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1993
Abstract In this chapter we will explore in more detail the geometry of a Euclidean space P of dimension 2 or Euclidean plane. We will begin by looking at the concept of orientation, which allows us to distinguish between ‘left-handed’ and ‘right-handed’ coordinate systems.
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Abstract In this chapter we will explore in more detail the geometry of a Euclidean space P of dimension 2 or Euclidean plane. We will begin by looking at the concept of orientation, which allows us to distinguish between ‘left-handed’ and ‘right-handed’ coordinate systems.
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1981
The properties in Chapter II belong to both absolute geometry and to Euclidean geometry, but the axioms there are sufficient to imply only a part of Euclidean geometry. For example, they do not imply that the angle sum of a triangle is 180°. To establish this, and many other facts of Euclidean geometry, some assumption equivalent to Euclid’s “parallell
Paul Kelly, Gordon Matthews
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The properties in Chapter II belong to both absolute geometry and to Euclidean geometry, but the axioms there are sufficient to imply only a part of Euclidean geometry. For example, they do not imply that the angle sum of a triangle is 180°. To establish this, and many other facts of Euclidean geometry, some assumption equivalent to Euclid’s “parallell
Paul Kelly, Gordon Matthews
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2002
Crisp plane geometry starts with points, then lines and parallel lines, circles, triangles, rectangles, etc. In fuzzy plane geometry we will do the same. Our fuzzy points, lines, circles, etc. will all be fuzzy subsets of R × R. We assume the standard xy— rectangular coordinate system in the plane.
James J. Buckley, Esfandiar Eslami
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Crisp plane geometry starts with points, then lines and parallel lines, circles, triangles, rectangles, etc. In fuzzy plane geometry we will do the same. Our fuzzy points, lines, circles, etc. will all be fuzzy subsets of R × R. We assume the standard xy— rectangular coordinate system in the plane.
James J. Buckley, Esfandiar Eslami
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1994
Let us divide a straight line p by a point O into two half-lines + p and − p (Fig. 5.1). Let us choose on p a unit of length. The coordinate x of a point M is defined to be the distance of the point M from the point O prefixed by a sign, plus or minus (the so-called directed distance) according as M belongs to + p or − p, respectively.
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Let us divide a straight line p by a point O into two half-lines + p and − p (Fig. 5.1). Let us choose on p a unit of length. The coordinate x of a point M is defined to be the distance of the point M from the point O prefixed by a sign, plus or minus (the so-called directed distance) according as M belongs to + p or − p, respectively.
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2010
context: no traces of ancient sources providing a context of similar problems, but: connection to non-trivial theorems/methodological devices with “potential” for future mathematical theories: (i) Diameters in configuration of tangent circles, theorem of Menelaus (→ points of similarity (projective geometry)) (ii) Arithmetical ...
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context: no traces of ancient sources providing a context of similar problems, but: connection to non-trivial theorems/methodological devices with “potential” for future mathematical theories: (i) Diameters in configuration of tangent circles, theorem of Menelaus (→ points of similarity (projective geometry)) (ii) Arithmetical ...
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