Kepler and Desargues used the gnomonic projection to relate a plane σ to points on a hemisphere tangent to it. The elliptic plane is the real projective plane provided with a metric. Analogously, a non-Euclidean plane is said to be elliptic or hyperbolic according as each of its lines contains no point at infinity or two points at infinity. A central conic is called an ellipse or a hyperbola according as it has no asymptote or two asymptotes. It does not imply any direct connection with the curve called an ellipse, but only a rather far-fetched analogy. The name "elliptic" is possibly misleading. The distance between a pair of points is proportional to the angle between their absolute polars. Such a pair of points is orthogonal, and the distance between them is a quadrant. Any point on this polar line forms an absolute conjugate pair with the pole. The reason for doing this is that it allows elliptic geometry to satisfy the axiom that there is a unique line passing through any two points.Įvery point corresponds to an absolute polar line of which it is the absolute pole. For example, this is achieved in the hyperspherical model (described below) by making the "points" in our geometry actually be pairs of opposite points on a sphere. This is because there are no antipodal points in elliptic geometry. However, unlike in spherical geometry, the poles on either side are the same. The perpendiculars on the other side also intersect at a point. In fact, the perpendiculars on one side all intersect at a single point called the absolute pole of that line. In elliptic geometry, two lines perpendicular to a given line must intersect. For example, the sum of the interior angles of any triangle is always greater than 180°. The appearance of this geometry in the nineteenth century stimulated the development of non-Euclidean geometry generally, including hyperbolic geometry.Įlliptic geometry has a variety of properties that differ from those of classical Euclidean plane geometry. Because of this, the elliptic geometry described in this article is sometimes referred to as single elliptic geometry whereas spherical geometry is sometimes referred to as double elliptic geometry. However, unlike in spherical geometry, two lines are usually assumed to intersect at a single point (rather than two). Instead, as in spherical geometry, there are no parallel lines since any two lines must intersect. This is an ancient impossibility - it is impossible to accomplish using a compass and an unmarked straightedge.Elliptic geometry is an example of a geometry in which Euclid's parallel postulate does not hold. Trisecting an Angle: To trisect an angle is to use the same procedure as bisecting an angle, but to use two lines and split the angle exactly in thirds. This is possible using a compass and an unmarked straightedge. They share the same degree value.īisecting an Angle: To bisect an angle is to draw a line concurrent line through the angle's vertex which splits the angle exactly in half. \(\measuredangle HRS, \, \measuredangle RST\) are alternate interior angles. They share the same degree value.Īlternate interior angles (Z property): Angles which share a line segment that intersects with parallel lines, and are in opposite relative positions on each respective parallel line, are equivalent. \(\measuredangle IRQ, \, \measuredangle KUQ\) are corresponding angles. They share the same degree value.Ĭorresponding angles (F property): Angles which share a line segment that intersects with parallel lines, and are in the same relative position on each respective parallel line, are equivalent. \(\measuredangle JSR, \, \measuredangle OST\) are vertical angles. Vertical angles (X property): Angles which share line segments and vertexes are equivalent. \(\measuredangle JSN, \, \measuredangle NSK\) are supplementary angles. \(\measuredangle PRQ, \, \measuredangle QRI\) are complementary angles. \(\measuredangle HRL, \, \measuredangle HRO\) are adjacent.Ĭomplementary angles: add up to 90°. Obtuse angle: Angles which measure > 90° - \(\measuredangle CDE\)Īcute angle: Angles which measure 180°, which adds to an angle to make 360° - \(\measuredangle CDE\)'s reflex angle is \(\measuredangle CDF \measuredangle FDE\)Īdjacent angles: Have the same vertex and share a side. Right angle: Angles which measure 90° - \(\measuredangle ABC\) Normally, Angle is measured in degrees (\(^0\)) or in radians rad).
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