JSON format. A spreadsheet needs to have the following aliases defined:
- a_cell -- String that contains the a formula.
- b_cell -- String that contains the b formula.
- c_cell -- String that contains the c formula.
- [optional] d1,d2,d3,d4, etc. -- d formula held as string list in Property "d".
- X -- String that contains the X formula.
- Y -- String that contains the Y formula.
- Z -- String that contains the Z formula.
- t_min -- Floating point that contains the starting value of t.
- t_max -- Floating point that contains the ending value of t.
- interval -- Floating point that contains the step value for t as it goes from t_min to t_max.
The feature python object can add these aliases for you to an existing spreadsheet or a new spreadsheet that it can also create. If there is no spreadsheet linked in the Spreadsheet property and you set the Update Spreadsheet property to True, the fp object will create the new spreadsheet, link it to itself, and add the aliases to the spreadsheet. It also sets the Use Spreadsheet property to True. If you manually select a spreadsheet by editing the Spreadsheet property to link to an existing spreadsheet, and that spreadsheet does not contain the required aliases you will be asked whether to create them or not. Doing this will clobber any existing data in cells A1 through B9.
You can save the fp object's state (the Equation Group and JSON group properties) to the JSON file, and then later load them back from the JSON file in this or in another Parametric Curve object. With JSON files you are able to have more than one formula per file, unlike spreadsheets, which are limited to one formula per spreadsheet. If you have many formulas you like to use you can save them all to a single JSON file.
Sample JSON file:
{"coil": {"a": "6", "b": "1", "c": "20", "d": [],
"X": "(a + b * cos(c*t))*cos(t)", "Y": "(a + b * cos(c*t))*sin(t)", "Z": "b*sin(c*t)",
"t_min": "0.0", "t_max": "6.283185307179586", "interval": "0.02"},
"holesaw": {"a": "12", "b": "6", "c": "5.000000", "d": [],
"X": "cos(t)*a", "Y": "sin(t)*a", "Z": "c*sin(b*t)",
"t_min": "0.0", "t_max": "6.283185307179586", "interval": "0.1"},
"amoeba": {"a": "36.3", "b": "12 #number of elements", "c": "1.5", "d": ["(a+c*sin(b*t))"],
"X": "cos(t)*d1", "Y": "sin(t)*d1", "Z": "0",
"t_min": "0.0", "t_max": "6.283185307179586", "interval": "0.1"},
"sawtooth": {"a": "40", "b": "10", "c": "0.8", "d": ["mod(b*t, 1)", "10"],
"X": "a*cos(2*pi*t)", "Y": "a*sin(2*pi*t)",
"Z": "d2*(lt(d1,c)*d1/c + gte(d1,c)*(1-d1)/(1-c))",
"t_min": "0.0", "t_max": "1.0", "interval": "0.01"}}
You can manually edit the JSON files in a text editor, but there are a few things to keep in mind.
- Make sure to follow the formatting properly or else the fp object will not be able to read the file
- Values for t_min, t_max, and interval must evaluate directly to float. They can't reference other values, such as a or X.
- All values must be strings (in quotes)
- Values may contain comments, 2 styles are supported:
** #comment (everything following # is ignore)
** {braced comment -- all things inside braces are ignored}
The Feature Python object (ParametricCurve) has a number of properties separated into different groupings.
The property sets the curve to either closed or not closed. If it's set to True the wire will close itself (connect the first vertex to the last vertex). This is sometimes required if you are to use the curve to create a solid with a Pad, Extrude, etc. THis was set to True by default, but since 0.2022.02.13 it is now False by default. This is because OCC can sometimes fail if this is True and the first and last points are coincident already.
You can add a document object to this property to create a dependency between that object and the ParametricCurve object. The ParametricCurve object will recompute when the Dependencies objects recompute. The purpose of this is for use with the fc(expr) evaluation function described below in the Functions section.
Choose your shape type here. Options are: BSpline, Polygon, Points.
Fixes a bug by adding one more iteration to the loop. But if this causes a problem with an existing model, this can be set to False to keep the current (buggy) behavior.
This is a list of vectors used to create the output shape. Note: if Shape Type is "Polygon" and Closed = True, the first point is also copied to the end of the points list.
