Sketch: Two-operand (binary) forms

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 to 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).