MATH 329 FINAL DEFINITIONS

• ·        
• 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 1-1 ( 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.

• 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

• ·        
• 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.

• 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.

• 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:

•                                                             PQ
• = P’Q’

• A positive orientation of a simple
• closed curve in the plane is the counter clockwise direction.

•             A negative
• orientation is the clockwise direction.

• 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.

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

• 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.