MATH 329 FINAL DEFINITIONS

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Jorge732
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266866
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MATH 329 FINAL DEFINITIONS
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2014-03-18 00:32:22
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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.

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