Rational trigonometry

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Divine Proportions: Rational Trigonometry to Universal Geometry is a book by Norman Wildberger, presenting his reformulation of trigonometry. It was first published in 2005, while the author was teaching and researching at the University of New South Wales, Australia.

Instead of distance and angle, rational trigonometry uses as its fundamental units quadrance (square of distance) and spread (square of sine of angle). This choice of variables enables calculations to produce output results whose complexity matches that of the input data. For instance, in a typical trigonometry problem if rational numbers are assigned to all quadrances and spreads, then the calculated results will be rational numbers (or roots of rational numbers).

This rationality is obtained at the expense of linearity. Unlike the traditional distance and angle units, doubling or halving a quadrance or spread does not double or halve a length or a rotation. Similarly, the sum of two lengths or rotations is not the sum of their individual quadrances or spreads. Although rational trigonometry does not use transcendental trigonometric functions, its results may not be appropriate for many science and engineering problems that rely on linear measurements. In addition, some calculations using rational trigonometry require solving simultaneous linear or quadratic equations or the use of the quadratic formula.

For distinction, Wildberger refers to the traditional trigonometry as classical trigonometry. It is otherwise broadly based on Cartesian analytic geometry, with a point defined as an ordered pair (x, y) and a line as a general linear equation

$Ax\; +\; By\; +\; C\; =\; 0.,$

The mathematics of rational trigonometry is, applications aside, a special instance of the description of geometry in terms of linear algebra (using rational methods such as dot products and quadratic forms), but students who are first learning trigonometry are often not taught about the use of linear algebra in geometry. Changing this state of affairs is a stated aim of Wildberger's book (to paraphrase his comments).

Trigonometry over finite fields

Rational trigonometry makes it possible to do trigonometry over finite fields in the same way as over the field of real numbers.

Quadrance

Compared to distance, quadrance = distance².

Quadrance and distance are concerned with the separation of points. Quadrance differs from standard distance in that it squares the distance. Most immediately, this means that calculating the distance (or, more accurately, quadrance) between two points in 2-dimensional space is easier, as there is no need to find the square root of the sum of the squares of the differences in the x and y coordinates.

In the (x,y)-plane, the quadrance $Q(A\_1,A\_2)$ for the points $A\_1$ and $A\_2$ is defined as

$Q(A\_1,\; A\_2)\; =\; (x\_2\; -\; x\_1)^2\; +\; (y\_2\; -\; y\_1)^2.,$

Spread

Spread, a measure of the separation of lines, is a dimensionless number in the range [0, 1]. The value of the spread is the square of the sine of the angle.

Suppose two lines, $l\_1$ and $l\_2$, intersect at the point A as shown at right. Choose a point $B\; neq\; A$ on $l\_1$ and let C be the foot of the perpendicular from B to $l\_2$. Then the spread s is

$s(ell\_1,\; ell\_2)\; =\; frac\{Q(B,\; C)\}\{Q(A,\; B)\}\; =\; frac\{Q\}\{R\}.$

See also spread polynomials.

Spread compared to angle

In rational trigonometry, spread is a fundamental concept, somewhat but not precisely corresponding to the concept in traditional geometry of angle. Spread describes a relationship between two lines, whereas angle describes a relationship between two rays emanating from a common point.

Compared to the corresponding angles, $mathrm\{spread\}\; =\; sin^2(mathrm\{angle\})$.

Degree	Radian	Spread
0	0	0
30	(1/6)π	1/4
45	(1/4)π	1/2
60	(1/3)π	3/4
90	(1/2)π	1
120	(2/3)π	3/4
135	(3/4)π	1/2
150	(5/6)π	1/4
180	π	0

Spread is not proportional to degrees or radians, and has a period of 180 degrees (π radians).

Laws of rational trigonometry

Wildberger states that there are five basic laws in rational trigonometry and these laws can be easily verified using high-school level mathematics. Some are equivalent to standard trigonometrical formulae with the variables expressed as quadrance and spread.

