Unity.Mathematics cheat sheet

Most of the methods in Mathematics have many overloads for different combinations of types. For example, math.abs() takes vector arguments, not just scalars, e.g. math.abs(new int3(5, -7, -1)) returns new int3(5, 7, 1). This cheat sheet does not exhaustively demonstrate all of the overloads. Consult the API reference for the full list.

In this page:

• Types
• Vector creation and copying
• Matrix creation and copying
• Vector and matrix operators
• Arithmetic
• Exponents and logarithms
• Rounding and signs
• Value checks
• Conversion
• Interpolation and clamping
• Picking between two values
• Hashing
• Bitwise and boolean operations
• Trig, degrees, and radians
• Vector geometry
• Cardinal direction vectors
• Rotations and transforms
• Generating random numbers
• Generating noise

Types

math	Class containing many static mathematical constants and methods.
noise	Class containing static methods for generating noise.
quaternion	Struct representing a rotation.
Random	Struct for generating random numbers.
RigidTransform	Struct representing a transform matrix.

Scalar types:

half	A 16-bit floating-point number.

Vector types:

bool2, bool3, bool4
int2, int3, int4
uint2, uint3, uint4
float2, float3, float4
half2, half3, half4
double2, double3, double4

Matrix types:

bool2x2, bool2x3, bool2x4, bool3x2, bool3x3, bool3x4, bool4x2, bool4x3, bool4x4
int2x2, int2x3, int2x4, int3x2, int3x3, int3x4, int4x2, int4x3, int4x4
uint2x2, uint2x3, uint2x4, uint3x2, uint3x3, uint3x4, uint4x2, uint4x3, uint4x4
float2x2, float2x3, float2x4, float3x2, float3x3, float3x4, float4x2, float4x3, float4x4
double2x2, double2x3, double2x4, double3x2, double3x3, double3x4, double4x2, double4x3, double4x4

Vector creation and copying

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Matrix creation and copying

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Vector and matrix operators

The vector and matrix operators (+, -, *, /, %, --, ++, ==, !=, <, >, <=, >=) operate upon corresponding component pairs:

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The integer vector and matrix types also have bitwise operators: &, |, ~, <<, >>.

Arithmetic

math.fmod	The floating point remainder of x/y.
math.mad	Componentwise (a * b + c) on three scalars or vectors.
math.modf	Modulus and fractional component.
math.csum	Horizontal sum of components of a vector.
math.rcp	Reciprocal (1 divided by the value).

Exponents and logarithms

math.ceillog2	Ceiling of the base-2 logarithm.
math.ceilpow2	Smallest power of two greater than or equal to the input.
math.exp	The constant e raised to a power.
math.exp10	The value 10 raised to a power.
math.exp2	The value 2 raised to a power.
math.floorlog2	Floor of the base-2 logarithm.
math.log	Natural logarithm.
math.log10	Base-10 logarithm
math.log2	Base-2 logarithm.
math.pow	Raised to a power.
math.rsqrt	Reciprocal of the square root.
math.sqrt	Square root.

Rounding and signs

math.abs	Absolute value.
math.ceil	Round up.
math.floor	Round down.
math.round	Round to nearest.
math.sign	The sign of a value: +1, 0 , or -1

Value checks

math.isfinite	Is finite floating-point value?
math.isinf	Is infinite floating-point value?
math.isnan	Is NaN?
math.ispow2	Is power of 2?

Conversion

math.asdouble	Reinterpret the bits of a 64-bit integer as a double.
math.asfloat	Reinterpret the bits of a 32-bit integer as a float.
math.asint	Reinterpret the bits of a float or uint as an int.
math.aslong	Reinterpret the bits of a double or 64-bit integer as a long.
math.asulong	Reinterpret the bits of a double or 64-bit integer as a ulong.
math.asuint	Reinterpret the bits of a float or uint as a uint.
math.f16tof32	The floating point representation of a half-precision floating-point value.
math.f32tof16	Nearest half-precision floating-point representation of a floating-point value.
math.frac	Fractional part of a floating-point value.
math.trunc	Integral part of a floating-point value (rounded towards zero).

Interpolation and clamping

math.clamp	Clamp a value into an interval.
math.lerp	Linear interpolation between two values.
math.nlerp	Normalized linear interpolation between two quaternions.
math.remap	Non-clamping, linear remapping of a value from a source range to a destination range.
math.saturate	Clamp a value into the interval [0, 1].
math.slerp	Spherical interpolation between two quaternions.
math.smoothstep	Smooth Hermite interpolation between 0.0f and 1.0f.
math.step	Return 1.0f if x >= y, otherwise returns 0.0f.
math.unlerp	Normalize a value into a range. (Opposite of lerp.)

Picking between two values

math.shuffle	Pick one or more specific components from two vectors.
math.select	Pick between two values based on a boolean. Similar to the ternary operator, but you can also pick specific components of two vectors.
math.cmax	Largest component of a vector.
math.cmin	Smallest component of a vector.
math.max	Largest of two values.
math.min	Smallest of two values.

Hashing

See the Collections package for more hashing options.

math.hash	Hash of a value.
math.hashwide	When hashing multiple values together, it's often more efficient to separately pass them to hashwide(), combine the results, then hash() the combination.

Bitwise and boolean operations

math.all	True if all booleans are true.
math.any	True if any boolean is true.
math.bitmask	Bitmask of a bool4: one bit per component (4 bits in total) in LSB order (lower to higher).
math.countbits	Count of the 1-bits.
math.compress	Packs mask-enabled components of a vector to the left.
math.lzcnt	Leading zero count of the bits.
math.reversebits	Reverse the order of the bits.
math.rol	Rotate bits left.
math.ror	Rotate bits right.
math.tzcnt	Trailing zero count of the bits.