This gives the version used to create this object (not necessarily the same as currently installed.) It is in the form of the date of last modification, e.g. 2021.08.27.
Gives you the Continuity property of the Curve. This is a readonly property for information only.
When a spreadsheet is linked and the Use Spreadsheet property is set to True, all of the feature python object's equation properties become readonly. You must modify them in the spreadsheet instead. Use Spreadsheet will create a new spreadsheet, add aliases, set them to the current formula's values, and link it if one is not already linked. Update Spreadsheet will push the current formula to the spreadsheet, overwriting any existing values in cells A1-B9. If Use Spreadsheet is set to True, any changes to the spreadsheet are automatically reflected in the related feature python property.
This is a link property, linking the fp object to a spreadsheet. By default, it is empty, but if you select a spreadsheet in the tree view, and then run the macro to create the fp object, it will link that spreadsheet automatically and import the data into the properties. See Usage section above for more details on required format of spreadsheet.
[Trigger] This property serves as a command button. When you set it to True it triggers the fp object to save the properties to the connected spreadsheet, overwriting any existing values. If no spreadsheet is connected it will create a new spreadsheet for you, link it in the Spreadsheet property, add the required aliases, and set their values. It's a way of saving the current properties to a new spreadsheet for later use. In all other cases the fp object pulls values from the connected spreadsheet (if Use Spreadsheet = True, see below), but in this case it is pushing those values to the spreadsheet. After it pushes the data to the spreadsheet it resets itself back to False.
When this is True, the formula properties (a,b,c,d,X,Y,Z,t,t_max, and interval) are all set to readonly. You won't be able to modify the properties in the fp object's property view when this is True. Instead, you must modify the appropriate cells in the spreadsheet. The fp object automatically updates its properties from the spreadsheet when they change. Set this to False if you would prefer to modify the properties in the fp object rather than in the spreadsheet. But if you set it to True again your property changes will be overwritten with the values from the spreadsheet. Use the Update Spreadsheet property/command trigger to push the data to the spreadsheet first if you don't want the property values to be clobbered.
You can link a text file to the feature python object. In such a text file you can save/load formulas in JSON format. Note: merely linking the file does nothing but set the file name in the File property. You must then trigger the Read File property to read in the contents (which will overwrite existing formulas). If you want to save your current formula to the file first, then use the Append File command to append the current formula to the JSON file before reading it in.
As of version 0.2022.03.08 there is a new formula editor, replacing some functions previously provided using boolean trigger properties. To access the editor, toggle the Edit Formulas property to True, select from the context menu in the tree, or double click the ParametricCurve object in the tree.
The formula editor dialog buttons are fairly self-explanatory, but I will give a brief introduction here anyway. Generally, the buttons along the right side at the top are for the formula names, and the buttons along the right side at the bottom are for the currently selected formula.
Sort the formulas alphabetically in the formula list if checked.
The + button creates a new formula, initially a copy of the currently selected formula, and adds it to the bottom of the list. It gets a default new name of "formula1", "formula2", etc., searching until a new unused name is available. It can be later renamed using the Rename button.
The - button deletes the currently selected formula. It cannot be undone except by using Reset, which sets the dialog back to its initial state when it was first opened, or Cancel, which does the same thing and also closes the dialog.
Rename button is used to rename the currently selected formula.
Clears all formulas from the editor. Use with caution. To undo, press Reset or Cancel, but in such cases you might lose some work because these options return you to the state when the dialog was first opened.
Open button opens and reads a JSON text file of the appropriate format. It only loads the data into memory in the editor, does not change the ParametricCurve object until the user presses OK or Apply. Current content in editor memory is all overwritten.
Save one or more formulas to a JSON file. This overwrites the contents of the file, if any.
Append one or more formulas to a JSON file. Does not overwrite the contents of the file. You will be prompted for new formula names if there are conflicts with existing formulas in the file.
Import one or more formulas from a JSON file. If the file only contains one formula you will not be prompted to select which formula to import. You will be prompted to select a new name for any imports that conflict with existing formula names in the editor. If you press Cancel when prompted for a new name the import will be aborted, but anything already imported will remain.
Copy button copies the currently selected formula to the clipboard as a text object that can be later pasted into another formula.
