Divine Proportions: Rational Trigonometry to Universal geometry
Abstract
This book initiates the study of rational trigonometry, and applies it to develop a purely algebraic form of Euclidean geometry valid over arbitrary fields. A new family of spread polynomials are introduced, which are similar to the Chebyshev polynomials but have a more remarkable factorization property.
There are also many applications to engineering problems, to surveying and to pure geometry, such as the metrical relations in the Platonic solids and new equations for curves. The book is available in hard copy and e book form at wildegg.com.
... In this text, we delve into an exploration of Norman Wildberger's Rational Trigonometry (RT) [1], specifically through the lens of Geometric Algebra (GA). Our focus centers on his innovative concepts of spread and quadrance as alternatives to traditional angular measurement and distance, respectively. ...
... Rational Trigonometry, as introduced by Wildberger [1], revolutionizes the conventional understanding of geometry by eliminating the need for trigonometric functions and instead utilizing simpler, more computationally efficient algebraic methods. This approach not only simplifies many geometric calculations but also provides greater precision and clarity, especially in computational contexts. ...
... Although Wildberger [1] suggests that spread is a dimensionless parameter, its units, akin to quadrance, are actually squared distance (m 2 ). ...
This paper explores the possibility of a combination of Rational Trigonometry (RT) and Geometric Algebra (GA) for the advancement of computational geometry. The proposed integration of RT's algebraic efficiency with the structural robustness of GA aims to enhance the speed and precision of geometric problem-solving. The exploration serves as a foundational step in assessing the potential of RT and GA within computational contexts.
... General works on group theory and topology are respectively [11] and [7]. We will in particular work with Euclidean three dimensional vector space R 3 as E 3 in the physical universe, and affine integer lattice sub-spaces of E 3 as E 3 , together with one dimensional scalar spaces of quadrance measure [23], this quantity being the definitely positive quadratic form of the square of Euclidean distance. ...
... In the elementary atom each pair of empty spheres in the void represents a separation of charge and hence carries a potential difference (PD); this physical structure is a capacitor whose adjacent but separated plates carry opposite polarity charges. 23 17 The proton has been conjectured to have structure as a combination particle, this inference based on high energy particle collision data. See Section 2.12 for a treatment of such particles, we note however that the existence of such collision products does not necessarily imply their presence in the proton, just that such nucleic decomposition under extreme stress is possible. ...
... 22 This is analogous to stacked and interleaved foil and paper layers in an old-fashioned electrical capacitor, which assembly clamps fiercely together on application of a potential difference. 23 An electronic capacitor contrary to popular expression does not store charge but rather a separation of charge. We could say that an inherent atomic capacitor stores structural energy. ...
A spatial model based on localised integer lattice sub-spaces embedded in a physical three dimensional universe is analysed for subatomic structures and correlated with the physical universe. Static structure determines physical qualities such as the relationships between mass and charge while providing an infrastructure that supports electronic energy levels and emission spectra. The internal structures of particles built in a local integer lattice, distinguishing between lattice points and point masses which may be located at such points, allows charge and structural energy to be associated with bounded voids in spherical particles. To facilitate the mathematical analysis, balls are decomposed into spheres and shells, each shell being a set of points determined by signed permutations of the coordinates of a representative lattice point and classified into one of seven shell classes according to point topology as defined by the solutions in integers to Diophantine equations. Inherent in the model is localised quantisation of space, such quantisation extending out to the boundaries of atoms and molecules, implying quantisation of electromagnetic and gravitational phenomena. The static structural model provides a mechanism for the existence of sub-atomic particle charges including fractional charges. There is quantitative agreement with other models of atomic structure despite huge differences in concept, our claim for the model presented here is that it successfully provides a mechanism for charge and structural energy indivisibly mated to a locally quantised but globally continuous environment. The implications however require careful consideration, the consistency with other theoretical models is achieved without the need to postulate strong nuclear interactions, anti-matter, mass-energy conversion or positional or existential uncertainty.
... For this reason the derivation of planar a¢ ne rational trigonometry of [21] does not immediately extend to three-dimensional space; one of our goals is to carefully set up exactly such an extension, but crucially not restricting to the Euclidean case. With a more ‡exible approach to metrics, we touch base with chromogeometry, which is a three-fold symmetry which unites Euclidean geometry with two relativistic geometries as laid out in [19], [20] and [22]. ...
... The concepts of B-quadrance and B-spread between points and lines were introduced in [21] for Euclidean planar geometry, which corresponds to B being the 2 2 identity matrix, and are valid in any dimension over an arbitrary symmetric bilinear form, but they are only the "one-dimensional" metrical notions in rational trigonometry. As we go up in dimension, we can expect a hierarchy of additional, and increasingly complicated (but fascinating!) invariants. ...
... These tools will then be the basis for a rigorous framework of vector trigonometry over the threedimensional vector space V 3 , following closely the framework of a¢ ne rational trigonometry in twodimensional Euclidean space formulated in [21], but working more generally with vector triangles in three-dimensional vector space, by which we mean an unordered collection of three vectors whose sum is the zero vector. ...
In three-dimensional Euclidean geometry, the scalar product produces a number associated to two vectors, while the vector product computes a vector perpendicular to them. These are key tools
of physics, chemistry and engineering and supported by a rich vector calculus of 18th and 19th century results. This paper extends this calculus to arbitrary metrical geometries on three-dimensional space, generalising key results of Lagrange, Jacobi, Binet and Cauchy in a purely algebraic setting which applies also to general fields, including finite fields. We will then apply these vector theorems to set up the basic framework of rational trigonometry in the three-dimensional affine space and the related two-dimensional projective plane, and show an example of its applications to relativistic geometry.
... Rational trigonometry, introduced in 2005 in [8], see also [9], is a purely algebraic approach to trigonometry which uses quadrance and spread instead of distance and angle for metrical measurements. This approach has now led to a range of new developments in planar Euclidean geometry [5], [6], chromogeometry [10], [11], and non-Euclidean geometries [12], [13], [14], [15], [1], and [2]. ...
... We now give a brief overview of the principal notions of Euclidean rational trigonometry ( [8] , [9]) and then introduce the corresponding ideas for vector trigonometry. Given a vector v (x; y), its quadrance is the number Q (v) ...
... the last equality since y 2 = r 2 x 2 . Alternatively, use (8) to see that ...