Trig, degrees, and radians

math.acos	Arccosine.
math.asin	Arcsine.
math.atan	Arctangent
math.atan2	Arctangent for 2 arguments.
math.cos	Cosine.
math.cosh	Hyperbolic cosine.
math.math.degrees	Degrees from radians.
math.radians	Radians from degrees.
math.sin	Sine.
math.sincos	Sinecosine.
math.sinh	Hyberbolic sine.
math.tan	Tangent.
math.tanh	Hyberbolic tangent.

Vector geometry

math.cross	Cross product.
math.distance	Distance between two points (1 to 4 dimensions).
math.distancesq	Square root of distance between two points (1 to 4 dimensions).
math.dot	Dot product.
math.faceforward	Flips a vector if two other vectors point in the same direction (i.e. the angle between them is less than or equal to 90 degrees).
math.length	Distance of a point from the origin.
math.lengthsq	Square root of the distance of a point from the origin.
math.normalize	Normalized vector.
math.normalizesafe	Normalized vector. Returns the default value if the normalized vector is not finite.
math.project	Project a vector on to another.
math.projectsafe	Project a vector on to another. Returns the default value if the projected vector is not finite.
math.reflect	Reflection of an incident vector and a normal vector.
math.refract	Refraction of an incident vector and a normal vector.
math.transform	Transform a 3-dimensional vector with a 4x4 matrix.

Cardinal direction vectors

math.forward	The forward axis in Unity coordinates.
math.back	The back axis in Unity coordinates.
math.up	The up axis in Unity coordinates.
math.down	The down axis in Unity coordinates.
math.left	The left axis in Unity coordinates.
math.right	The right axis in Unity coordinates.

Matrix ops

math.determinant	Determinant of a matrix.
math.fastinverse	Fast matrix inverse for rigid transforms (orthonormal basis and translation).
math.inverse	Inverse of a matrix or quaternion.
math.mul	Matrix multiplication.
math.transpose	Transpose of a matrix.
math.unitlog	Natural logarithm of a unit length quaternion.

Rotations and transforms

math.conjugate	Conjugate a quaternion. (Flips the signs of x, y, and z but not w.)
math.orthonormalize	Orthonormalize a float3x3 matrix.
math.rotate	Rotate a vector by a unit quaternion.
math.unitexp	Natural exponent of a quaternion. (Assumes w is zero.)
quaternion.AxisAngle	Quaternion representation of an axis-angle rotation.
quaternion.Euler	Quaternion representation of an Euler angle rotation (axis order specified by argument).
quaternion.EulerXYZ	Quaternion representation of an Euler angle rotation (axis order XYZ).
quaternion.EulerXZY	Quaternion representation of an Euler angle rotation (axis order XZY).
quaternion.EulerYXZ	Quaternion representation of an Euler angle rotation (axis order YXZ).
quaternion.EulerYZX	Quaternion representation of an Euler angle rotation (axis order YZX).
quaternion.EulerZXY	Quaternion representation of an Euler angle rotation (axis order ZXY).
quaternion.EulerZYX	Quaternion representation of an Euler angle rotation (axis order ZYX).
quaternion.LookRotation	Quaternion representing a rotation derived from a unit-length forward vector and a unit-length upwards vector.
quaternion.LookRotationSafe	Quaternion representing a rotation derived from a forward vector and an upwards vector.
quaternion.RotateX	Quaternion representation of a rotation around the X axis.
quaternion.RotateY	Quaternion representation of a rotation around the Y axis.
quaternion.RotateZ	Quaternion representation of a rotation around the Z axis.
float4x4.LookAt	View matrix derived from an eye position, a target point, and a uni-length upwards vector.
float4x4.Ortho	Orthographic projection matrix
float4x4.OrthoOffCenter	Off-center orthographic projection matrix.
float4x4.PerspectiveFov	Perspective projection matrix based on field of view.
float4x4.PerspectiveOffCenter	Off-center perspective projection matrix
float4x4.Scale	Matrix representing a scale transform.
float4x4.Translate	Matrix representing a translation transform.
float4x4.TRS	Matrix representing a combined translation, rotation, and scale transform.
float3x3.Scale	Matrix representing a scale transform.
RigidTransform.Translate	Matrix representing a translation transform.

RigidTransform, float3x3, and float4x4 also have most of the same rotation methods as quaternion.

Generating random numbers

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Generating noise

noise.cellular(float2)	2D Cellular noise ("Worley noise") with standard 3x3 search window for good feature point values.
noise.cellular(float3)	3D Cellular noise ("Worley noise") with 3x3x3 search region for good F2 everywhere, but a lot slower than the 2x2x2 version.
noise.cellular2x2	2D Cellular noise ("Worley noise") with a 2x2 search window.
noise.cellular2x2x2	3D Cellular noise ("Worley noise") with a 2x2x2 search window.
noise.cnoise	Classic Perlin noise.
noise.pnoise	Classic Perlin noise, periodic variant.
noise.psrdoise	2-D tiling simplex noise with fixed or rotating gradients and analytical derivative.
noise.psrnoise	2-D tiling simplex noise with fixed or rotating gradients, but without the analytical derivative.
noise.snoise	Simplex noise.
noise.srdnoise	2-D non-tiling simplex noise with fixed or rotating gradients and analytical derivative.
noise.srnoise	2-D non-tiling simplex noise with fixed or rotating gradients, without the analytical derivative.