Paste button copies the text from the clipboard (previously inserted into the clipboard using Copy) into a new formula. If there is a naming conflict between text in clipboard and the formula names already open in the editor, then a default formula name is used for the name of the new formula and a warning is shown.
Clear button clears the currently selected formula. Cannot be undone except with Reset, which resets the entire set of formulas to what it was when the dialog was initially opened. Cancel will also undo, and close the dialog.
Save the currently selected formula to a JSON file. All contents of the file, if any, will be overwritten.
Appends the current selected formula to the selected file. Use Append All if you want to append all the formulas to a JSON file. If the file currently contains a formula with the same name as the currently selected formula, then you will be prompted to enter a new name.
close the dialog, apply changes. Can be undone.
Apply changes, leave the dialog open. Can be undone.
Resets the dialog back to its initial state when it was first opened.
Closes the dialog, ignoring all changes. (Changes that have been applied are not undone, but they can be undone using Undo.)
The property is now hidden. Use the formula editor instead.
Default is False. Sorts formulas alphabetically in the Formulas enumeration. When the editor is opened it takes its sorted checkbox setting from this property and also updates it when the editor is closed with OK or Apply.
These properties are now gone as of v0.2022.03.09, except for File, which is hidden. Use the Formula editor dialog to open, save, and append JSON files.
String variables: a, b, c, d1-dN, X, Y, and Z may contain python style comments. These begin with a pound symbol (#). The # and the rest of the line are ignored by the evaluation mechanism. For example, a value for a could be 12 #radius. In such a case only the 12 is evaluated, not the # or anything following it.
These are string properties that hold the formulas for creating the curve. Math expressions, like cos(), sin(), atan(), etc. can be used in the formulas. Most things in the math package, such as math.pi can be used (use it as simply pi and not as math.pi). See the list of supported functions below. In all of these you can refer to t. For property a you cannot refer to b or c (because these variables haven't been created yet). In b you can refer to a, but not c. In c you can refer to both a and b. In X,Y, and Z you can refer to a, b, or c. All references are case-sensitive, meaning, for example, x doesn't work, but X does.
a -> only refer to t
b -> only refer to a and t
c -> only refer to a, b, and t
d -> d1 only refer to a, b, c, and t, d2 only refer to a, b, c, d1, and t, d3 only refer to a, b, c, d1, d2, and t...
X -> only refer to a, b, c, d1-dN, and t
Y -> only refer to a, b, c, d1-dN, X, and t
Z -> only refer to a, b, c, d1-dN, X, Y, and t
The d variables are all held in a single property ("d"). This is a string list, so d is really a list of variables d1, d2, d3, d4, d5, etc., however many you want
to have you can have until you run out of computer memory or get tired of typing them all in. To add them you must use FreeCAD's string list property editor. Click on the value field for the d property, and then click the [...] button to open the editor. There you have list of lines numbered 1 through n. Click on line 1, enter your value, press Enter on the keyboard to register that entry and move down to line 2, and so on. Line 1 represents d1, Line 2 represents d2, etc. You must only reference d1 from d2, d1 and d2 from d3, etc.
d1 -> a, b, c, t
d2 -> a, b, c, t, d1
d3 -> a, b, c, t, d1, d2
d4 -> a, b, c, t, d1, d2, d3
d5 -> a, b, c, t, d1, d2, d3, d4
...