Rational trigonometry is a purely algebraic approach to trigonometry which uses quadrance and spread instead of distance and angle for metrical measurements. In this paper we introduce a variant called vector trigonometry, which is useful for planar applied engineering problems where vector quantities are involved. We derive basic trigonometric laws involving rotor coordinates of length and half-slope.
... This paper repositions and extends triangle geometry by developing it in the wider framework of Rational Trigonometry and Universal Geometry ([10], [11]), valid over arbitrary …elds and with general quadratic forms. Our main focus is on strong concurrency results for quadruples of lines associated to the Incenter hierarchy. ...
... We identify the resulting centers in Kimberling's list. Our basic technology is simple but powerful: we propose to replace the a¢ ne study of a general triangle under a particular bilinear form with the study of a particular triangle under a general bilinear form— analogous to the projective situation as in ([14]), and using the framework of Rational Trigonometry ([10], [11]). By choosing a very elementary standard Triangle— with vertices the origin and the two standard basis vectors— we get reasonably pleasant and simple formulas for various points, lines and constructions. ...
... We begin with some terminology and concepts for elementary a¢ ne geometry in a linear algebra setting, following [10]. Fix a …eld F; of characteristic not two, whose elements will be called numbers. ...
We develop a generalized triangle geometry, using an arbitrary bilinear form in an affine plane over a general field. By introducing standardized coordinates we find canonical forms for some basic centers and lines. Strong concurrencies formed by quadruples of lines from the incenter hierarchy are investigated, including joins of corresponding incenters, Gergonne, Nagel, Spieker points, Mittenpunkts and the New points we introduce. The diagrams are taken from relativistic (green) geometry.
... As a reward, we find that rational trigonometry falls into our laps, essentially for free. Our reformulation extends to a general field (not of characteristic two), to higher dimensions, and even with an arbitrary quadratic form, as in [3] and [4]. We then apply these ideas to give a completely algebraic solution to probably the most famous classical problem in surveying: the resection problem of Snellius and Pothenot (see for example [1]). ...
... Rational trigonometry, developed in [3], see also [6], shows how to enrich and simplify the subject at the same time, leading to greater accuracy and quicker computations. In what follows, we show how the basic ideas follow naturally from our presentation of the theorems of Pythagoras, Euclid and Archimedes. ...
... As demonstrated at some length in [3], these formulas and a few additional secondary ones suffice to solve the majority of trigonometric problems, usually more simply, more accurately and more elegantly than the classical theory involving sin θ, cos θ , tan θ and their inverse functions. As shown in [4], [5] and [6], the same formulas extend to geometry over general fields and with arbitrary quadratic forms, and as shown recently in [7] and [8], the main laws of rational trigonometry also in hyperbolic and elliptic geometry are closely related. ...
Pythagoras' theorem, Euclid's formula for the area of a triangle as one half the base times the height, and Heron's or Archimedes' formula are amongst the most important and useful results of ancient Greek geometry. Here we look at all three in a new and improved light, replacing distance with quadrance, and angle with spread. As an application of this simpler and more elegant rational trigonometry , we show how the famous surveying problem of Snellius and Pothenot, also called resection, can be simplified by a purely algebraic approach.
... To find the length h of the altitude A 3 F is somewhat more work. If we set the origin to be at A 1 , then the line A 1 A 2 has Cartesian equation 5x − 7y = 0 while A 3 = [2, 8] . A well-known result from coordinate geometry states that then the ...
... You will marvel at how generations of mathematicians accepted this theory with scarcely a peep of protest! See [2] for a complete development of this exciting new theory. In what follows, we show how the basic ideas follow naturally from our presentation of the theorems of Pythagoras, Euclid and Archimedes. ...
... These are implicitly contained in the geometrical work of the ancient Greeks. As demonstrated at some length in [2], these formulas and a few additional secondary ones suffice to solve the majority of trigonometric problems, usually more simply, more accurately and more elegantly than the classical theory involving sin θ, cos θ, tan θ and their inverse functions. As shown in [3], the same formulas extend to geometry over general fields and with arbitrary quadratic forms. ...
Pythagoras' theorem, Euclid's formula for the area of a triangle as one half the base times the height, and Heron's or Archimedes' formula are amongst the most important and useful results of ancient Greek geometry. Here we look at all three in a new and improved light, giving a dramatically simpler and more elegant trigonometry.
... Rational trigonometry relies on the idea that algebra is more basic than analysis [27], and that the true measurements in elementary geometry should be quadratic rather than linear. In contrast to classical trigonometry, rational trigonometry does not require using transcendental functions [28]. We have included some rational concepts equivalent to distance and angle in Appendix A. ...
... 1y − e 2 1z ) + 2y(e 1x e 1y − t 1 e 1z ) + 2z(e 1x e 1z + t 1 e 1y ) y = y(t 2 1 − e 2 1x + e 2 1y − e 2 1z ) + 2z(e 1y e 1z − t 1 e 1x ) + 2x(e 1x e 1y + t 1 e 1z ) z = z(t 2 1 − e 2 1x − e 2 1y + e 2 1z ) + 2x(e 1x e 1z − t 1 e 1y ) + 2y(e 1y e 1z + t 1 e 1x ) (27) x = x (t 2 2 + e 2 2x − e 2 2y − e 2 2z ) + 2y (e 2x e 2y − t 2 e 2z ) + 2z (e 2x e 2z + t 2 e 2y ) y = y (t 2 2 − e 2 2x + e 2 2y − e 2 2z ) + 2z (e 2y e 2z − t 2 e 2x ) + 2x (e 2x e 2y + t 2 e 2z ) z = z (t 2 2 − e 2 2x − e 2 2y + e 2 2z ) + 2x (e 2x e 2z − t 2 e 2y ) + 2y (e 2y e 2z + t 2 e 2x ) (28) x = x (t 2 3 + e 2 3x − e 2 3y − e 2 3z ) + 2y (e 3x e 3y − t 3 e 3z ) + 2z (e 3x e 3z + t 3 e 3y ) y = y (t 2 3 − e 2 3x + e 2 3y − e 2 3z ) + 2z (e 3y e 3z − t 3 e 3x ) + 2x (e 3x e 3y + t 3 e 3z ) z = z (t 2 3 − e 2 3x − e 2 3y + e 2 3z ) + 2x (e 3x e 3z − t 3 e 3y ) + 2y (e 3y e 3z + t 3 e 3x ) (29) Equations (27)- (29) could be considered as the description of the orientation with Euler angles, but in a rational way, since they do not contain any transcendental functions. The parameters involved in the calculation of the orientation are t n and e n , where t n is a parameter obtained with Equation (2). ...
In recent years, the recreational and commercial use of flight and driving simulators has become more popular. All these applications require the calculation of orientation in either two or three dimensions. Besides the Euler angles notation, other alternatives to represent rigid body rotations include axis-angle notation, homogeneous transformation matrices, and quaternions. All these methods involve transcendental functions in their calculations, which represents a disadvantage when these algorithms are implemented in hardware. The use of transcendental functions in software-based algorithms may not represent a significant disadvantage, but in hardware-based algorithms, the potential of rational models stands out. Generally, to calculate transcendental functions in hardware, it is necessary to utilize algorithms based on the CORDIC algorithm, which requires a significant amount of hardware resources (parallel) or the design of a more complex control unit (pipelined). This research presents a new procedure for model orientation using rational trigonometry and quaternion notation, avoiding trigonometric functions for calculations. We describe the orientation of a gimbal mechanism presented in many applications, from autonomous vehicles such as cars or drones to industrial manipulators. This research aims to compare the efficiency of a rational implementation to classical modeling using the techniques mentioned above. Furthermore, we simulate the models with software tools and propose a hardware architecture to implement our algorithms.
... Viete's remarkable table [21] has been largely forgotten, but it is exactly a heroic compilation of exact right triangles also obtained from Pythagorean triples to avoid the use of angles. And in 2005, Wildberger introduced the modern form of planar rational trigonometry [23], valid over a general …eld, and then in 2011 [26] went on to extend this rational form to hyperbolic geometry. ...
... where V 3 , as given above, is its associated vector space. These include the B-quadrance and B-spread, as well as the B-quadrea of a triangle in A 3 which extends the de…nition of quadrea in [23]. ...
The Euclidean vector product can be generalised for an arbitrary non-degenerate symmetric bilinear form, which will then be called the B-vector product. This operation, and the properties that follow from it, can be applied to set up a framework for the trigonometry of a general tetrahedron using Wildberger's framework of rational trigonometry. While defining the fundamental trigonometric invariants in three-dimensional space, we can also prove some interesting results pertaining to the ratios of various trigonometric invariants of the general tetrahedron.
... Each plane is represented by a vector , in general. Now, for the plane we can write: (11) and for the plane we can write ...
... The rest is analogues to the case. It can be proved, that extended cross-product is equivalent to a solution of a linear system of equations [7] if projective representation is used [2,11,12,13]. ...
There are many applications in which a bounding sphere containing the given triangle E
3 is needed, e.g. fast collision detection, ray-triangle intersecting in raytracing etc. This is a typical geometrical problem in E
3 and it has also applications in computational problems in general.
In this paper a new fast and robust algorithm of circumscribed sphere computation in the n-dimensional space is presented and specification for the E
3 space is given, too. The presented method is convenient for use on GPU or with SSE or Intel’s AVX instructions on a standard CPU.
... Rational Trigonometry is a theory of trigonometry that presents a mostly algebraic toolkit for solving geometric, surveying and engineering problems involving triangles in a more e¢ cient and elegant fashion. It was developed by the author in his 2005 book Divine Proportions: Rational Trigonometry to Universal Geometry ([4]) which is now available as a free download at Research Gate. This theory uses quadrances, spreads and quadreas instead of distances, angles and areas, and gives a much simpler, more powerful and accurate theory of trigonometry, in which computations run faster, and can often be full precision. ...
... The geometry of these may happily be studied with RT. RT also leads to chromogeometry: a remarkably intertwined triple of planar geometries which transcends Klein's Erlangen program (see [5], [4]). It turns out that Euclidean planar geometry (blue) combine with two relativistic geometry (red and green) in a surprising way that interlinks them and touches almost all aspects of classical Euclidean planar geometry. ...
This is an expository article that introduces the basic framework of Rational Trigonometry (RT), a purely algebraic alternative to the usual classical trigonometry involving transcendental (i.e. infinite) processes. The paper also shows how chromogeometry, a profound three-fold symmetry in planar geometry, arises from RT applied to Euclidean and two relativitistic geometries. This paper will appear in G.
... , we showed that both these metrical notions can also be reformulated projectively and rationally using suitable cross ratios (and no transcendental functions!) The following formula, introduced in [12], is is given in a more general setting in [13]. ...
... In [14], we showed that both these metrical notions can also be reformulated projectively and rationally using suitable cross ratios (and no transcendental functions!) The following formula, introduced in [12], is is given in a more general setting in [13]. ...
We introduce the new notion of sydpoints into projective triangle geometry with respect to a general bilinear form. These are analogs of midpoints, and allow us to extend hyperbolic triangle geometry to non-classical triangles with points inside and outside of the null conic. Surprising analogs of circumcircles may be defined, involving the appearance of pairs of twin circles, yielding in general eight circles with interesting intersection properties. For Parts I–III of this paper see [Zbl 1277.51018; Zbl 1217.51009 and Zbl 1262.51016], respectively.
... In [11], Wildberger introduces a remarkable new approach to trigonometry and Euclidean geometry by replace distance by quadrance and angle by spread, thus allowing the development of Euclidean geometry over any field. The following definition follows from [11]. ...
... In [11], Wildberger introduces a remarkable new approach to trigonometry and Euclidean geometry by replace distance by quadrance and angle by spread, thus allowing the development of Euclidean geometry over any field. The following definition follows from [11]. ...
The quadrance between two points $A_1 = (x_1, y_1)$ and $A_2 = (x_2, y_2)$ is the number $Q (A_1, A_2) = (x_1 - x_2)^2 + (y_1 - y_2)^2$. Let $q$ be an odd prime power and $F_q$ be the finite field with $q$ elements. The unit-quadrance graph $D_q$ has the vertex set $F_q^2$, and $X, Y \in F_q^2$ are adjacent if and only if $Q (A_1, A_2) = 1$. In this paper, we study some colouring problems for the unit-quadrance graph $D_q$ and discuss some open problems.
... In fact, angular functions are new mathematical functions that produce a rectangular signal, in which period is function of angles. Similar to trigonometric functions, the angular functions have the same properties as the precedent, but the difference is that a rectangular signal is obtained instead of a sinusoidal signal [14],[15],[16] and moreover, one can change the width of each positive and negative alternate in the same period. This is not the case of any other trigonometric function. ...