Triple quad formula

The three points $A\_\{1\},\; A\_\{2\},\; A\_\{3\}$ are collinear precisely when:

$(Q\_\{1\}\; +\; Q\_\{2\}\; +\; Q\_\{3\})^2\; =\; 2(Q\_\{1\}^\{2\}\; +\; Q\_\{2\}^\{2\}\; +\; Q\_\{3\}^\{2\}).,$

This is equivalent to using Heron's formula, the condition for collinearity being that the triangle formed by the three points has zero area.

It can either be proved by analytic geometry (the preferred means within rational trigonometry) or derived from Heron's formula, using the condition for collinearity that the triangle formed by the three points has zero area.

The line $AB,$ has the general form:

$ax\; +\; by\; +\; c\; =\; 0,$

where the (non-unique) parameters $a,$, $b,$ and $c,$, can be expressed in terms of the coordinates of points $A,$ and $B,$ as:

$a\; =\; A\_y\; -\; B\_y,$

$b\; =\; B\_x\; -\; A\_x,$

$c\; =\; A\_xB\_y\; -\; A\_yB\_x,$

so that, everywhere on the line:

$(A\_y\; -\; B\_y)x\; +\; (B\_x\; -\; A\_x)y\; +\; (A\_xB\_y\; -\; A\_yB\_x)\; =\; 0.,$

But the line can also be specified by two simultaneous equations in a parameter $t,$, where $t\; =\; 0,$ at point $A,$ and $t\; =\; 1,$ at point $B,$:

$x\; =\; (B\_x\; -\; A\_x)t\; +\; A\_x,$ and $y\; =\; (B\_y\; -\; A\_y)t\; +\; A\_y,$

or, in terms of the original parameters:

$x\; =\; bt\; +\; A\_x,$ and $y\; =\; -at\; +\; A\_y.,$

If the point $C,$ is collinear with points $A,$ and $B,$, there exists some value of $t,$ (for distinct points, not equal to 0 or 1), call it $lambda,$, for which these two equations are simultaneously satisfied at the coordinates of the point $C,$, such that:

$C\_x\; =\; blambda\; +\; A\_x$ and $C\_y\; =\; -alambda\; +\; A\_y.,$

Now, the quadrances of the three line segments are given by the squared differences of their coordinates, which can be expressed in terms of $lambda,$:

where use was made of the fact that $(-lambda\; +\; 1)^2\; =\; (lambda\; -\; 1)^2$.

Substituting these quadrances into the equation to be proved:

$(Q(AB)\; +\; Q(BC)\; +\; Q(AC))^2\; =\; 2(Q(AB)^\{2\}\; +\; Q(BC)^\{2\}\; +\; Q(AC)^\{2\}),$

$((a^2\; +\; b^2)\; +\; (a^2\; +\; b^2)(lambda\; -\; 1)^2\; +\; (a^2\; +\; b^2)lambda^2)^2\; =\; 2((a^2\; +\; b^2)^2\; +\; ((a^2\; +\; b^2)(lambda\; -\; 1)^2)^2\; +\; ((a^2\; +\; b^2)lambda^2)^2),$

$(a^2\; +\; b^2)^2(1\; +\; (lambda\; -\; 1)^2\; +\; lambda^2)^2\; =\; 2(a^2\; +\; b^2)^2(1\; +\; ((lambda\; -\; 1)^2)^2\; +\; (lambda^2)^2),$

Now, if $A,$ and $B,$ represent distinct points, such that $(a^2\; +\; b^2),$ is not zero, we may divide both sides by $Q(AB)^2\; =\; (a^2\; +\; b^2)^2,$:

$(1\; +\; lambda^2\; -2lambda\; +\; 1\; +\; lambda^2)^2\; =\; 2(1\; +\; (lambda^2\; -2lambda\; +\; 1)^2\; +\; lambda^4),$

$(2lambda^2\; -\; 2lambda\; +\; 2)^2\; =\; 2(1\; +\; lambda^4\; -\; 2lambda^3\; +\; lambda^2\; -\; 2lambda^3\; +\; 4lambda^2\; -\; 2lambda\; +\; lambda^2\; -\; 2lambda\; +\; 1\; +\; lambda^4),$

