Algebra with Matrices, Rectangles and Points

For the Matrix, Rect, IRect and Point classes algebraic operations are possible to some extent. Following we will describe what they are and how they can be used.

Matrix Algebra

Matrix objects m, m1, m2 can be added, subtracted, negated, inverted or multiplied. The result of these operations always is a new matrix. In addition the "absolute value" of a matrix can be calculated - its Euclidean norm - which is a non-negative floating point number.

• m1 + m2 is defined as the matrix [m1.a + m2.a, m1.b + m2.b, ..., m1.f + m2.f].
• m1 - m2 analogous to addition.
• m1 * m2 is defined by m1.concat(m2).
• -m is defined by [-m.a, -m.b, ..., -m.f].
• ~m is defined by invert(m). If m is not invertible (degenerate), then ~m = [0, 0, 0, 0, 0, 0].
• abs(m) is defined by math.sqrt(m.a**2 + m.b**2 + ... + m.f**2).

Following are some examples to illustrate the above.

• check if a matrix is invertible: if not ~m: print("m is not invertible")
• check whether two matrices are (almost) euqal: if abs(m1 - m2) < 1E-6: print("m1 almost equal m2")
• do a complex matrix calculation: m = m1*(m2 + ~m3)*m4

Rectangle Algebra

Rect and IRect objects can be added and subtracted to / from other rectangles or numbers, they can be negated and multiplied with matrices. In addition, inclusion | and intersection & of rectangles with other objects are supported as binary operations. The result of these operations always is a new rectangle with the same type (Rect or IRect) as the left operand.

• r1 + r2 and r1 - r2 are defined component-wise as with matrices. Any of the two operands can be a Rect or an IRect. In addition, r2 can also be a number.
• -r is defined component-wise (Rect or IRect).
• r * m is a new rectangle resulting from r transformed with matrix m. Works for Rect and IRect.
• r1 | r2 (inclusion) is the smallest rectangle containing both, r1 (Rect or IRect) and r2 (Rect, IRect or Point).
• r1 & r2 (intersection) is the largest rectangle that is contained in both, r1 (Rect or IRect) and r2 (Rect or IRect).

Example ("rectangle hull"):

If plist = [p0, p1, ..., pn] is a list of Point objects then r computed by

r = fitz.Rect(p0, p0)
for p in plist:
    r = r | p

is the smallest rectangle containing all points in the list. Note: if you want an IRect perform a r.round() afterwards.

Point Algebra

Point objects can be added, subtracted and negated just like rectangles, yielding new point objects.

• p1 + p2, p1 - p2, -p are defined component-wise like the rectangle operations. p2 can also be a number.
• abs(p) is the Eucldean norm math.sqrt(p.x**2 + p.y**2) as with matrices.
• p * m is a new point resulting from p transformed with matrix m.