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Rational Trigonometry to Universal Geometry by Norman Wildberger
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**About this book :- **
**Rational Trigonometry to Universal Geometry ** written by
** Norman Wildberger **

This text introduces a remarkable new approach to trigonometry and Euclidean geometry, with dramatic implications for mathematics teaching, industrial applications and the direction of mathematical research in geometry.

The key insight is that geometry is a quadratic subject. So rational trigonometry replaces the quasi-linear notions of distance and angle with the related, but more elementary, quadratic concepts of quadrance and spread, thus allowing the development of Euclidean geometry over any field. This text covers the key definitions and results of this new theory in a systematic way, along with many applications including Platonic solids, projectile motion, Snell’s law, the problems of Snellius-Pothenot and Hansen, and three-dimensional volumes and surface areas.

**Book Detail :- **
** Title: ** Rational Trigonometry to Universal Geometry
** Edition: **
** Author(s): ** Norman John Wildberger
** Publisher: ** Wild Egg Books
** Series: **
** Year: ** 2005
** Pages: ** 320
** Type: ** PDF
** Language: ** English
** ISBN: ** 097574920X,9780975749203
** Country: ** Australia
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**About Author :- **

The author ** Norman John Wildberger ** is Canadian Australian mathematician. He is an Associate Professor in mathematics at University of New South Wales (UNSW) in Sydney Australia. He has also taught at Stanford University and the University of Toronto.

He earned his Ph.D. from Yale University in 1984. His main research interests are Lie theory, representation theory and hypergroups but he has also done work in number theory, combinatorics and mathematical physics.

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**Book Contents :- **
**Rational Trigonometry to Universal Geometry ** written by
** Norman Wildberger **
cover the following topics.
**Overview **

Introducing quadrance and spread, Laws of rational trigonometry, Why classical trigonometry is hard, Why rational trigonometry is easier, Comparison example, Ancient Greek triumphs and difficulties, Modern ambiguities
**Background**

Fields, Proportions, and determinants, Linear equations, Polynomial functions and zeroes, Quadratic equations
**Cartesian coordinate geometry**

Points and lines, Collinear points and concurrent lines, Parallel and perpendicular lines, Parallels and altitudes, Sides, vertices and triangles, Quadrilaterals, Affine combinations, Perpendicular bisectors
**Reflections**

Affine transformations, Lineations and reflection sequences
**Quadrance **

Quadrances of triangles and quadrilaterals, Triple quad formula, Pythagoras’ theorem, Quadrance to a line, Quadrea, Archimedes’ formula, Quadruple quad formula
**Spread **

Spreads of triangles and quadrilaterals, Cross, Twist, Ratio theorems, Complementary spreads, Spread law, Cross law, Spreads in coordinates, Vertex bisectors
**Triple spread formula**

Triple spread formula, Triple cross formula, Triple twist formula, Equal spreads, Spread reflection theorem, Examples using different fields, Quadruple spread formula
**Spread polynomials**

Combining equal spreads, Spread polynomials, Special cases, Explicit formulas, Orthogonality, Composition of spread polynomials, Cross polynomials
**Oriented triangles and turns **

Oriented sides, vertices and triangles, Turns of oriented vertices, Signed areas
**Triangles **

Isosceles triangles, Equilateral triangles, Right triangles, Congruent and similar triangles, Solving triangles
**Laws of proportion**

Triangle proportions, Quadrilateral proportions, Two struts theorem, Stewart’s theorem, Median quadrance and spread, Menelaus’ and Ceva’s theorems
**Centers of triangles**

Perpendicular bisectors and circumcenter, Formulas for the circumcenter, Altitudes and orthocenter, Formulas for the orthocenter, Incenters
**Isometries**

Translations, rotations, reflections, Classifying isometries
**Regular stars and polygons**

Regular stars, Order three stars, Order five stars, Order seven stars, Regular polygons
** Conics**

Centers of conics, Circles and ribbons, Parabolas, Quadrolas, GrammolasIntersections with lines
**Geometry of circles**

Diameters and chords, Spreads in a circle, Parametrizing circles
**Quadrilaterals**

Cyclic quadrilaterals, Circumquadrance formula, Cyclic quadrilateral quadrea, Ptolemy’s theorem, Four point relation
**Euler line and nine point circle**

Euler line, Nine point circle
**Tangent lines and tangent conics **

Translates and Taylor conics, Tangent lines, Higher order curves and tangents, Folium of Descartes, Lemniscate of Bernoulli
**Triangle spread rules **

Spread ruler, Line segments, rays and sectors, Acute and obtuse sectors, Acute and obtuse triangles, Triangle spread rules
**Two dimensional problems **

Harmonic relation, Overlapping triangles, Eyeball theorem, Quadrilateral problem
**Three dimensional problems**

Planes, Boxes, Pyramids, Wedges, Three dimensional Pythagoras’ theorem, Pagoda and seven-fold symmetry
**Physics applications**

Projectile motion, Algebraic dynamics, Snell’s law, Lorentzian addition of velocities
**Surveying**

Height of object with vertical face, Height of object with inaccessible base, Height of a raised object, Regiomontanus’ problem, Height from three spreads, Vertical and horizontal spreads, Spreads over a right triangle, Spherical analogue of Pythagoras’ theorem
**Resection and Hansen’s problem**

Snellius-Pothenot problem, Hansen’s problem
** Platonic solids**

Tetrahedron, Cube, Octahedron, Icosahedron, Dodecahedron
**Rational spherical coordinates**

Polar spread and quadrance, Evaluating π2/16, Beta function, Rational spherical coordinates, Surface measure on a sphere, Four dimensional rational spherical coordinates, Conclusion
**Rational polar equations of curves**
**Ellipson **
**Theorems with pages and Important Functions**

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