# MATH 329 FINAL DEFINITIONS

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1. 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 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.
2. 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
3. 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.
4. 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.
5. 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:

•                                                             PQ
• = P’Q’
6. 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.
7. 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.
8. 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.
9. 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.
 Author: Jorge732 ID: 266866 Card Set: MATH 329 FINAL DEFINITIONS Updated: 2014-03-18 04:32:22 Tags: MATH329 Folders: Description: MATH 329 FINAL DEFINITIONS Show Answers: