Expressions

Overview

It is possible to define properties using mathematical expressions. In the GUI, spin boxes or input fields that are bound to properties contain a blue icon . Clicking on the icon or typing the equal sign = brings up the expression editor for that particular property.

A FreeCAD expression is a mathematical expression using the standard mathematical operators, functions and predefined constants as described below. In addition, the expression may reference object properties, and also use conditionals. Numbers in an expression may have an optional unit attached to them.

Numbers may use either a comma , or a decimal point . to separate whole digits from decimals. When the decimal marker is used, it must be followed by at least one digit. Thus, the expressions 1. + 2. and 1, + 2, are invalid, but 1.0 + 2.0 and 1,0 + 2,0 are valid.

Operators and functions are unit-aware, and require valid combinations of units, if supplied. For example, 2mm + 4mm is a valid expression, while 2mm + 4 is not. This also applies to references to object properties that have units, such as Length properties. Thus Pad001.Length + 1 is invalid since it adds a pure number to a property with length units, it requires Pad001.Length + 1mm.

Some unit related errors can seem unintuitive, with expressions either being rejected or producing results that do not match the units of the property being set. Here are some examples:

1/2mm

is not interpreted as half a millimeter but as 1/(2mm), resulting in: 0.5 mm^-1.

sqrt(2)mm

is not valid because the function call is not a number. This has to be entered as sqrt(2) * 1mm.

Function arguments

Multiple arguments to a function may be separated by either a semicolon followed by a space , . In the latter case, the comma is converted to a semicolon after entry. When a semicolon is used, no trailing space is necessary.

Arguments may include references to cells in a spreadsheet. A cell reference consists of the cell's uppercase row letter followed by its column number, for example A1. A cell may also be referenced by using the cell's alias instead, for example Spreadsheet.MyPartWidth.

Referencing objects

As already shown above, you can reference an object by its Name. But you can also use its Label. In the case of a Label, it must be enclosed in double << and >> symbols, such as <<Label>>.

You can reference any property of an object. For example, to reference a Cylinder's height, you may use Cylinder.Height or <<Label_of_cylinder>>.Height.

For more information about referencing objects, see Reference to CAD data.

Supported constants

The following constants are supported:

Constant Description

e Euler's number pi Pi

Supported operators

The following operators are supported:

Operator Description

+ Addition - Subtraction * Multiplication / Floating point Division % Remainder ^ Exponentiation

Supported functions

General mathematical functions

The following mathematical functions are supported:

Trigonometric functions

Trigonometric functions use degree as their default unit. For radians add first value in an expression. So e.g. cos(45) is the same as cos(pi rad / 4). Expressions in degrees can use either deg or °, e.g. 360deg - atan2(3; 4) or 360° - atan2(3; 4). If an expression is without units and needs to be converted to degrees or radians for compatibility, multiply by 1deg, 1° or 1rad as appropriate, e.g. (360 - X) * 1deg; (360 - X) * 1°; (0.5 + pi / 2) * 1rad.

++++ | Function | Description | Input range | +========================+===================================================================================================================================================================================+============================================+ | | Arc cosine | -1 <= x <= 1 | | acos(x) | | | | | | | ++++ | | Arc sine | -1 <= x <= 1 | | asin(x) | | | | | | | ++++ | | Arc tangent, return value in the range -90° < value < 90° | all | | atan(x) | | | | | | | ++++ | | Arc tangent of y/x accounting for quadrant, return value in the range -180° < value <= 180° | all, the invalid input x = y = 0 returns 0 | | atan2(y; x) | | | | | | | ++++ | | Cosine | all | | cos(x) | | | | | | | ++++ | | Hyperbolic cosine | all | | cosh(x) | | | | | | | ++++ | | Sine | all | | sin(x) | | | | | | | ++++ | | Hyperbolic sine | all | | sinh(x) | | | | | | | ++++ | | Tangent | all, except x = n*90 with n = odd integer | | tan(x) | | | | | | | ++++ | | Hyperbolic tangent | all | | tanh(x) | | | | | | | ++++ | | Pythagorean addition (hypotenuse), e.g. hypot(4; 3) = 5 | x and y >= 0 | | hypot(x; y) | | | | | | | ++++ | | Given hypotenuse, and one side, returns other side of triangle, e.g. cath(5; 3) = 4 | x >= y >= 0 | | cath(x; y) | | | | | | | ++++

Exponential and logarithmic functions

++++ | Function | Description | Input range | +================================+==============================================================================================+=============+ | | Exponential function | all | | exp(x) | | | | | | | ++++ | | Natural logarithm | x > 0 | | log(x) | | | | | | | ++++ | | Common logarithm | x > 0 | | log10(x) | | | | | | | ++++ | | Exponentiation | all | | pow(x; y) | | | | | | | ++++ | | Square root | x >= 0 | | sqrt(x) | | | | | | | ++++ | | Cubic root | all | | cbrt(x) | | | | | | | | (v0.21) | | | ++++

Rounding, truncation and remainder functions

++++ | Function | Description | Input range | +======================+==================================================================================================================================+===================+ | | Absolute value | all | | abs(x) | | | | | | | ++++ | | Ceiling function, smallest integer value greater than or equal to x | all | | ceil(x) | | | | | | | ++++ | | Floor function, largest integer value less than or equal to x | all | | floor(x) | | | | | | | ++++ | | Remainder after dividing x by y, sign of result is that of the dividend. | all, except y = 0 | | mod(x; y) | | | | | | | ++++ | | Rounding to the nearest integer | all | | round(x) | | | | | | | ++++ | | Truncation to the nearest integer in the direction of zero | all | | trunc(x) | | | | | | | ++++

Statistical / aggregate functions

Aggregate functions take one or more arguments. Individual arguments to aggregate functions may consist of ranges of cells. A range of cells is expressed as two cell references separated by a colon {{Incode|:}}, for example {{Incode|average(B1:B8)}} or {{Incode|sum(A1:A4; B1:B4)}}. The cell references may also use cell aliases, for example {{Incode|average(StartTemp:EndTemp)}}.

The following aggregate functions are supported:

++++ | Function | Description | Input range | +==================================+====================================================================================================================================+=============+ | | Average value of the arguments, same as sum(a; b; c; ...) / count(a; b; c; ...) | all | | average(a; b; c; ...) | | | | | | | ++++ | | Count of the arguments, typically used for cell ranges | all | | count(a; b; c; ...) | | | | | | | ++++ | | Maximum value of the arguments | all | | max(a; b; c; ...) | | | | | | | ++++ | | Minimum value of the arguments | all | | min(a; b; c; ...) | | | | | | | ++++ | | Standard deviation of the values of the arguments | all | | stddev(a; b; c; ...) | | | | | | | ++++ | | Sum of the values of the arguments, typically used for cell ranges | all | | sum(a; b; c; ...) | | | | | | | ++++

String manipulation

String identification

Strings are identified in expressions by surrounding them with opening/closing double chevrons (as are labels).

In following example, "TEXT" is recognized as a string : <<TEXT>>

String concatenation

Strings can be concatenated using the '+' sign.

Following example <<MY>> + <<TEXT>> will be concatenated to "MYTEXT".

String conversion

Numerical values can be converted to strings with the str function:

String formatting

String formatting is supported using the (old) %-style Python way.

All %-specifiers as defined in Python documentation.

As an example, supposing you have a default 10mm-side cube named 'Box' (default FreeCAD naming), the following expression <<Cube length : %s>> % Box.Length will expand to "Cube length : 10.0 mm"

For more than one %-specifier use the following syntax: <<Cube length is %s and width is %s>> % tuple(Box.Length; Box.Width). Or use concatenation: <<Cube length is %s>> % Box.Length + << and width is %s>> % Box.Width. Both will expand to "Cube length is 10.0 mm and width is 10.0 mm".