$4(lambda^2\; -\; lambda\; +\; 1)^2\; =\; 2(2lambda^4\; -\; 4lambda^3\; +\; 6lambda^2\; -\; 4lambda\; +\; 2),$

$4(lambda^4\; -\; lambda^3\; +\; lambda^2\; -\; lambda^3\; +\; lambda^2\; -\; lambda\; +\; lambda^2\; -\; lambda\; +\; 1)\; =\; 4(lambda^4\; -\; 2lambda^3\; +\; 3lambda^2\; -\; 2lambda\; +\; 1),$

$lambda^4\; -\; 2lambda^3\; +\; 3lambda^2\; -\; 2lambda\; +\; 1\; =\; lambda^4\; -\; 2lambda^3\; +\; 3lambda^2\; -\; 2lambda\; +\; 1,$

Q.E.D.

Pythagoras' theorem

The lines $A\_\{1\}A\_\{3\}$ and $A\_\{2\}A\_\{3\}$ are perpendicular precisely when:

$Q\_\{1\}\; +\; Q\_\{2\}=\; Q\_\{3\}.,$

This is equivalent to the Pythagorean theorem.

There are many classical proofs of Pythagoras's theorem; this one is framed in the terms of rational trigonometry.

The spread of an angle is the square of its sine. Given the triangle ABC with a spread of 1 between sides AB and AC,

$Q(AB)\; +\; Q(AC)\; =\; Q(BC),$

where Q is the "quadrance", i.e. the square of the distance.

Construct a line AD dividing the spread of 1, with the point D on line BC, and making a spread of 1 with DB and DC. The triangles ABC, DBA and DAC are similar (have the same spreads but not the same quadrances).

This leads to two equations in ratios, based on the spreads of the sides of the triangle:

$s\_C\; =\; frac\{Q(AB)\}\{Q(BC)\}\; =\; frac\{Q(BD)\}\{Q(AB)\}\; =\; frac\{Q(AD)\}\{Q(AC)\}.$

$s\_B\; =\; frac\{Q(AC)\}\{Q(BC)\}\; =\; frac\{Q(DC)\}\{Q(AC)\}\; =\; frac\{Q(AD)\}\{Q(AB)\}.$

Now in general, the two spreads resulting from dividing a spread into two parts, as line AD does for spread CAB, do not add up to the original spread since spread is a non-linear function. So we first prove that dividing a spread of 1, results in two spreads that do add up to the original spread of 1.

For convenience, but with no loss of generality, we orient the lines intersecting with a spread of 1 to the coordinate axes, and label the dividing line with coordinates $(x\_1,\; y\_1)$ and $(x\_2,\; y\_2)$. Then the two spreads are given by:

$s\_1\; =\; frac\{(x\_2\; -\; x\_2)^2\; +\; (y\_2\; -\; y\_1)^2\}\{(x\_2\; -\; x\_1)^2\; +\; (y\_2\; -\; y\_1)^2\}$

= frac{(y_2 - y_1)^2}{(x_2 - x_1)^2 + (y_2 - y_1)^2},

$s\_2\; =\; frac\{(x\_2\; -\; x\_1)^2\; +\; (y\_2\; -\; y\_2)^2\}\{(x\_2\; -\; x\_1)^2\; +\; (y\_2\; -\; y\_1)^2\}$

= frac{(x_2 - x_1)^2}{(x_2 - x_1)^2 + (y_2 - y_1)^2}.

Hence:

$s\_1\; +\; s\_\; 2\; =\; frac\{(x\_2\; -\; x\_1)^2\; +\; (y\_2\; -\; y\_1)^2\}\{(x\_2\; -\; x\_1)^2\; +\; (y\_2\; -\; y\_1)^2\}$

= 1.,

So that:

$s\_C\; +\; s\_B\; =\; 1.,$

Using the first two ratios from the first set of equations, this can be rewritten:

$frac\{Q(AB)\}\{Q(BC)\}\; +\; frac\{Q(AC)\}\{Q(BC)\}\; =\; 1.,$

Multiplying both sides by $Q(BC)$:

$Q(AB)\; +\; Q(AC)\; =\; Q(BC).,$

Q.E.D.

