Rational trigonometry
Divine Proportions: Rational Trigonometry to Universal Geometry is a book by Dr. Norman Wildberger of The
University of New South Wales, presenting the author's reformulation of
trigonometry.
Instead of distance and
angle, rational trigonometry uses as its fundamental units
quadrance and
spread . 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 .
Encyclopedia
Divine Proportions: Rational Trigonometry to Universal Geometry is a book by Dr. Norman Wildberger of The
University of New South Wales, presenting the author's reformulation of
trigonometry.
Instead of distance and
angle,
rational trigonometry uses as its fundamental units
quadrance and
spread . 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 .
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 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 , 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 .
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 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 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 -plane, for the points and , is defined as:
Spread
Spread, a measure of the separation of lines, is a dimensionless number in the range [0, 1].
At an origin , being the point at which two lines converge, and picking two points on each line; on one line and on the other, such that the line formed between and is perpendicular to the line upon which lies.
For the lines and , is defined as:
See also spread polynomials.
Spread compared to the traditional geometrical concept of 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:
Spread is not proportional to degrees or radians, and has a period of 180 degrees .
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 are
collinear precisely when:
This is equivalent to using
Heron's formula, the condition for collinearity being that the triangle formed by the three points has zero area.
Triple quad formula proof.
Pythagoras' theorem
The lines and are perpendicular precisely when:
This is equivalent to the
Pythagorean theorem.
Pythagoras' theorem proof
Spread law
For any triangle with non zero quadrances:
This is equivalent to the
law of sines.
Cross law
For any triangle define the cross . Then:
This is equivalent to the
law of cosines.
Triple spread formula
For any triangle
This corresponds to the
angle sum formulae for sine and cosine.
Calculating quadrance and spread
Given the coordinates of two points and , the quadrance between them is:
Given the coordinates of two points on each of two lines , and , the spread between them can be calculated as:
or
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:
See also
References
- , including downloadable papers and sections of his book
External links
