3.1.4.1 Two-operand (binary) forms and precedence
Most two-operand binary
forms have meanings dependent on the types of
their arguments. An exhaustive summary of the possibilities is given
in the following table.
| Left | Op | Right | Result | Description
|
|---|
| scalar | + | scalar
| scalar | Scalar sum.
|
| vector | + | vector
| vector | Vector sum.
|
| point | + | vector
| point | Point-vector affine sum.
|
| vector | + | point
| " | "
|
| scalar | - | scalar
| scalar | Scalar difference.
|
| vector | - | vector
| vector | Vector difference.
|
| point | - | point
| vector | Point-point affine difference.
|
| point | - | vector
| point | Point-vector affine difference.
|
| scalar | * or
. | scalar
| scalar | Scalar product.
|
| scalar | * or
. | vector
| vector | Scalar-vector product.
|
| vector | * or
. | scalar
| " | "
|
| vector | * | vector
| vector | Vector cross-product.
|
| vector | . | vector
| scalar | Vector dot product.
|
| scalar | ^ | scalar
| scalar | Raise scalar to scalar power.
|
| transform | ^ | integer
| transform | Raise transform or integer power.
|
| transform | * or
. | point
| point | Affine point transform (right-to-left).
|
| transform | * or
. | vector
| vector | Affine vector transform (right-to-left).
|
| transform | * or
. | transform
| transform | Transform composition (right-to-left).
|
| point | then | transform
| point | Affine point transform (left-to-right).
|
| vector | then | transform
| vector | Affine vector transform (left-to-right).
|
| transform | then | transform
| transform | Transform composition (left-to-right).
|
| scalar | / | scalar
| scalar | Scalar division.
|
| vector | / | scalar
| vector | Vector component-wise division by scalar.
|
| point | ' | x, y, or z
| scalar | Point component extraction.
|
| vector | ' | x, y, or z
| scalar | Vector component extraction.
|
Operator precedence
is shown in this table.
| Op | Precedence
|
|---|
' | highest (most tightly binding)
|
^ |
|
- | (unary negation)
|
*
.
/ |
|
+
- |
|
then | lowest (least tightly binding)
|
All operations are left-associative
except for ^.
Parentheses ( )
are used for grouping to override precedence in the usual way.
As you can see, the dot operator .
is usually a synonym for run-of-the-mill multiplication, *.
The meanings differ only for vector operands. The then
operator
merely reverses the operand
order with respect to normal multiplication *. The intent
here is to make compositions read more naturally. The code
(1,2,3) then scale(2) then rotate(30) then translate([1,3,0])
expresses a series of successive modifications to the point,
whereas the equivalent form
translate([1,3,0]) * rotate(30) * scale(2) * (1,2,3)
will be intuitive only to mathematicians (and perhaps Arabic
language readers).