Supported math functions:
"sin": math.sin
"sinh": math.sinh
"cos": math.cos
"cosh": math.cosh
"tan": math.tan
"tanh": math.tanh
"exp": math.exp
"atan": math.atan
"atanh": math.atanh
"atan2": lambda a, b: math.atan2(a,b)
"acos": math.acos
"acosh": math.acosh
"asin": math.asin
"asinh": math.asinh
"sqrt": math.sqrt
"ceil": math.ceil
"floor": math.floor
"log": math.log
"factorial":math.factorial
"abs": abs
"degrees": math.degrees
"degree": math.degrees
"deg": math.degrees
"lgamma": math.lgamma
"gamma": math.gamma
"radians": math.radians
"rad": math.radians
"trunc": int
"round": round
"mod": lambda a, b: a % b
"sgn": lambda a: -1 if a < -epsilon else 1 if a > epsilon else 0
"interval": lambda a, b, t: 1 * ((t >= a) and (t < b))
"lt":lambda a, b: 1 * (a < b)
"lte": lambda a, b: 1 * (a <= b)
"gt": lambda a, b: 1 * (a > b)
"gte": lambda a, b: 1 * (a >= b)
"isequal": lambda a, b: 1 * (a == b)
"isclose": lambda a, b: 1 * (math.isclose(a, b, abs_tol = 1e-9))
"isclosetol": lambda a, b, tol: 1 * (math.isclose(a, b, abs_tol = tol))
"floordiv": lambda a, b: a // b
"copysign(a, b)": lambda a, b: 1 * math.copysign(a, b)
"any": lambda *a: 1 * any(a) (returns 1 if any item in list is not 0, else 0)
"all": lambda *a: 1 * all(a) (returns 1 if all items in list are not 0, else 0)
"sum": lambda *a: sum(a)
"avg": lambda *a: sum(a)/float(len(a))
"mean": lambda *a: sum(a)/float(len(a))
"gmean": lambda *a: statistics.geometric_mean(a)
"hmean": lambda *a: statistics.harmonic_mean(a)
"mode": lambda *a: statistics.mode(a)
"median": lambda *a: statistics.median(a)
"stdev": lambda *a: statistics.stdev(a)
"prod": lambda *a: math.prod(a) (all items in list multiplied by each other)
"perm": lambda a, b: math.perm(a,b) (ways to arrange a items into lists of length b)
"ternary": lambda a, b, c: b if a else c (if a is non-zero, returns b, else c)
"fc(expr)": evaluated as FreeCAD.DocumentObject.evalExpression(expr)
The function fc(expr) was added with version 0.2022.03.10.rev3. The argument expr can be any expression you would enter into the FreeCAD expression engine. For example, if there is a cylinder object in the document and you want to set "a" variable to its height, you could use fc(Cylinder.Height) in the "a" field. But this does not create a dependency because FreeCAD treats the "a" property as a string property and fc(Cylinder.Height) is just another string value. Thus, when the Cylinder's Height changes, the ParametricCurve object will not automatically update. You must manually update the ParametricCurve object. Alternatively, you can create this dependency by adding the Cylinder to the Dependencies property.
To do basic adding, subtracting, multiplying, dividing, use standard "+-*/". For exponents instead of 3**7 standard python syntax use 3^7 to do "3 to the power of 7".
t_min,t_max,interval (t_min was renamed from t in v0.2022.03.06, but t is still used for the current value of t in the loops)
The way the macro works is it creates points in a loop, and then at the end of the loop it uses those points to create the BSpline / Polygon. The t is the looping index. It starts the loop initialized at t_min and at the end of the loop t = t_max (max_t in the spreadsheet). The interval is the amount by which t is increased each time through the loop. The lower the interval the more points get produced. The properties in this group are type Float, whereas the other properties are type String. The others have to be Strings in order for you to be able to use variables in the formulas. These string formulas get evaluated by some code using the pyparsing module. It's slower, but more secure than using eval().
These are readonly, information only, floating point values based on the string properties of the same names. For example F_a = evaluate(a). This only works if a can be evaluated as a compile time constant. Otherwise, F_a = 0.0. For example, if c = a+b, then F_c = 0.0 because a+b is not a compile time constant. It's possible a or b could be based on the value of t at some point during the loop, which makes it a runtime variable. In other words, evaluate() knows nothing of the values of the other variables and c must be able to be evaluated strictly based on the value of the string in c. c = pi * 5 + cos(30) is fine, F_c = ~2.42, but c = cos(t) is not, then F_c = 0.0. Currently, there is no way to tell if c was able to be evaluated or if 0.0 is the proper evaluation.
