# Expressions/es

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## Overview

It is possible to define properties using mathematical expressions. From 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 following notation for the standard mathematical operators and functions as described below. In addition, the expression may reference other 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 '.' separating 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 (the reason for this is that an expression like 1in + 4 will most likely be interpreted as 1in + 4in by humans, but all units are converted to the SI system internally, and the system is not able to guess this). These units are currently recognized.

You can use predefined constants and functions.

### Referencing objects

You can reference to an object by its `Name` or by its `Label`. In the case of a `Label`, it must be enclosed in double `<<` and `>>` symbols, such as `<<Label>>`.

You can reference any numerical property of this object. For example, to refer to a Cylinder's height, you may use `Cylinder.Height` or `<<Long_name_of_cylinder>>.Height`.

To reference list objects, the syntax is `<<object_label>>.list[list_index]` or `object_name.list[list_index]`. If you want for example reference a constraint in a sketch, do it this way <<MySketch>>.Constraints. If you are in the same sketch you can omit its name and just use Constraints.
Note: The index starts with 0, therefore constraint 17 has the index 16.

The following operators are supported:

Operator Description
- Subtraction
* Multiplication
/ Floating point Division
% Remainder
^ Exponentiation

## Supported Constants

The following constants are supported:

Constant Description
e Euler's number
pi Pi

## Supported Functions

### General Mathematical Functions

The mathematical functions listed below are available.

Multiple arguments to a function may be separated by either a semicolon ';' or a comma 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.

Trigonometric functions use degree as their default unit. For radian measure, add rad following the 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 1 deg, 1 ° or 1 rad as appropriate, e.g. (360 - X) * 1deg; (360 - X) * 1°; (0.5 + pi / 2) * 1rad.
These trigonometric functions are supported:

Function Description Value range
acos(x) Arc cosine -1 <= x <= 1
asin(x) Arc sine -1 <= x <= 1
atan(x) Arc tangent all
atan2(x, y) Arc tangent of x/y all, except y = 0
cos(x) Cosine all
cosh(x) Hyperbolic cosine all
sin(x) Sine all
sinh(x) Hyperbolic sine all
tan(x) Tangent all, except of x = n·90 with n = integer
tanh(x) Hyperbolic tangent all

These functions for exponentiation and logarithmization are supported:

Function Description Value range
exp(x) Exponential function all
log(x) Natural logarithm x > 0
log10(x) Common logarithm x > 0
pow(x, y) Exponentiation all
sqrt(x) Square root x >= 0

These functions for rounding, truncation and remainder are supported:

Function Description Value range
abs(x) Absolute value all
ceil(x) Ceiling function smallest integer value greater than or equal to x all
floor(x) Floor function, largest integer value less than or equal to x all
mod(x, y) Remainder after dividing x by y all, except y = 0
round(x) Rounding to the nearest integer all
trunc(x) Truncation to the nearest integer all

### Statistical / Aggregate Functions

Aggregate functions take one or more arguments, separated by a semicolon ';' or a comma and a space ', '.
Arguments may include references to cells in a spreadsheet. Cell references consist of the (CAPITAL) row letter followed by the column number.
Arguments may include ranges of cells (two cell references separated by a colon), for example average(B1:B8).

These aggregate functions are supported:

Function Description Value range
average(x:y) Arithmetic mean of values in cells x through y; sum(x:y) / count(x:y) all
count(x:y) Counting of cells from x through y all
max(x:y) Maximum value in cells x through y all
min(x:y) Minimum value in cells x through y all
stddev(x:y) Standard deviation of values in cells x through y all
sum(x: y) Sum of values in cells x through y all

## 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 directly be used 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 with a name of a unit you must put the variable into << >> to prevent that it will be recognized as unit. For example if you have the dimension 'Sketch.Constraints.A' it would be recognized as 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 °
S Second of arc
Second of arc; alternative to the unit S
M Minute of arc
Minute of arc; alternative to the unit M

Current:

Unit Description
mA Milliampere
A Ampere
kA Kiloampere
MA Megaampere

Electrical Capacitance:

Unit Description
pF Picofarad, introduced in version 0.19
nF Nanofarad, introduced in version 0.19
uF Microfarad; alternative to the unit µF, introduced in version 0.19
µF Microfarad; alternative to the unit uF, introduced in version 0.19
mF Millifarad, introduced in version 0.19
F Farad; 1 F = 1 s^4·A^2/m^2/kg, introduced in version 0.19

Electrical Conductance:

Unit Description
uS Microsiemens; alternative to the unit µS, introduced in version 0.19
µS Microsiemens; alternative to the unit uS, introduced in version 0.19
mS Millisiemens, introduced in version 0.19
S Siemens; 1 S = 1 s^3·A^2/kg/m^2, introduced in version 0.19

Electrical Inductance:

Unit Description
nH Nanohenry, introduced in version 0.19
uH Microhenry; alternative to the unit µH, introduced in version 0.19
µH Microhenry; alternative to the unit uH, introduced in version 0.19
mH Millihenry, introduced in version 0.19
H Henry; 1 H = 1 kg·m^2/s^2/A^2, introduced in version 0.19