... The traditional trigonometry contains only 6 principal functions: Cosine, Sine, Tangent, Cosec, Sec, Cotan. [15],[16]. But in the Elliptical Trigonometry, there are 32 principal functions and each function has its own characteristics. ...
In industrial electronic systems, power converters with power components are used. Each controlled component has its own control circuit. In this paper, the author proposes an original control circuit for each function in order to replace the different existing circuits. The proposed circuit is the representation of an elliptical trigonometric function as "Elliptic Mar" and "Elliptic Jes-x" that are particular cases of the elliptical trigonometry. Thus, with one function, by varying the values of its parameters, the output waveform will change and can describe more than 12 different waveforms. Finally for each function, a block diagram, a model of the circuit and a programming part are treated using Matlab/Simulink. The results of the studied circuit are presented and discussed.
... Six principal functions (e.g. cosine, sine, tangent …) are used to produce signals that have an enormous variety of applications in all scientific domains [6],[12],[13],[15],[16],[17]. It can be considered as the basis and foundation of many domains as electronics, signal theory, astronomy, navigation, propagation of signals and many others… [17]. ...
... E.g.: the function �í µí°ºí µí±í µí±í µí± (í µí»¼)� " General Jes " and the Elliptical trigonometry �Ejes(α)� is not the same [4], neither in the Rectangular Trigonometry [3] �Rjes(α)� nor in the General Triangular Trigonometry �í µí°ºí µí±í µí±í µí±í µí± (í µí»¼)�. The same for others functions… This variation gives the General Trigonometry a new world vision in all scientific domains and makes a new challenge in the reconstruction of the science especially when working on the economical side of the electrical power, power electronics, electrical transmission, Signal theory and many others domains [15],[16], [17]. The General trigonometric function in written using the following abbreviation " í µí°ºí µí±í µí±í µí±¢í µí±(í µí»¼) " : -the first letter " G " represents the type of the specific trigonometry if it is general case (Fig. 3.c), if not this letter is eliminated (Fig. 3.b), i.e. í µí°ºí µí°¸í µí±í µí±¢í µí±(í µí»¼) is the General Elliptical function, í µí°¸í µí±í µí±¢í µí±(í µí»¼) is the Elliptical function. ...
The General Trigonometry is a new trend of trigonometry introduced by the author into the mathematical domain. It is introduced to replace the traditional trigonometry; it has huge advantages ahead the traditional one. It gives a general concept view of the trigonometry and forms an infinite number of trigonometry branches and each branch has its own characteristics and features. The concept of the General Trigonometry is completely different from the traditional one in which the study of angles will not be the relation between sides of a right triangle that describes a circle as the previous one, but the idea here is to use the relation between angles and sides of a geometrical form (e.g.: circle, elliptic, rectangle, quadrilateral …) with the internal and external circles formed by the intersection of the geometrical form and the positive parts of x'ox and y'oy axis in the Euclidian 2D space and their projections. This new concept of relations will open a huge gate in the mathematical domain and it can resolve many complicated problems that are difficult or almost impossible to solve with the traditional trigonometry, and it can describe a huge number of multi form periodic signals. The most remarkable trigonometry branches are the "Elliptical trigonometry" and the "Rectangular trigonometry" introduced by the author and published by WSEAS. The importance of these trigonometry branches is that with one function, we can produce multi signal forms by varying some parameters. In this paper, an original study is introduced and developed by the author and some few examples are discussed only to give an idea about the importance of the General Trigonometry and its huge application in all scientific domains especially in Mathematics, Power electronics, Signal theory and processing and in Energy Economic Systems.
... In fact, angular functions are new mathematical functions that produce a rectangular signal, in which period is function of angles. Similar to trigonometric functions, the angular functions have the same properties as the precedent, but the difference is that a rectangular signal is obtained instead of a sinusoidal signal [14],[15],[16] and moreover, one can change the width of each positive and negative alternate in the same period. This is not the case of any other trigonometric function. ...
... The traditional trigonometry contains only 6 principal functions: Cosine, Sine, Tangent, Cosec, Sec, Cotan. [15],[16]. But in the Elliptical Trigonometry, there are 32 principal functions and each function has its own characteristics. ...
In industrial electronic systems, specific electronic circuits are used to produce specific signals. In general, one circuit can"t produce more than two or three different signals; this is not the case of the Elliptical Trigonometry, in which one simple circuit can produce more than 12 different signals. In this paper, the author proposes an original control circuit for each function in order to replace the different existing circuits. The proposed circuit is the representation of an elliptical trigonometric function as "Elliptic Mar" and "Elliptic Jes-x" that are particular cases of the elliptical trigonometry. Thus, with one function, by varying the values of its parameters, the output waveform will change and can describe more than 12 different waveforms. Finally, for each function, a block diagram, a model of the circuit and a programming part are treated using Matlab/Simulink. The results of the studied circuit are presented and discussed.
... The results described here are just the tip of an iceberg, leading to many rich generalizations of results of Euclidean geometry, with much waiting to be discovered and explored, see for example [4] for applications to conics and [3] for connections with one dimensional metrical geometry. The basic structure of all three geometries are the same—they are ruled by the laws of rational trigonometry as developed recently in [1], which hold over a general field not of characteristic two. Although over the rational numbers (or the 'real numbers') there are significant differences between the Euclidean (blue) version and the other two (red and green), it is the interaction of all three which yields the biggest surprises. ...
... In universal geometry one regards the quadratic form as primary, not its square root, and by expressing everything in terms of the algebraic concepts of quadrance and spread, Euclidean geometry can be built up so as to allow generalization to the relativistic framework, and indeed to geometries built from other quadratic forms. This approach was introduced recently in [1], see also [5], and works over a general field with characteristic two excluded for technical reasons, as shown in [2]. The possibility of relativistic geometries over other fields seems particularly attractive. ...
Chromogeometry brings together Euclidean geometry (called blue) and two relativistic geometries (called red and green), in a surprising three-fold symmetry. We show how the red and green `Euler lines' and `nine-point circles' of a triangle interact with the usual blue ones, and how the three orthocenters form an associated triangle with interesting collinearities. This is developed in the framework of rational trigonometry using quadrance and spread instead of distance and angle. The former are more suitable for relativistic geometries.