A FreeCAD sample file using string formatting is available in the forum

Object creation functions

The following objects may be created in expressions using the following functions:

++++ | Type | Function | Description | +================================================================+====================================================================================+=========================================================================================================================================================================================================================================================================================================================================+ | | | Example: tuple(2; 1; 2) | | Tuple | tuple(a; b; ...) | | | | | | ++++ | | | Example: list(2; 1; 2) | | List | list(a; b; ...) | | | | | | ++++ | Vector | | Create a vector using three unit-less or Length unit values. Example: vector(2; 1; 3) | | | vector(x; y; z) | | | | | | ++++ | | | | | | create(<<vector>>; x; y; z) | | | | | | ++++ | Matrix | matrix( | Create a 4x4 matrix in row-major order: $\begin{bmatrix} | | | a11; a12; a13; a14; | a_{11} & a_{12} & a_{13} & a_{14} \ | | | a21; a22; a23; a24; | a_{21} & a_{22} & a_{23} & a_{24} \ | | | a31; a32; a33; a34; | a_{31} & a_{32} & a_{33} & a_{34} \ | | | a41; a42; a43; a44 | a_{41} & a_{42} & a_{43} & a_{44} \ | | | ) | \end{bmatrix}$ | | | | | | | | A minimum of 1 argument can be supplied such as matrix(1) which creates an identity matrix. | | | | | | | | Example: matrix(1; 2; 3; 4; 5; 6; 7; 8; 9; 10; 11; 12; 13; 14; 15; 16) | ++++ | | | | | | create(<<matrix>>; a11; a12; ...; a44) | | | | | | ++++ | | | Create a Rotation by specifying its axis (Vector) and angle (Angle unit or unit-less), or three Euler angles α, β, γ. Examples: | | Rotation | rotation(axis; angle) | | | | | - | | | | rotation(vector(0; 1; 0); 45) | | | | | | | | | | | | - | | | | create(<<rotation>>; 30; 30; 30) | | | | | ++++ | | | | | | rotation(α; β; γ) | | | | | | ++++ | | | | | | create(<<rotation>>; axis; angle) | | | | | | ++++ | | | | | | create(<<rotation>>; α; β; γ) | | | | | | ++++ | Placement | | Create a Placement with various parameters, including: | | | placement(base; rotation) | | | | | - | | | | base | | | | | | | | : base location (Vector) | | | | | | | | - | | | | center | | | | | | | | : center location (Vector) | | | | | | | | - | | | | rotation | | | | | | | | : Rotation | | | | | | | | - | | | | axis | | | | | | | | : Rotation axis (Vector) | | | | | | | | - | | | | angle | | | | | | | | : Rotation angle (unit-less or Angle unit value) | | | | | | | | - | | | | matrix | | | | | | | | : Matrix | | | | | | | | Examples: | | | | | | | | - | | | | placement(vector(2; 1; 3); rotation(vector(0; 0; 1); 45)) | | | | | | | | | | | | - | | | | create(<<placement>>; create(<<vector>>; 2; 1; 2); create(<<rotation>>; create(<<vector>>; 0; 1; 0); 45)) | | | | | ++++ | | | | | | placement(base; rotation; center) | | | | | | ++++ | | | | | | placement(base; axis; angle) | | | | | | ++++ | | | | | | placement(matrix) | | | | | | ++++ | | | | | | create(<<placement>>; ...) | | | | | | ++++

Vector functions

Functions: (v0.22) .

+++ | Function / Operator | Description | +=====================================+=====================================================================================================================================================================================+ | | Add two vectors. | | v1 + v2 | | | | | +++ | | Subtract two vectors. | | v1 - v2 | | | | | +++ | | Uniformly scale a vector by s. | | v * s | | | | | +++ | | Angle between two vectors in degrees. | | vangle(v1; v2) | | | | | +++ | | Cross product of two vectors $v_1 \times v_2$. | | vcross(v1; v2) | | | | | +++ | | Dot product of two vectors $v_1 \cdot v_2$. | | v1 * v2 | | | | | +++ | | | | vdot(v1; v2) | | | | | +++ | | Distance between vector v1 and a line through v2 in direction v3. | | vlinedist(v1; v2; v3) | | | | | +++ | | Distance between vector v1 and the closest point on a line segment from v2 to v3. | | vlinesegdist(v1; v2; v3) | | | | | +++ | | Project vector v1 on a line through v2 in direction v3. | | vlineproj(v1; v2; v3) | | | | | +++ | | Normalize a vector to a unit vector. | | vnormalize(v) | | | | | +++ | | Distance between vector v1 and a plane defined by a point v2 and a normal v3. | | vplanedist(v1) | | | | | +++ | | Project vector v1 on a plane defined by a point v2 and a normal v3. | | vplaneproj(v1) | | | | | +++ | | Non-uniformly scale a vector by sx in the X direction, sy in the Y direction, and sz in the Z direction. | | vscale(v; sx; sy; sz) | | | | | +++ | | | | vscalex(v; sx) | | | | | +++ | | | | vscaley(v; sy) | | | | | +++ | | | | vscalez(v; sz) | | | | | +++

