OpenVCAD Math Expression Reference

This document covers the operators, functions, and coordinate variables available in OpenVCAD math expression strings. These expressions are used by FloatAttribute, Vec4Attribute, and VolumeFractionsAttribute to define spatially varying fields.

OpenVCAD uses the Exprtk library for parsing math strings. The syntax follows conventions common on most graphing calculators, but Exprtk supports many advanced features. See their documentation for the full specification.

---

Coordinate Systems

The following variables are available in any math expression. You can freely mix variables from different coordinate systems in a single expression.

Coordinate System	Form	Variable Names
Cartesian	f(x, y, z)	x, y, z
Cylindrical	f(rho, phi, z)	rho, phic, z
Spherical	f(r, theta, phi)	r, theta, phis
Signed Distance	f(d)	d

Note: phic is the azimuthal angle in the cylindrical system; phis is the polar angle in the spherical system. Both are in radians.

Note: d is the signed distance to the surface of the object. Zero is the surface, negative values are inside, and positive values are outside. This can be used to create conformal (surface-following) gradients.

Cylindrical Coordinate Definitions

• rho = sqrt(x^2 + y^2) — radial distance from the Z axis

• phic = atan2(y, x) — azimuthal angle in the XY plane (radians, range -pi to +pi)

• z = z — same as the Cartesian Z coordinate

Spherical Coordinate Definitions

• r = sqrt(x^2 + y^2 + z^2) — radial distance from the origin

• theta = atan2(y, x) — azimuthal angle in the XY plane (radians, range -pi to +pi)

• phis = acos(z / r) — polar angle from the +Z axis (radians, range 0 to pi)

---

Operators

Arithmetic & Assignment

Operator	Definition
+	Addition. (x + y)
-	Subtraction. (x - y)
*	Multiplication. (x * y)
/	Division. (x / y)
%	Modulus. (x % y)
^	Exponentiation. (x ^ y)
:=	Assignment. (y := x)
+=	Increment-assign. (x += expr)
-=	Decrement-assign. (x -= expr)
*=	Multiply-assign. (x *= expr)
/=	Divide-assign. (x /= expr)
%=	Modulo-assign. (x %= expr)

Equalities & Inequalities

Operator	Definition
== or =	Equal to. (x == y)
<> or !=	Not equal to. (x != y)
<	Less than. (x < y)
<=	Less than or equal. (x <= y)
>	Greater than. (x > y)
>=	Greater than or equal. (x >= y)

Boolean Operations

Operator	Definition
true	True state (any non-zero value).
false	False state (exactly zero).
and	Logical AND. (x and y)
or	Logical OR. (x or y)
not	Logical NOT. (not x)
xor	Logical XOR.
nand	Logical NAND.
nor	Logical NOR.
mand	Multi-input AND.
mor	Multi-input OR.

---

Functions

General Purpose

Function	Definition
abs(x)	Absolute value of x.
avg(x, y, ...)	Average of all inputs.
ceil(x)	Smallest integer >= x.
floor(x)	Largest integer <= x.
sqrt(x)	Square root (x >= 0).
log(x)	Natural logarithm.
exp(x)	e raised to x.
pow(x, y)	x to the power y.
max(x, y, ...)	Largest of all inputs.
min(x, y, ...)	Smallest of all inputs.
clamp(lo, x, hi)	Clamp x between lo and hi.

Trigonometry

Function	Definition
sin(x)	Sine of x (radians).
cos(x)	Cosine of x (radians).
tan(x)	Tangent of x (radians).
asin(x)	Arc sine (returns radians).
acos(x)	Arc cosine (returns radians).
atan(x)	Arc tangent (returns radians).
deg2rad(x)	Convert degrees to radians.
rad2deg(x)	Convert radians to degrees.

---

Tips

• Parentheses work as expected for grouping: (x + y) * z.

• Constants: Use numeric literals directly: 3.14159, 2.71828. Exprtk also provides pi as a built-in constant.

• Mixing coordinate systems: An expression like rho * sin(phis) + z is perfectly valid — it uses cylindrical rho, spherical phis, and Cartesian z simultaneously.

• Clamping outputs: Use clamp(0, expr, 1) to keep values in [0, 1], which is especially important for color channels and volume fractions.