F_d is a list of floats pointing to the d string list. It is 1-indexed to make it simpler (hopefully). For example, F_d[3] points to d3. F_d[0] is always 0.0 unless d = [] an empty list, in which case F_d is also [] empty. If there is anything in d, even if none can be interpreted as compile time constants, then F_d will contain some values. All that cannot be evaluated as compile time constants get a value of 0.0. Let's say d = [1, 2*c, 3], then F_d[0] = 0.0, F_d[1] = 1.0, F_d[2] = 0.0, F_d[3] = 3.0. F_d in the property view would show F_d [0.0, 1.0, 0.0, 3.0]
- 2024.05.08b ** fix typo (missing double quote character)
- 2024.05.08 ** accept edwilliams pr to add some hyperbolic function support
- 2023.05.06 ** add copysign() function, document sgn() function in readme thanks edwilliams
- 2022.12.16 ** fix bug in updating from spreadsheet related to d1, d2, etc. thanks mike-554
- 2022.05.04 ** Use FreeCAD.ActiveDocument.Objects[0] instead of FreeCAD.ActiveDocument.ActiveObject with evalExpression() because if the active object is deleted there won't be one.
- 2022.04.04.rev2 ** accept pr to fix typo ** update version information
- 2022.04.04 ** accept pr to fix issue with realthunder branch ** update version
- 2022.03.16 ** when copying to clipboard, include also the name of the formula ** when pasting from clipboard, create a new formula and paste into it rather than into the currently selected formula ** fix regression: after deleting formula, select the next item instead of row 0
- 2022.03.13.rev2 ** bug fix -- instead of updating formula editor memory with each line change, update when certain buttons are clicked
- 2022.03.13 ** bug fix -- update formula editor memory when line edit text changes ** add new comment style: {braced}
- 2022.03.12 ** improve Save, Import, Append commands in dialog by allowing multiple selections ** add helix formula, make it first formula --thanks edwilliams16 for this formula
- 2022.03.11 ** bug fix fc(expr) better handle both unit and unitless expressions ** check for nested(fc(fc(expr)), which are not supported
- 2022.03.10.rev6 ** after deleting formula select next item in list instead of first item.
- 2022.03.10.rev5 ** fix fc(expr) so it can handle unit property types
- 2022.03.10.rev4 ** fix bug in reset button handler in dialog
- 2022.03.10.rev3 ** add fc(expr) function
- 2022.03.10.rev2 ** bug fixes ** added warning messages when unable to create shape from formula ** check for compatibility with formula editor for objects created with older versions ** add version information to formula editor panel title, change name from Edit formulas to Formula editor
- 2022.03.10 ** New buttons in dialog. ** Sorted option. ** Fix context menu ** Require 1 formula in editor ** Return null shape from makeCurve() if a shape cannot be made
- 2022.03.09.rev2
** add ternary(a,b,c) function
reads "if a, then b, else c" ** add hammax's param_curve formula ** add ellipse formula - 2022.03.09 ** Add JSON file buttons to editor, remove related properties ** New buttons: Open, Save, Append ** File property retained, but hidden ** Allow comments in JSON files for t_min, interval, and t_max fields
- 2022.03.08
** add formula editor ** remove some formula editing properties - 2022.03.06.rev2
** add some more math functions ** rename t to t_min in default JSON text - 2022.03.06
** F_??? are now evaluated after stripping comments ** renamed t property to t_min ** allow for such as "2 * pi" in JSON formulas for t_min, t_max, and interval ** added 3 more formulas, 2 new braids and "formula1", a variation on the amoeba formula - 2022.03.05.rev3
** added F_??? float property aliases for a, b, c, d, X, Y, and Z, so these may be accessed via the expression engine as floating point values where they can be evaluated as such. (Must be evaluatable as a constant or else 0.0). These alias properties are: F_a, F_b, F_c, F_d, F_X, F_Y, and F_Z. F_d is a list of floats, 1-indexed, so F_d[1] = d1, F_d[2] = d2. F_d[0] is always 0.0 unless d=[] in which case F_d is [] empty list.