Matrix functions

Rotation

and Placement can each be represented by a Matrix. The following functions all take in a Matrix, Rotation, or Placement as their first parameter denoted in the table below by m. The type of the returned object is the same as the object supplied in the first argument except when using mtranslate on a Rotation, in which case a Placement will be returned.

Conditional expressions

Conditional expressions are of the form condition ? resultTrue : resultFalse. The condition is defined as an expression that evaluates to either 0 (false) or non-zero (true).

The following relational operators are defined:

Unit Description

== equal to != not equal to > greater than < less than >= greater than or equal to <= less than or equal to

Units

Units can be used directly in expressions. The parser connects them to the previous value. So 2mm or 2 mm is valid while mm is invalid because there is no preceding value.

All values must have a unit. Therefore you must in general use a unit for values in spreadsheets. In some cases it works even without a unit, for example if you have e.g. in spreadsheet cell B1 just the number 1.5 and refer to it for a pad height. This only works because the pad height predefines the unit mm that is used if no unit is given. It will nevertheless fail if you use for the pad height e.g. Sketch1.Constraints.Width - Spreadsheet.B1 because Sketch1.Constraints.Width has a unit and Spreadsheet.B1 has not.

Units with exponents can directly be entered. So e.g. mm^3 will be recognized as mm³ and m^3 will be recognized as m³.

If you have a variable whose name is that of a unit you must put the variable between << >> to prevent it from being recognized as a unit. For example if you have the dimension Sketch.Constraints.A it would be recognized as the unit ampere. Therefore you must write it in the expression as Sketch.Constraints.<<A>>.

The following units are recognized by the expression parser:

Amount of substance

Unit Description

mmol Millimole mol Mole

Angle

Unit Description

° Degree; alternative to the unit deg deg Degree; alternative to the unit ° rad Radian gon Gradian S Second of arc; alternative to the unit ″ ″ Second of arc; alternative to the unit S M Minute of arc; alternative to the unit ′ ′ Minute of arc; alternative to the unit M

Current

Unit Description

mA Milliampere A Ampere kA Kiloampere MA Megaampere

Electric capacitance

Unit Description

pF Picofarad nF Nanofarad uF Microfarad; alternative to the unit µF µF Microfarad; alternative to the unit uF mF Millifarad F Farad; 1 F = 1 s^4·A^2/m^2/kg

Electric charge

Unit Description

C Coulomb; 1 C = 1 A*s

Electric conductivity

Unit Description

uS Microsiemens; alternative to the unit µS µS Microsiemens; alternative to the unit uS mS Millisiemens S Siemens; 1 S = 1 s^3·A^2/kg/m^2 kS KiloSiemens MS MegaSiemens

Electric inductance

Unit Description

nH Nanohenry uH Microhenry; alternative to the unit µH µH Microhenry; alternative to the unit uH mH Millihenry H Henry; 1 H = 1 kg·m^2/s^2/A^2

Electric potential

Unit Description

mV Millivolt V Volt kV Kilovolt

Electric resistance

Unit Description

Ohm Ohm; 1 Ohm = 1 kg·m^2/s^3/A^2 kOhm Kiloohm MOhm Megaohm

Energy/work

Unit Description

mJ Millijoule J Joule kJ Kilojoule eV Electronvolt; 1 eV = 1.602176634e-19 J keV Kiloelectronvolt MeV Megaelectronvolt kWh Kilowatt hour; 1 kWh = 3.6e6 J Ws Watt second; alternative to the unit Joule VAs Volt-ampere-second; alternative to the unit Joule CV Coulomb-volt; alternative to the unit Joule cal Calorie; 1 cal = 4.184 J kcal Kilocalorie

