OpenSCAD User Manual/Mathematical Operators
Scalar Arithmetical Operators
The scalar arithmetical operators take numbers as operands and produce a new number.
The "-" can also be used as prefix operator to negate a number.
Relational operators produce a Boolean result from two operands.
If both operands are simple numbers, the meaning is self-evident.
If both operands are strings, alphabetical sorting determines equality and order. E.g., "ab" > "aa" > "a".
If both operands are Booleans, true > false. If one operand is Boolean, the other operand is converted to Boolean before the comparison is made.
If both operands are vectors, OpenSCAD performs an element-by-element comparison and can only result in true if the vectors are equal in size and each and every pair of elements results in true upon the comparison. Otherwise, false is returned.
Vectors of different sizes are treated as unequal for '==' and '!=' operators, and always result in false for '>', '>=', '<' and '<=' operators. In fact the same principle applies for all comparison between dissimilar types of operand, e.g. comparing a string with a number.
Note that  ≠ 1.
undef doesn't equal anything but undef. undef compares ('>' etc.) anything result in false.
nan doesn't equal anything. See Numbers.
All logical operators take Booleans as operands and produce a Boolean. Non-Boolean quantities are converted to Booleans before the operator is evaluated.
|!||logical unary NOT|
false || [false] is also
Note that how logical operators deal with vectors is different than relational operators:
[1, 1] > [0, 2] is
[false, false] && [false, false] is
The ?: operator can be used to conditionally evaluate one or another expression. It works like the ?: operator from the family of C-like programming languages.
|? :||Conditional operator|
a=1; b=2; c= a==b ? 4 : 5;
If a equals b, then c is set to 4, else c is set to 5.
The vector-number operators take a vector and a number as operands and produce a new vector.
|*||multiply all vector elements by number|
|/||divide all vector elements by number|
L = [1, [2, [3, "a"] ] ]; echo(5*L); // ECHO: [5, [10, [15, undef]]]
The vector operators take vectors as operands and produce a new vector.
The "-" can also be used as prefix operator to element-wise negate a vector.
L1 = [1, [2, [3, "a"] ] ]; L2 = [1, [2, 3] ]; echo(L1+L1); // ECHO: [2, [4, [6, undef]]] echo(L1+L2); // ECHO: [2, [4, undef]]
Vector Dot-Product Operator
If both operands of multiplication are simple vectors, the result is a number
according to the linear algebra rule for dot product.
c = u*v; results in . If the operands'
sizes don't match, the result is
If one or both operands of multiplication are matrices, the result is a simple vector or matrix according to the linear algebra rules for matrix product. In the following, A, B, C... are matrices, u, v, w... are vectors. Subscripts i, j denote element indices.
For A a matrix of size n × m and
B a matrix of size m × p, their product
C = A*B; is a matrix of size n × p with elements
C = B*A; results in
undef unless n = p.
For A a matrix of size n × m and
v a vector of size m, their product
u = A*v; is a vector of size n with elements
In linear algebra, this is the product of a matrix and a column vector.
For v a vector of size n and
A a matrix of size n × m, their product
u = v*A; is a vector of size m with elements
In linear algebra, this is the product of a row vector and a matrix.
Matrix multiplication is not commutative: , .