... Trotzdem passiert immer wieder Erstaunliches, etwa mit der Einführung von verdichteten Mengen durch Peter Scholze (*1987) und Dustin Clausen, die ein neues Licht auf die reellen Zahlen und damit auf die Grundlage der Analysis werfen [11]. Ebenso gibt es Versuche in Bereichen der Mathematik auf reelle Zahlen ganz zu verzichten, wie es zum Beispiel Norman Wildberger in seiner Rationalen Trigonometrie versucht, die im Wesentlichen den reellen Winkel durch ein rationales Streckenverhältnis ausdrückt und so die gesamte Trigonometrie ohne reelle Sinus-und Kosinusfunktion reproduziert [12]. [14]. ...
Betrachtungen über das unendlich Kleine und Große in Mathematik, Naturwissenschaft und Philosophie. Sind unendlich große Mengen als Begründung für das unendlich Kleine in der Infinitesimalrechnung sinnvoll? Kann man vernünftig über die Frage diskutieren, ob das Universum endlich oder unendlich groß ist? Sind der Raum und die Zeit kontinuierlich? Und: ist das alles praktisch überhaupt relevant? Fern von finalen Antworten wollen wir zumindest den Fragen ein wenig näherkommen.
... Although messy, it showed that it was indeed possible to establish mathematics without the problematic notion of infinity. In more recent times, Associate Professor Norman Wilderberger of the University of South Wales has followed the same constructivist line and has demonstrated and proposed mathematics that can be done using purely rational numbers (17). ...
The world could be an illusion, just a projection of consciousness. Naturally, we have spent the 5 best part of our known history categorizing the world around us. We have developed mathematics to such a degree that it is difficult to question its fundamental assumptions, let alone looking at our fundamental axioms for a reappraisal, audit, and maybe a fresh start. Nevertheless, as flaws have been discovered in the past, flaws in our mathematical systems will undoubtedly be discovered in the future, potentially bringing numerical discovery to a halt. We should continually search for 10 possible errors in our systems in order to improve, like many before us have. This paper examines the many inconsistencies which exist across a diverse range of knowledge domains to explain the need for such a theory. It provides plausibility conditions to propose a new universal law called The Universal Law of Homeostasis discovered by the author to lay at the intersection of neuroscience, psychology, philosophy, mathematics, logic, computer science and physical theory.
... The exceptionality of zero stands out in a rational context more than in Z because it can be a fraction's numerator, not a fraction's denominator. A way to sort out this algebraic disruption is the convention that a rational expression will be assumed to be vacuous if a choice of variables involves a zero denominator [166]. An alternative solution is seeking an interpretation of "division by zero", but this approach also assumes that zero demands differentiated handling. ...
Zero signifies absence or an amount of no measure. This mathematical object purportedly exemplifies one of humanity’s most splendid insights. Endorsement of the continuum consolidated zero as a cultural latecomer that, at present, everybody uses daily as an indispensable number. Zero and infinity represent symmetric and complementary concepts; why did algebra embrace the former as a number and dismiss the latter? Why is zero an unprecedented number in arithmetic? Is zero a cardinal number? Is it an ordinal number? Is zero a “real” point? Has it a geometrical meaning? To what extent is zero naturalistic?
A preliminary analysis indicates that zero is short of numerical competence, contrived, and unsolvable. We find it elusive when we dig into zero’s role in physics, especially in thermodynamics, quantum field theory, cosmology, and metrology. A minimal fundamental extent is plausible but hard to accept due to zero’s long shade. In information theory, the digit 0 is inefficient; we should replace standard positional notation with bijective notation. In communication theory, the transmission of no bits is impossible, and information propagation is never error-free. In statistical mechanics, the uniform distribution is inaccessible. In set theory, the empty set is ontologically paradoxical. Likewise, other mathematical zeroes are semantically vacuous (e.g., the empty sum, zero vector, zero function, unknot). Because division by zero is intractable, we advocate for the nonzero rational numbers, Q-{0}, to build a new physics that reflects nature’s countable character. We provide a zero-free and unique rational-based representation of the algebraic numbers punctured at the origin, A-{0}, the computable version of the complex numbers.
In a linear scale, we must handle zero as the limit of an asymptotically vanishing sequence of rationals or substitute it for the smallest possible nonzero rational. Zero, as such, is the predetermined power indicating the beginning of logarithmically encoded data via log(1). The exponential function decodes the logarithmic scale’s beables back to the linear scale. The exponential map is crucial to understand advanced algebraic concepts such as the Lie algebra-group correspondence, the Laplace transform, and univariate rational functions in cross-ratio form. Specifically, linear fractional transformations over a ring lead to the critical notion of conformality, the property of a projection or mapping between spaces that preserves angles between intersecting conics. Ultimately, we define “coding space” as a doubly conformal transformation domain that allows for zero-fleeing hyperbolic (logarithmic) geometry while keeping relationships of structure and scale.
... The exceptionality of zero stands out in a rational context more than in Z because it can be a fraction's numerator, not a fraction's denominator. A way to sort out this algebraic disruption is the convention that a rational expression will be assumed to be vacuous if a choice of variables involves a zero denominator [165]. An alternative solution is seeking an interpretation of "division by zero", but this approach also assumes that zero demands differentiated handling. ...
Zero signifies absence or an amount of no magnitude and allegedly exemplifies one of humanity’s most splendid insights. Nonetheless, it is a questionable number. Why did algebra embrace zero and dismiss infinity despite representing symmetric and complementary concepts? Why is zero exceptional in arithmetic? Is zero a “real” point? Has it a geometrical meaning? Is zero naturalistic? Digit 0 is unnecessary in positional notation (e.g., bijective numeration). The uniform distribution is unreachable, transmitting nill bits of information is impossible, and communication is never error-free. Zero is elusive in thermodynam-ics, quantum field theory, and cosmology. A minimal fundamental extent is plausible but hard to accept because of our acquaintance with zero. Mathematical zeroes are semantically void (e.g., empty set, empty sum, zero vector, zero function, unknot). Because “division by zero” and “iden-tically zero” are uncomputable, we advocate for the nonzero algebraic numbers to build new physics that reflects nature’s countable character. In a linear scale, we must handle zero as the smallest possible nonzero rational or the limit of an asymptotically vanishing sequence of rationals. Zero, as such, is a logarithmic scale’s pointer to a being’s property via log(1)). The exponential function, which decodes the encoded data back to the linear scale, is crucial to understanding the Lie algebra-group correspondence , the Laplace transform, linear fractional transformations, and the notion of conformality. Ultimately, we define a “coding space” as a doubly conformal transformation realm of zero-fleeing hyperbolic geometry that keeps universal relationships of structure and scale.