Electrical Resistance:

Unit Description
Ohm Ohm; 1 Ohm = 1 kg·m^2/s^3/A^2, introduced in version 0.19
kOhm Kiloohm, introduced in version 0.19
MOhm Megaohm, introduced in version 0.19

Electric Charge:

Unit Description
C Coulomb; 1 C = 1 A·s, introduced in version 0.19

Electric Potential:

Unit Description
mV Millivolt
V Volt
kV Kilovolt

Energy / Work:

Unit Description
mJ Millijoule
J Joule
kJ Kilojoule, introduced in version 0.19
eV Electronvolt; 1 ev = 1.602176634e-19 J, introduced in version 0.19
keV Kiloelectronvolt, introduced in version 0.19
MeV Megaelectronvolt, introduced in version 0.19
kWh Kilowatt hour; 1 kWh = 3.6e6 J, introduced in version 0.19
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, introduced in version 0.19
kcal Kilocalorie, introduced in version 0.19

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 mu
mm Millimeter
cm Centimeter
mm Millimeter
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
ft Foot; alternative to the unit '
' Foot; alternative to the unit ft
yd Yard
mi Mile

Luminous Intensity:

Unit Description
cd Candela

Magnetic Field Strength:

Unit Description
Oe Oersted; 1 Oe = 79.57747 A/m, introduced in version 0.19

Magnetic Flux:

Unit Description
Wb Weber; 1 Wb = 1 kg*m^2/s^2/A, introduced in version 0.19

Magnetic Flux Density:

Unit Description
G Gauss; 1 G = 1 e-4 T, introduced in version 0.19
T Tesla; 1 T = 1 kg/s^2/A, introduced in version 0.19

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, introduced in version 0.19
VA Volt-ampere

Pressure:

Unit Description
Pa Pascal
kPa Kilopascal
MPa Megapascal
GPa Gigapascal
mbar MilliBar, introduced in version 0.19
bar Bar, introduced in version 0.19
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
Mpsi Megapound-force per square inch, introduced in version 0.19

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, introduced in version 0.19
kHz Kilohertz, introduced in version 0.19
MHz Megahertz, introduced in version 0.19
GHz Gigahertz, introduced in version 0.19
THz Terahertz, introduced in version 0.19

Torque:

Unit Description
Nm Newton metre

Volume:

Unit Description
ml Milliliter, introduced in version 0.19
l Liter
cft Cubicfoot, introduced in version 0.19

Special imperial units:

Unit Description
mph Miles per hour, introduced in version 0.19
sqft Square foot, introduced in version 0.19

The following commonly used units are not yet supported:

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
lm Lumen not directly
lx Lux not directly
px Pixel not directly

It is possible to use data from the model itself in an expression. To reference a property use "object.property". If the property is a compound of fields, the individual fields can be accessed as "object.property.field".

The following table shows some examples:

Parametric Length of a Part-Workbench Cube `Cube.Length` `Length` with units mm
Volume of the Cube `Cube.Shape.Volume` `Volume` in mm³ without units
Type of the Cube-shape `Cube.Shape.ShapeType` String: Solid
Label of the Cube `Cube.Label` String: Label
x-coordinate of center of mass of the Cube `Cube.Shape.CenterOfMass.x` x-coordinate in mm without units
Value of constraint in a sketch `Constraints.Width` Numeric value of the named constraint '`Width`' in the sketch, if the expression is used in the sketch itself.
Value of constraint in a sketch `MySketch.Constraints.Width` Numeric value of the named constraint '`Width`' in the sketch, if the expression is used outside of the sketch.
Value of a spreadsheet alias `Spreadsheet.Depth` Value of the alias "`Depth`" in the spreadsheet "`Spreadsheet`"
Value of a local property `Length` Value of the `Length` property in e.g a Pad object, if the expression is used in e.g Length2 in the same object.

## 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.

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:

```master#modelConstants.Length
```

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

Unfortunately, the integrated checker sometimes claims that a valid name doesn't exist. Continue typing anyway. When you have completed the full reference, the OK button will become active.

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

## Known issues / remaining tasks

• The dependency graph is 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. As a work-around, use multiple spreadsheets -- one to drive your model, and one for reporting.
• The expression parser does not handle parentheses well, and is unable to properly parse some expressions. For example: "= (A1 > A2) ? 1 : 0" results in an error, while "= A1 > A2 ? 1 : 0" is accepted. The expression "= 5 + ((A1>A2) ? 1 : 0)" cannot be entered in any form.
• As stated above, unfortunately, the integrated checker sometimes claims that a valid name doesn't exist. Continue typing anyway. When you have completed the full reference, the OK button will become active.
• There is no expression manager implemented where all expressions in a document are listed, and can be created, deleted, queried, etc.
• The names of Sketcher constraints must not contain any blanks when the value is calculated by an expression, see forum discussion.
• Open bugs/tickets for Expressions can be found in the FreeCAD Bugtracker Expressions category