Spread law

For any triangle $overline\{A\_\{1\}\; A\_\{2\}\; A\_\{3\}\}$ with non zero quadrances:

$frac\{s\_\{1\}\}\{Q\_\{1\}\}=frac\{s\_\{2\}\}\{Q\_\{2\}\}=frac\{s\_\{3\}\}\{Q\_\{3\}\}.,$

This is equivalent to the law of sines.

Cross law

For any triangle $overline\{A\_\{1\}\; A\_\{2\}\; A\_\{3\}\}$ define the cross $c\_3$ as $c\_\{3\}\; =\; 1\; -\; s\_\{3\}$. Then:

$(Q\_\{1\}+Q\_\{2\}-Q\_\{3\})^\{2\}=4Q\_\{1\}Q\_\{2\}c\_\{3\}.,$

This is equivalent to the law of cosines.

Triple spread formula

For any triangle $overline\{A\_\{1\}\; A\_\{2\}\; A\_\{3\}\}$

$(s\_\{1\}+s\_\{2\}+s\_\{3\})^\{2\}=2(\{s\_\{1\}\}^\{2\}+\{s\_\{2\}\}^\{2\}+\{s\_\{3\}\}^\{2\})+4s\_\{1\}s\_\{2\}s\_\{3\}.$

This corresponds (roughly) to the angle sum formulae for sine and cosine.

Calculating quadrance and spread

Given the coordinates of two points ($x\_\{1\},\; y\_\{1\}$) and ($x\_\{2\},\; y\_\{2\}$), the quadrance between them is

$Q\; =\; (x\_\{2\}-x\_\{1\})^\{2\}\; +\; (y\_\{2\}-y\_\{1\})^\{2\}.,$

Given two line segments $A\_\{1\}A\_\{3\}$ and $A\_\{2\}A\_\{3\}$ (forming an angle at point $A\_\{3\}$), the spread between them is

$s\; =\; 1\; -\; frac\{(Q\_\{1\}+Q\_\{2\}-Q\_\{3\})\; ^\{2\}\}\{4Q\_\{1\}Q\_\{2\}\}.,$

Given the coordinates of two points on each of two lines ($x\_\{11\},\; y\_\{11\}$), ($x\_\{12\},\; y\_\{12\}$) and ($x\_\{21\},\; y\_\{21\}$) ($x\_\{22\},\; y\_\{22\}$), the spread $s$ between them can be calculated as:

$s\; =\; frac\{((x\_\{12\}\; -\; x\_\{11\})(y\_\{22\}\; -\; y\_\{21\})\; -\; (x\_\{22\}\; -\; x\_\{21\})(y\_\{12\}\; -\; y\_\{11\}))^2\}\{((x\_\{12\}-x\_\{11\})^2+(y\_\{12\}-y\_\{11\})^2)\; ((x\_\{22\}-x\_\{21\})^2+(y\_\{22\}-y\_\{21\})^2)\}$

or

$s\; =\; frac\{(Delta\; x\_1\; ,\; Delta\; y\_2\; -\; Delta\; x\_2,\; Delta\; y\_1)^\{2\}\}\{Q\_1\; Q\_2\}.,$

If the lines described by the points emanate from or are shifted to the origin by subtracting the coordinates of the first point from each line, as illustrated on the right, the computation simplifies to

$s\; =\; frac\{(x\_1\; y\_2\; -\; x\_2\; y\_1)^\{2\}\}\{Q\_\{1\}Q\_\{2\}\}.,$

See also

• Triple quad formula proof
• Spread polynomials
• Pythagorean theorem proof (rational trigonometry)

References

• Wildberger's rational trigonometry site, including downloadable papers and sections of his book
• A comparison of classical and rational trigonometry
• Alexander Bogomolny A brief introduction to Rational Trigonometry. Cut-the-knot. (2007). .

External links

• Print a protractor to measure spread
• N. J. Wildberger's YouTube channel—lectures on rational trigonometry

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