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Translations of Encyclopedia about Mathematics

 

Abscissae, Rays and Bisectors

An abscissa is the shortest connection between two points, always starting at one point and ending at another.

If an abscissa starts at point A and ends at point B, we write it as [AB]. Abscissae are single dimensional, which means they only have length but no width.

Lines are abscissae which continue past the points A and B and which we write as AB. A bisector is defined by two points but continues past them into infinity. Unless parallel to one another, two bisectors intersect each other at a single point on a two-dimensional field.

Rays continue on into infinity only on one side past one of the two points. For example, it might start at point A and continue past point B and is written as [AB]. Such a ray describes for example a sun ray, which starts at some point yet continues on into infinity.

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Plain

A plain is a two-dimensional surface which extends unlimitedly in every direction. We can imagine a plain in the following manner: take a rectangular sheet of paper and imagine the following:

  1. add two other sheets of paper to its left and right sides
  2. continue this process forever to get an unlimited length of one row sheets of papers
  3. add other sheets of papers above and below this row so that the row widen
  4. continue this process forever.

As with a point, a plain has a depth of zero.

A plain is defined by three points which do not lie on the same line or bisector. A plain can also be defined by a line and a point which does not lie on that line. Plains have no bends.

In geometry, plains serve in constructing other two-dimensional objects, such as triangles, circles and polygons. A simple, two-dimensional system of coordinates naturally defines a plain.

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Angle

An angle is defined by the difference in inclines of two rays which meet together at a single point. The point is referred to as the cusp and the rays are referred to as the angle legs.

According to the size of an angle, we can call it a zero, full, acute, right, obtuse, straight and salient angle.
We usually write angles using small Greek alphabets, such as alfa (
a ), beta (b ), gama () and so on.

Angles:

zero : a = 0°
acute: 0°
< a < 90°
right:
a = 90°
obtuse: 90°
< a < 180°
straight:
a = 180°
salient: 180°
< a < 360°
full:
a = 360°

We measure angles with the help of squares, and the units of measurement of angles are the °, gon or radian. Their usual abbreviations are as follows: zero degrees as 0°, acute angles as those having less than 90°, obtuse angles as those more than 90° yet less than 180°, right angles as equal to 90°, straight angles as 180° and salient angles as those greater than 180°.

The gon is an analogue measurement in normal degrees except for that 90° relates to 100 gon. This means that the full angle is equal to 400 gon and the straight angle 200 gon.

The radian is derived from the length of a circle’s cross-section, where its cusp is the centre of the circle. A full angle in this sense would relate to the actual circumference of a unit circle (a circle with radian of 1) and is equal to 2p . To convert between radian and degrees, the following applies: 1 radian = 57.29579 (or 360/2p ).

For converting radian to degrees and back, the following equations may be used:

a °/180° = a (rad)/p
a (rad) = (p /180°) .a °
a ° = (180°/p ) . a (rad)

Measuring angles is important for certain technical tasks, such as defining a given point on the planet. Thanks to trigonometric functions, we can determine the height of a mountain without expending great amounts to drill measuring holes.

We use a right angle every day: walls are built at right angles to the ground, floors are built at right angles to the floor, and so on.

In the same manner, postcards, books, disks and newspapers are rectangular. Determining right angles therefore has its practical use, for which reason leads and levels are used in constructing homes.

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