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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).

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