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Definition 16.2
A transformation in Absolute Geometry is a
function(or mapping) f that associates with each point P with another point P’; we denote this f(P) = P’ such that
A mapping f is said to preserve
If a transformation preserves collinearity then
 ·
 A transformation in Absolute Geometry is a
 function(or mapping) f that associates with each point P with another point P’; we denote this f(P) = P’ such that
o f is 11 ( i.e. P ≠ Q ⇒ f(P) ≠ f(Q))
o f is onto
· A mapping f is said to preserve collinearity if given three collinear points, their images under f are also three collinear points.
·If a transformation preserves collinearity then it is called a linear transformation.

Definition 16.6 A transformation f, defined by f(x,y)= (f1(x,y), f2(x,y)), is
 A transformation f, defined by f(x,y)= (f1(x,y),
 f2(x,y)), is linear iff f1 and f2 are of the form
 f1 = ax + by + c and f2 = dx + ey + g for some real numbers a,b,c,d,e,gELE R

Definition 16.7
Given a transformation of the plane f, A is said
to be a fixed point for f if
A transformation of the plane is called the
identity mapping, iff
 ·
 Given a transformation of the plane f, A is said
 to be a fixed point for f if f(A)=A.
 ·
 A transformation of the plane is called the
 identity mapping, iff every point of the plane is a fixed point of the
 transformation. This transformation is denoted e.

Definition 17.1 Linear Reflection IMPORTANT!!! MIGHT BE ON TEST
 If a transformation f has the
 property that some fixed line l is the perpendicular bisector of each segment
 linePP’ for any point P on the plane, where P’ = f(P), then f is a linear
 reflection on the line l and l is called the line of reflection.

Definition 17.4
Any mapping of the plane that preserves
distances is called
 Any mapping of the plane that
 preserves distances is called an isometry (or motion, rigid motion, or
 Euclidean motion).
 This implies, f is an isometry iff
 for any points P, Q, with P’ = f(P), and Q’ = f(Q) , the new segment has the
 same measure as the old:

Definition 17.6
Orientation
 A positive orientation of a simple
 closed curve in the plane is the counter clockwise direction.
 A negative
 orientation is the clockwise direction.

Definition 17.7A linear transformation of the plane is called
 A linear transformation of the
 plane is called direct iff it preserves the orientation of any triangle, and
 opposite iff it reverses the orientation of any triangle.

Definition 18.1 Translation
A translation in the plane is
 A translation in the plane is the
 product of two reflections sl and sm, where l and m are parallel lines.

Definition 18.3 Rotation
 A rotation is the product of two reflections sl and sm,
 where l and m are nonparallel lines, i.e. they meet at some point P.
 P is called
 the center of rotation.