Force

Unit Description

mN Millinewton N Newton kN Kilonewton MN Meganewton lbf Pound of force

Length

Unit Description

nm Nanometer um Micrometer; alternative to the unit µm µm Micrometer; alternative to the unit um mm Millimeter cm Centimeter dm Decimeter m Meter km Kilometer mil Thousandth of an inch; alternative to the unit thou thou Thousandth of an inch; alternative to the unit mil in Inch; alternative to the unit " " Inch; alternative to the unit in ft Foot; alternative to the unit ' ' Foot; alternative to the unit ft yd Yard mi Mile

Luminous intensity

Unit Description

cd Candela

Magnetic flux

Unit Description

Wb Weber; 1 Wb = 1 kg*m^2/s^2/A

Magnetic flux density

Unit Description

G Gauss; 1 G = 1 e-4 T T Tesla; 1 T = 1 kg/s^2/A

Mass

Unit Description

ug Microgram; alternative to the unit µg µg Microgram; alternative to the unit ug mg Milligram g Gram kg Kilogram t Tonne oz Ounce lb Pound; alternative to the unit lbm lbm Pound; alternative to the unit lb st Stone cwt Hundredweight

Power

Unit Description

W Watt kW Kilowatt

Pressure

Unit Description

Pa Pascal kPa Kilopascal MPa Megapascal GPa Gigapascal uTorr Microtorr; alternative to the unit µTorr µTorr Microtorr; alternative to the unit uTorr mTorr Millitorr Torr Torr; 1 Torr = 133.32 Pa psi Pound-force per square inch; 1 psi = 6.895 kPa ksi Kilopound-force per square inch

Temperature

Unit Description

uK Microkelvin; alternative to the unit µK µK Microkelvin; alternative to the unit uK mK Millikelvin K Kelvin

Time

Unit Description

s Second min Minute h Hour Hz (1/s) Hertz kHz Kilohertz, MHz Megahertz GHz Gigahertz THz Terahertz

Volume

Unit Description

ml Milliliter l Liter cft Cubicfoot

Special imperial units

Unit Description

mph Miles per hour sqft Square foot

Unsupported units

The following commonly used units are not yet supported, for some an alternative is provided:

Unit Description Alternative

°C Celsius [°C] + 273.15 K °F Fahrenheit; ([°F] + 459.67) × 5/9 u Atomic mass unit; alternative to the unit Da 1.66053906660e-27 kg Da Dalton; alternative to the unit u 1.66053906660e-27 kg sr Steradian not directly lm Lumen not directly lx Lux not directly px Pixel not directly

Invalid characters and names

The expression feature is very powerful but to achieve this power it has some limitations concerning some characters. To overcome this, FreeCAD offers to use labels and reference them instead of the object names. In labels you can use almost all special characters.

In cases where you cannot use a label, such as the name of a sketch's constraints, you must be aware what characters are not allowed.

Labels

For labels there are no invalid characters, however some characters need to be escaped:

+++ | Characters | Description | +==========================================================+===========================================================================+ | | Need to be escaped by adding \ in front of them. | | ' | | | | | | , \, " | | +++

For example, the label Sketch\002 must be referenced as <<Sketch\\002>>.

Names

Names of objects like dimensions, sketches, etc. may not have the characters or character sequences listed below, otherwise the name is invalid:

Characters / Character sequences Description

+, -, *, /, ^, _, <, >, (, ), {, }, [, ], ., ,, = Characters that are math operators or part of mathematical constructs A, kA, mA, MA, J, K, ', ft, °, and many more! Characters and character sequences that are units (see the Units paragraph) #, !, ?, §, $, %, &, :, ;, \, |, ~, ∆, ¿, and many more! Characters used as placeholder or to trigger special operations pi, e Mathematical constants ´, ****, ', " Characters used for accents space A space defines the end of a name and can therefore not be used

For example, the following name is valid: <<Sketch>>.Constraints.T2üßµ@. While these are invalid names: <<Sketch>>.Constraints.test\result_2 (\r means "carriage return") or <<Sketch>>.Constraints.mol (mol is a unit).