... The exceptionality of zero stands out in a rational context more than in Z because it can be a fraction's numerator, not a fraction's denominator. A way to sort out this algebraic disruption is the convention that a rational expression will be assumed to be vacuous if a choice of variables involves a zero denominator [166]. An alternative solution is seeking an interpretation of "division by zero", but this approach also assumes that zero demands differentiated handling. ...
Zero signifies absence or an amount of no dimension. This mathematical object purportedly exemplifies one of humanity's most splendid insights. Endorsement of the continuum consolidated zero as a cultural latecomer that, at present, everybody uses daily as an indispensable number. Why did algebra embrace zero as a number and dismiss infinity, fully aware that they represent symmetric and complementary concepts? Why does arithmetic deal with zero ad hoc, unlike the rest of the whole numbers? Is zero a cardinal number? Is it an ordinal number? Is zero a "real" point? Has it a geometrical meaning? To what extent is zero naturalistic or universal? In a preliminary analysis, we apprehend that zero is short of numerical competence, contrived, and unsolvable. We find it elusive when we dig into the role zero plays in physics, especially in thermodynamics, quantum field theory, cosmology, and metrology. A minimal fundamental extent is plausible but hard to accept due to zero's long shade. In information theory, digit 0 is inefficient in positional notation, the uniform distribution is inaccessible, the transmission of no bits is impossible, and communication is never error-free. In set theory, the empty set is ontologically disturbing and paradoxical. Likewise, other mathematical zeroes are semantically vacuous and superfluous (e.g., the empty sum, zero vector, zero function, unknot). Because division by zero is uncom-putable, we advocate for the nonzero rational numbers,Q ≡ Q − {0}, to build new physics capable of reflecting nature's countable character. We provide a zero-free and uniqueQ-based representation of the algebraic numbers punctured at the origin,Ǎ ≡ A − {0}, the computable version of the complex numbers. In a linear scale, we must handle zero as the smallest possible nonzero rational or the limit of an asymptotically vanishing sequence of rationals. Zero is the predetermined power indicating the beginning of logarithmically encoded data via log (1). The exponential function, which decodes the logarithmic scale's beables back to the linear scale, is crucial to understanding advanced algebraic concepts such as the Lie algebra-group correspondence, the Laplace transform, and univariate rational functions in cross-ratio form. Specifically, linear fractional transformations over a ring lead to the critical notion of conformality, the property of a projection or mapping between spaces that preserves angles between intersecting conics. Ultimately, we define "coding space" as a doubly conformal transformation framework that keeps structural and scaling relationships to allow for zero-fleeing hyperbolic (logarithmic) geometry.
... Quadrance matrix [18,22] Q AB "¨0 ...
Slides of the talk to the paper 'On the Use of Ternary Products to Characterize the Dexterity of Spatial Kinematic Chains'
... While physicists have long pondered the question of the physical nature of the "continuum", mathematicians have struggled to similarly understand the corresponding mathematical structure. In the last decade, we have seen the emergence of rational trigonometry [1,2] as a viable alternative to traditional geometry, built not over a continuum of "real numbers", but rather algebraically over a general field, so also over the rational numbers, or over finite fields. ...
Universal hyperbolic geometry gives a purely algebraic approach to the subject that connects naturally with Einstein's special theory of relativity. In this paper, we give an overview of some aspects of this theory relating to triangle geometry and in particular the remarkable new analogues of midpoints called sydpoints. We also discuss how the generality allows us to consider hyperbolic geometry over general fields, in particular over finite fields.
... There is also more than one way to do trigonometry. In 2005 I introduced rational trigonometry as a simple yet powerful alternative to classical trigonometry, eliminating the need for transcendental functions and calculators, simplifying many problems, and allowing a more careful and logical derivation of Euclidean geometry ([5], see also [7], [9]). Rational trigonometry uses quadrance and spread instead of distance and angle, giving a purely algebraic approach to the subject. ...
... We note that Wildberger's method "...works over a general field not of characteristic two, and the main laws can be viewed as deformations of those from planar rational trigonometry...". We propose for study the book [76] in which first arose the thought of "universal geometry" and the paper [49] containing a detailed list of Wildberger's work in this direction. ...
... The inequalities in (5) are straightforward because of Proposition 22. The inequality (6) is obtained by applying (11) of Theorem 7. References on the Chebyshev and spread polynomials are [66, 111, 155, 171]. ...
... Therefore a user should take an attention to the correctness of operations. Another interesting application of the projective representation is the rational trigonometry [19]. ...
Many problems, not only in computer vision and visualization, lead to a system of linear equations Ax = 0 or Ax = b and fast and robust solution is required. A vast majority of computational problems in computer vision, visualization and computer graphics are three dimensional in principle. This paper presents equivalence of the cross–product operation and solution of a system of linear equations Ax = 0 or Ax = b using projective space representation and homogeneous coordinates. This leads to a conclusion that division operation for a solution of a system of linear equations is not required, if projective representation and homogeneous coordinates are used. An efficient solution on CPU and GPU based architectures is presented with an application to barycentric coordinates computation as well.
... Indeed, geometry enables visual settings to the understanding of concepts of trigonometry, but once these concepts are retained they are normally incorporated to formulas, i.e., to the algebraic universe. In spite of trigonometry had born before algebra [9], its learning have got gain from this fusion, both for the conceptualizations addressed by the geometry (in benefice of algebra) and for the elimination of the overstated algebraic formalism when addressing to the geometry. Under this point of view, the resolution of triangles is the only subject genuinely belonging to trigonometry because everything else may bijectively be incorporated to algebra and to geometry. ...
The learning of geometric concepts by Visually Impaired People(VIP)is a huge challenge. This paper presents a new dynamic computer-based environment for the learning of geometric concepts through adaptive technology. A case study on learning of geometric concepts in VIP classrooms using the proposed environment is detailed. Several experiments carried out with signed subjects (control group) and VIP subjects (experimental group) using the proposed method is also discussed. The results of this case study have shown that: i. the learning of geometric concepts by the VIP students was done through a peremptory and autonomous way; ii. the VIP students improved their ability to learn, retain and apply obtainedconcepts in othercontexts; iii. the environment innovated the VIPgeometry learning and increased their logical reasoning iv. the continuous use of the environment have enabled them to improve their spatial positioning and motions; v. the environment exhibited a superior performance than the classical geometry teaching inVIP classrooms.The main result of the experiments is that VIP students required (in average) only 20% of the time that was required in classical classes for solve correctly all proposed exercises.
