Angles and Geometric Similarity

Geometrical Similarity

Two figures are said to be geometrically similar if a one-to-one correspondence can be set up between their points such that the ratio of any two lengths in one figure is the same as the ratio of the two corresponding lengths in the other figure. i.e. AB/CD = A'B'/C'D'. Another way of writing this equation is A'B'/AB = C'D'/CD, that is the ratio of any distance in one figure to the corresponding distance in the other figure is the same. This constant is called the scale factor (or ratio of similitude).

If k is the scale factor then A'B' = k.AB. If k = 1 the similarity is a congruence. If k>1 it is a magnification (scaling up). If k<1 it is a diminution (scaling down). Similarity is an equivalence relation. Similar figures are of the same shape.

Since an angle is measured as a ratio of distances, similarity transformations preserve angles. They also preserve parallelism (i.e. lines with zero angle between them). They also preserve ratios of distances along lines, in particular midpoints remain midpoints.

A magnification or diminution (rescaling) has exactly one invariant point. [Proof: There cannot be two invariant points since the distance between them would not have been rescaled. If P is a variant point then it transforms into P', then P' into P'' and so on. If we follow this sequence in its decreasing direction it will approach a limit point that must be invariant.]

All points on a circle centred at the invariant point are transformed into points on a circle k times its radius with the same centre. The order of points on the circle is either preserved or reversed. These two types of scaling are referred to as rotational and reflective.

Angles and Rotations

The assumption that distances in different directions can be compared enables us to define the concept of angle and the related concept of rotation. This assumption is known as the isotropy of space.

The angles round a point add up to one cycle (also termed a complete rotation or Hertz or revolution). Angles of greater than one cycle may be encountered when dealing with rotatory motion, and negative angles may arise in comparing rotations in opposite senses. By an awkward tradition, anticlockwise rotations are counted positive and clockwise negative. A cycle is traditionally divided into 360 degrees (denoted °).

A straight line through a point divides the angles there into two equal half-cycles (or hemicycles) of 180°, one on either 'side' of the line. If angles at a point add to a half-cycle then the outer arms of the figure form one straight line. Two angles adding to a half-cycle are termed supplementary. We will show that the sum of the internal angles in any triangle is a half-cycle.

Two intersecting lines divide the plane into four angles. The 'adjacent' pairs of angles combine to form a half-plane and are supplementary. The 'vertically opposite' angles are equal.

If all four angles are equal, they are quarter-planes or quadrants. A quarter of a cycle is called a right angle of 90°. Angles that add to a right angle are called complementary. Angles less than a right angle are acute, while those greater than a right angle are obtuse. An angle between 180° and 360° is sometimes termed reflex. If two straight lines cross so that all four angles they make are equal the lines are said to be orthogonal or perpendicular (to each other).

There is a unique straight line through a given point, perpendicular to a given line. This remains true when the given point is on the given line. The perpendicular from a point to a line is the shortest line segment from the point to any point on the line and is termed the distance of the point from the line.

If a circle of radius r is drawn, then the compasses can be stepped round the circumference of the circle six times, thus dividing the cycle into six parts, called sextants of 60°. Six circular coins can be placed round one of the same type so that each touches the central coin and its two neighbours.

A line that divides a given angle into two equal parts is an angle bisector. The bisectors of the angles formed by two crossing lines are mutually perpendicular. If lines x, y make an angle A then lines x', y' perpendicular to x, y will make the same angle A.

Bisection of a right angle produces an octant, an angle of 45°. This angle is important in mechanics since it is the angle of elevation of a gun to give the greatest horizontal range for a given initial velocity.

Combining the divisions into sextants and octants we arrive at a division of the cycle into 6×4 = 8×3 = 24 which are known as hours (of arc or of rotation) of 15°. This is related to the way we traditionally measure the lapse of time, since the Earth completes one rotation in 24 hours (of time). This rotation can be measured relative to the sun or stars, and the results are slightly different.

An hour is traditionally divided into 60 minutes, and a minute into 60 seconds. This leads us to a division of the cycle into 24×60 = 1440 minutes of arc and 1440×60 = 86400 seconds of arc.

So that we can speak of the angle between any two lines in the same plane we must allow that parallel lines have a zero angle between them. Through any point there is a unique line parallel to a given line. We allow the case of a line being parallel to itself, so as to include the case when the point is on the line. Parallelism is then an equivalence relation. Lines parallel to a given line are parallel to each other. The equivalence class of all lines parallel to a given line constitutes a direction: they are all the lines 'in a given direction'.

If a line cuts two parallel lines then it makes the same corresponding angles with both. Conversely if these angles are equal the lines are parallel. The alternate angles are also equal. The angles between the parallels and the transversal are supplementary.

The Trigonometric Functions

In a right-angled triangle ONP with right angle at N, angle q at O, and ON = x, NP = y, OP = r, and in the similar triangle ON'P' with ON' = x', N'P' = y', OP' = r' the ratios y/r = y'/r', x/r = x'/r', y/x = y'/x', x/y = x'/y', r/x = r'/x', r/y = r'/y' are functions of the angle q, that is they depend only on the angle and are independent of the size of the triangle.

These ratios are called respectively the sine, cosine, tangent, cotangent, secant, cosecant of the angle q, and these trigonometric functions are written in the shorter notation: sinq, cosq, tanq, cotq, secq, cscq.

It follows from the definitions that:
tanq = sinq/cosq
secq =1/cosq
cscq = 1/sinq
cotq = 1/tanq =cosq/sinq

By the theorem of Pythagoras x² + y² = r². From this we can deduce the relations:
(cosq)² + (sinq)² = 1
(secq)² = (tanq)² + 1
(cscq)² = (cotq)² + 1

Many texts use the misleading notation cos²q for (cosq)².


If we are able to measure a distance of length r along the arc of a circle of radius r then the resulting angle, slightly less than a sextant, is known as a radian. The number of radians in a half-cycle is traditionally denoted by the Greek letter pi (p). This is not a rational number, because the length of an arc of a circle is not a rational concept but can only be approximated. 2p = 6.2831853...

For rational arithmetic we need to find a unit in which a cycle and a radian can be expressed in whole numbers, as cycle/n = radian/m. That is n/m = 2p. Converting the decimal to a continued fraction gives 6 + 1(/3 + 1/(1 + 1/(1 + 1/(7 + 1/(2 + 1/(144 ... Successive approximations are: 6/1 (6), 19/3(6.3...), 25/4 (6.25), 44/7 (6.2857...), 333/53 (6.2830...), 710/113 (6.2831858...), 102573/16325, ...

The last fraction here = 27×3799/25×653 = (27/25)×5.8177642... The continued fraction for this decimal is 5 + 1/(1 + 1/(4 + 1/(2 + 1/(19 + 1/(3 ... which yields an approximation 64/11. Hence 2p = (27×64)/(25×11) = 1728/275 (6.2836...). Multiplying by 50 and adding 1 to the denominator gives: 86400/13751 (6.283179...). Thus a radian is approximately 13751 seconds of arc. The 1728 = 12^3 part of a cycle (50 seconds of arc) is sometimes called a point of rotation. Then a radian is 275 points.