Since shorter names (especially if they have only one or two characters) can easily result in invalid names, consider using longer names and/or establishing a suitable naming convention.

Cell aliases

See Spreadsheet SetAlias.

Reference to CAD data

It is possible to use data from the model itself in an expression. To reference a property use object_name.property or <<object_label>>.property, labels must be enclosed in << and >>. If you want to use labels they must be unique.

All next examples reference the object by its name, but in all cases the object label can also be used.

If the property is a compound of fields, the individual fields can be accessed as object_name.property.field.

To reference list objects use object_name.list[list_index]. If you want to reference a constraint in a sketch, use Sketch.Constraints[16]. If you are in the same sketch you may omit its name and just use Constraints[16]. Note that the index starts with 0, therefore Constraint17 has to be referenced as Constraints[16].

To reference the object itself use the _self pseudo property: object_name._self.

The following table shows some more examples:

Cyclic dependencies

FreeCAD checks dependencies based on the relationship between document objects, not properties. This means that you cannot provide data to an object and query that same object for results. For example, even though there are no cyclic dependencies when the properties themselves are considered, you may not have an object which gets its dimensions from a spreadsheet and then display the volume of that object in the same spreadsheet. You have to use two spreadsheets, one to drive your model and the other for reporting.

As a workaround it is possible to display a cell range from the second spreadsheet in the first (or vice versa) by creating a cell binding with the Hide dependency of binding option.

Another way to workaround cyclic dependencies is to hide the reference by using the href or hiddenref function for individual expressions, for example: href(Box.Length).

Please note that both mentioned workarounds should be used with caution, and that they do not work if the properties that are reported depend on dimensions that are driven from the same spreadsheet.

Document-wide global variables

There is no concept of global variables in FreeCAD at the moment. Instead, arbitrary variables can be defined as cells in a spreadsheet using the Spreadsheet workbench, and then be given a name using the alias property for the cell (right-click on cell). Then they can be accessed from any expression just as any other object property.

Cross-document linking

It is possible (with limitations) to define a Property of an object in your current document (".FCstd" file) by using an Expression to reference a Property of an object contained in a different document (".FCstd" file). For example, a cell in a spreadsheet or the Length of a Part Cube, etc. in one document can be defined by an Expression that references the X Placement value or another Property of an object contained in a different document.

A document's name is used to reference it from other documents. When saving a document the first time, you choose a file name; this is usually different from the initial default "Unnamed1" (or its translated equivalent). To prevent links being lost when the master document is renamed upon saving, it is recommended that you first create the master document, create a spreadsheet inside it, and save it. Subsequently, you can still make changes to the file and its spreadsheet but you should not rename it.

Once the master document with the spreadsheet is created and saved (named), it is safe to create dependent documents. For example, assuming you name the master document master, the spreadsheet modelConstants, and give a cell an alias-name Length, you can then access the value as:

Note that the master document must be loaded for the values in the master to be available to the dependent document.

Of course, it's up to you to load the corresponding documents later when you want to change anything.

Known issues / remaining tasks

FreeCAD does not yet have a built-in expression manager where all expressions in a document are listed, and can be created, deleted, queried, etc. But an addon is available: fcxref expression manager.
Open bugs/tickets for Expressions can be found on GitHub.

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Expressions

Overview

Function arguments

Referencing objects

Supported constants

Supported operators

Supported functions

General mathematical functions

Trigonometric functions

Exponential and logarithmic functions

Rounding, truncation and remainder functions

Statistical / aggregate functions

String manipulation

String identification

String concatenation

String conversion

String formatting

Object creation functions

Vector functions

Matrix functions

Conditional expressions

Units

Amount of substance

Angle

Current

Electric capacitance

Electric charge

Electric conductivity

Electric inductance

Electric potential

Electric resistance

Energy/work

Force

Length

Luminous intensity

Magnetic flux

Magnetic flux density

Mass

Power

Pressure

Temperature

Time

Volume

Special imperial units

Unsupported units

Invalid characters and names

Labels

Names

Cell aliases

Reference to CAD data

Cyclic dependencies

Document-wide global variables

Cross-document linking

Known issues / remaining tasks