... Ultimately this theory is a natural consequence of Rational Trigonometry ([15], [16], [17]). ...
We initiate a triangle geometry in the projective metrical setting, based on the purely algebraic approach of universal geometry, and yielding in particular a new form of hyperbolic triangle geometry. There are three main strands: the orthocenter, incenter and circumcenter hierarchies, with the last two dual. Formulas using ortholinear coordinates are a main objective. Prominent are five particular points, the b, z, x, h and s points, all lying on the orthoaxis A. A rich kaleidoscopic aspect colours the subject. [For part I, II of the paper see the author [N. J. Wildberger “Universal hyperbolic geometry I: Trigonometry”, (submitted), arXiv.org/abs/0909.1377v1; KoG 14, 3–24 (2010; Zbl 1217.51009)].]
... This example has shown that rational trigonometry often involves solving quadratic equations, and that two possible solutions may arise from the same initial data. In [2] there are some additional laws, called the Triangle spread rules, that exploit convexity over the 'real numbers' to specify exactly which solution to take. They are a bit too involved to state here. ...
Diese kumulative Habilitationsschrift handelt von diskreten Strukturen, den zugehörigen Algorithmen und Anwendungsproblemen in denen diskrete Strukturen vorkommen bzw. zur Lösung nützlich sind. Als Leitfrage im Hintergrund stand: "Wie kann man auf diskreten Strukturen optimieren?" Da dies eine sehr umfassende Frage ist haben wir uns im Rahmen dieser Arbeit, auf einige Anwendungsbeispiele und ausgewählte diskrete Strukturen beschränkt. * Polyominoes * Ganzzahlige Punktmengen * Minimale Orientierungen von Graphen * Vektorapproximation bzw. Optimierung bei einem Textildiscounter * Modellierung bzw. Optimierung von Meinungsbildungsdynamiken
Many algorithms used are based on geometrical computation. There are several criteria in selecting appropriate algorithm from already known. Recently, the fastest algorithms have been preferred. Nowadays, algorithms with a high stability are preferred. Also technology and computer architecture, like GPU etc., plays a significant role for large data processing. However, some algorithms are ill-conditioned due to numerical representation used; result of the floating point representation. In this paper, relations between projective representation, duality and Plucker coordinates will be explored with demonstration on simple geometric examples. The presented approach is convenient especially for application on GPUs or vector-vector computational architectures
In the last half century, several philosophies of mathematics situate themselves in the orbit of materialism. However, their divergences are as important as their commonalities. Some examples include Mario Bunge’s systemic emergentism, Philip Kitcher’s naturalized empiricism, Eric Livingston’s ethnomethodology, the physicalist account of mathematics, the philosophy of mathematical practice, and the cognitive-anthropological approaches of Reviel Netz and Ian Hacking. These philosophies share common arguments, such as the denial of ontological Platonism and the partial or total refutation of the traditional aprioristic epistemology of mathematics. However, only a handful of these materialist approaches emphasize the constitutive role played by the corporeity of mathematicians and the materiality of logical and mathematical “ideograms” (to use the semiotic concept coined independently by Javier de Lorenzo and Brian Rotman). Following Gustavo Bueno’s formalist materialism, this chapter explores how a reconsideration of the role of mathematical glyphs—in Greek geometry, topology, and modern algebra, among other—blurs the boundary between the formal sciences and the natural or empirical sciences. There are no formal sciences, no sciences engaging in the study of pure and immaterial forms (be they logical or mathematical). The so-called formal sciences are, in fact, material sciences, because their construction requires scientists to perform operations with physical objects, such as signs and corporeal lines and circles. Logic thus emerges as a particular kind of mathematics dealing with glyphs that operate in a self-forming way. To end, the chapter offers an alternative (materialist) explanation of the “miracle” of the effectiveness of mathematics in the world.
Neste artigo discutimos um experimento de ensino que versou sobre periodicidade de funções trigonométricas e foi aplicado a dezesseis estudantes do primeiro ano de um curso de licenciatura em matemática. Uma Trajetória Hipotética de Aprendizagem – THA, baseada em Simon e Tzur, foi desenhada contemplando o mecanismo cognitivo centrado na relação atividade-efeito, a partir da ideia de abstração reflexiva, de Piaget. Na pesquisa qualitativa, com elementos do Design Based Research, investigamos como a tarefa matemática promoveu a aprendizagem dos estudantes. Destacamos como os licenciandos utilizaram applets no software GeoGebra e como caracterizaram as funções em estudo e seus respectivos períodos utilizando as linguagens analítica e geométrica. Os resultados indicaram que houve coordenação de registros enquanto os licenciandos modificaram parâmetros das expressões algébricas das funções e os relacionavam com os períodos das mesmas. Identificamos fatores relevantes para a aprendizagem propiciadas pelo experimento de ensino, os quais explicaram a relação entre a aprendizagem conceitual e as tarefas matemáticas propostas na THA. Concluímos que o mecanismo ofereceu uma estrutura para os licenciandos pensarem e avançarem na aprendizagem conceitual.
We study relations between the eight projective quadrangle centroids of a quadrangle in universal hyperbolic geometry.
The Elliptic Jes window form 1 is an original study introduced by the first author in Mathematics
and in Signal Processing. Similar to other windows used in signal processing such as: Hamming, Hanning,
Blackman, Kaiser, Lanczos, Tukey and many other windows, the main goal of introducing the Elliptic Jes
window form 1is to improve the convergence of the Fourier Series at the discontinuity. The different points
between the proposed window function and the previous ones are: -The proposed window function is variable
in form; it can take more than 12 different forms by varying only one parameter.-It can help the Fourier series to
converge more rapidly compared to the traditional ones. –It can be used in both analog design of filters and
digital design of filters. –It is used to truncate the Fourier series with a variable window shape that keep the
necessary information about the signal even after truncation.
In fact, the Elliptic Jes window form 1is an application of the Elliptic Trigonometry in Signal Processing. The
Elliptical Trigonometry is an original study introduced also by the first author in mathematics in 2004, and it
has an ultimate importance in all fields related to the Trigonometry topics such as Mathematics, Electrical
engineering, Electronics, Signal Processing, Image Processing, Relativity, Physics, Chemistry, and many other
domains. The Elliptical Trigonometry is the general case of the traditional trigonometry in which an Ellipse is
used instead of a Circle, so the Elliptical Trigonometry functions are much more important compared to the
traditional trigonometry functions. Therefore, all topics related to the traditional trigonometry will be ultimately
improved by using the Elliptical Trigonometry functions including Signal Processing and Specifically the
design of windows and filters. As a consequence, the Elliptic Jes window form 1 will replace all traditional
window functions.