** fixed a bug where when changing formulas if expressions were used, then the expression overrided the new value. So, now when changing formulas all expressions entered for a, b, c, X, Y, Z, t, t_max, interval are cleared. (But not d.) - 2022.03.05.rev2
** allow python style comments in variable definitions, e.g. a = 12 #number of elements - 2022.03.05
** improve amoeba formula -- thanks, ed - 2022.03.03
** added new default JSON formulas
** added ability to drag/drop Text document onto Parametric Curve object to import as interpreted JSON text
** added ability to select Text document object while creating ParametricCurve to use its text interpreted as JSON text
** Due to bug in FreeCAD, make sure file has been saved since last edit of Text document object - 2022.03.02
** added these functions to evaluate():
"mod": lambda a, b: a % b,
"interval": lambda a, b, t: 1 * ((t >= a) and (t < b)),
"lt":lambda a, b: 1 * (a < b),
"lte": lambda a, b: 1 * (a <= b),
"gt": lambda a, b: 1 * (a > b),
"gte": lambda a, b: 1 * (a >= b),
"isequal": lambda a, b: 1 * (a == b),
"isclose": lambda a, b: 1 * (math.isclose(a, b, abs_tol = 1e-9)),
"isclosetol": lambda a, b, tol: 1 * (math.isclose(a, b, abs_tol = tol)),
"floordiv": lambda a, b: a // b, - 2022.03.01
** workaround a conflict with another addon (cadquery_module workbench) - 2022.02.14
** make code slightly more robust when setting number of iterations - 2022.02.13.rev3
** check also (t_max - t) * Interval != 0. - 2022.02.13.rev2
** bug fix, restore buggy behavior on PlusOneIteration = False. We want to keep existing behavior to not break existing formulas. ** detect and prevent infinite loop where (t_max - t) * Interval < 0 (thanks to edwilliams16 for his help with this revision) - 2022.02.13
** rename a variable to better reflect its usage (increment becomes intervals) ** set t to t_max if it's not equal to t_max in the last iteration (only if PlusOneIteration = True) ** dynamically set plus1 variable based on needs for the loop, might be 0, might be 1, might be 2. - 2022.02.12
** Add PlusOneIteration property - 2021.09.15.rev3
** Eliminate need for Make_Parametric_Curve.FCMacro. You can now run Parametric_Curve_FP.py directly. - 2021.09.15.rev2
** spreadsheet integration bug fixes - 2021.09.15
- Add d variables d1, d2, d3, etc.
- Setup code in prepration to be able to put the macro in the wiki for availability in addon manager.
- The new code will allow created objects to remain parametric after saving file, restarting FreeCAD, reloading file
- all while only using a single .py file. We can do this now, but only with 2 files, 1 to create the object, 1 to hold the class information.
- But this is not yet complete, so for now continue using Make_Parametric_Curve.FCMacro until next release.
- 2021.09.01
** Fix bug in renameFormula() so now we don't switch to the first formula in Formulas, but rather remain in that formula. ** Do away with MakeBSpline boolean (breaks existing objects) ** Add new ShapeType enumeration with "BSpline,"Polygon","Points" as options ** Add Points property (list of vectors used to create shape) - 2021.08.31.rev2
** This revision breaks all compatability with all existing objects, spreadsheets, and JSON files created with earlier versions.
It might even crash FreeCAD if you try to open a file containing an existing FP object. The workaround for this, if needed, is to temporarily delete the macro, restart FreeCAD, open the file and delete the old object. Then re-install the macro and create a new FP object with the same formulas. JSON files can be manually edited by renaming "min_t" and "max_t" keys to "t" and "t_max", respectively. Spreadsheet aliases should be renamed to "t_min" and "t_max" because "t" is not a valid alias.
** Presets property renamed to Formulas ** RenamePresets property renamed to RenameFormulas - 2021.08.31
** fixed bug in fnumber regex, credit: edwilliams16 at FreeCAD forum ** removed unnecessary fixDots() preparser - 2021.08.30
** bug fixes ** Changed names of some properties (breaks existing models) ** Added support for undo / redo for some operations - 2021.08.29.rev2
** improve performance by inhibiting recomputes for some operations
** preparse to add "0" in front of cases like ".5" or "3 + .5" -> "0.5" "3 + 0.5" - 2021.08.29
** bug fixes ** renamed some internal functions and properties, probably breaks any existing models - 2021.08.28.rev3
** Adds JSON support - 2021.08.28, rev.2
** Allow for unused variables to be blank, empty string.
evaluate("") -> 0
** Add Continuity (read only) property in Curve group ** Remove unused hidden property "d" - 2021.08.28
** Generalize parser for more general use. To use:
from Parametric_Curve_FP import evaluate
evaluate("1+2") -> 3
evaluate("a+2",{"a":5}) -> 7
The dictionary object can contain any number of variables and their values. - 2021.08.27
** Initial upload.