Poster prepared for presentation at UNSW School of Mathematics & Statistics Postgraduate Conference 2016
So far we have presumed a minimal knowledge of linear algebra on the part of the reader. However in this chapter, we shall use basic properties of determinants and the fact that any invertible matrix is a product of ‘elementary’ matrices.
The transformation formula
that justifies the so called ‘substitution’
or ‘change of variables’
rule for evaluating a Riemann integral is fairly easy to establish in ℝ. In higher dimensions however, the corresponding formula is far more difficult to prove. This is the task we take up in this chapter.
Background:
One of the hardest tasks in developing or selecting grafts for bone substitution surgery or tissue engineering is to match the structural and mechanical properties of tissue at the recipient site, because of the large variability of tissue properties with anatomical site, sex, age and health conditions of the patient undergoing implantation. We investigated the feasibility of defining a quantitative bone structural similarity score based on differences in the structural properties of synthetic grafts and bone tissue.
Methods:
Two biocompatible hydroxyapatite porous scaffolds with different nominal pore sizes were compared with trabecular bone tissues from equine humerus and femur. Images of samples' structures were acquired by high-resolution micro-computed tomography and analyzed to estimate porosity, pore size distribution and interconnectivity, specific surface area, connectivity density and degree of anisotropy. Young's modulus and stress at break were measured by compression tests. Structural similarity distances between sample pairs were defined based on scaled and weighted differences of the measured properties. Their feasibility was investigated for scoring structural similarity between considered scaffolds or bone tissues.
Results:
Manhattan distances and Quadrance generally showed sound and consistent similarities between sample pairs, more clearly than simple statistical comparison and with discriminating capacity similar to image-based scores to assess progression of pathologies affecting bone structure.
Conclusions:
The results suggest that a quantitative and objective bone structural similarity score may be defined to help biomaterials scientists fabricate, and surgeons select, the graft or scaffold best mimicking the structure of a given bone tissue.
1 What's wrong with trigonometry? Trigonometry begins with the study of triangles. A triangle has three side lengths, three vertex angles, and an area. Classical trigonometry studies these seven quantities and the relations between them. It then applies this understanding to more complicated figures such as quadrilaterals and other polygons, along with three dimensional boxes, pyramids and wedges. Then it solves numerous problems in surveying, navigation, engineering, construction, physics, chemistry and other branches of mathematics. Surely understanding a triangle cannot be hard. But each year millions of students around the world are turned off further study in mathematics because of problems learning classical trigonom-etry. Somehow the subject is a lot more complicated than you would at first guess. Why is this? Is it necessarily so? Are there any alternatives? Let's describe the subject of classical trigonometry in a bit more detail. In keeping with tradition, precise definitions will be avoided, because they are invariably too subtle. Even so, you'll perhaps agree that classical trigonometry is difficult, and that it's not surprising that students don't grasp the material well.
The Angular functions are new mathematical functions introduced by the author, they produce rectangular signals, in which period is function of angles and not of time as the previous functions. Similar to trigonometric functions, the angular functions have the same properties as the precedent, but the difference is that a rectangular signal is obtained instead of a sinusoidal signal, and moreover, one can change the width of each positive and negative alternate in the same period. This is not the case of any other trigonometric function. In other hand, one can change the frequency, the amplitude and the width of any period of the signal at any position by using the general form of the angular function. In this paper, an original study is introduced. Thus, the definition of the original part is presented. The angular functions are also defined. These functions are very important in technical subjects. They will be widely used in mathematics and in engineering domains, especially in power electronics, signal theory, propagation of signals and many other topics. Moreover, the Angular functions are the basis of the Elliptical trigonometry and the rectangular trigonometry in which they are new domains introduced in mathematics by the author.
The thinking for the last two thousand years rests on the false assumptions that distance is the best way of measuring the separation of points, and that angle is the best way of measuring the separation of lines. So in order to study triangles students must first understand circles; they learn about π, lengths of circular arcs and the transcendental circular functions such as cosθ and sinθ that relate arc length on a circle to x and y projections. They study the relations between the circular functions and their inverse functions, ponder complicated graphs, and try to remember lots of special values. Advanced students see the infinite power series that calculators use to approximate true values. This is complicated stuff, especially if you try to do it correctly. Yet a triangle itself is seemingly quite an elementary object. Why should the theory of a triangle be so complicated? Until now there has been no reasonable alternative. So educators have resigned themselves to the difficulties, and each year millions of students memorize the formulas, pass the tests (or not), and then promptly forget the unpleasant experience. And mathematicians wonder why the general public regards their beautiful subject with distaste bordering on hostility. In this article, I am going to explain to you the right approach-called rational trigonometry. It is based on the idea that algebra is more basic than analysis, and that the true measurements in elementary geometry should be quadratic rather than linear in nature. Because this is a short paper, and I want to impart to you a working knowledge of the subject, the proofs are short. Giving more details is not hard, and the results are simple enough for a high school course. Much more information can be found in 'Divine Proportions: Rational trigonometry to Universal Geometry' ((Wildberger)). There the theory is developed over a general field (not of characteristic two) and many useful new formulas and applications appear. This is then used to develop Euclidean geometry in a general and powerful way which fully exploits the power of the Cartesian coordinates. This paper is an introduction to these ideas, accessible to high school students.
Trigonometry is a branch of mathematics that deals with relations between sides and angles of triangles. It has some relationship to geometry, though there is disagreement on exactly what that relationship is. For some, trigonometry is just a subtopic of geometry. The trigonometric functions are very important in technical subjects like science, engineering, architecture, and even medicine. In this paper, the elliptical trigonometry is introduced in order to be in the future a part of the trigonometry topic. Thus, the definition of this original part is presented. The elliptic trigonometric functions are also defined. The importance of these functions in producing different signals and forms are analyzed and discussed.
Trigonometry is a branch of mathematics that deals with relations between sides and angles of triangles. It has some relationship to geometry, though there is disagreement on exactly what that relationship is. For some, trigonometry is just a subtopic of geometry. The trigonometric functions are very important in technical subjects like Astronomy, Relativity, science, engineering, architecture, and even medicine. In this paper, the rectangular trigonometry is introduced in order to be in the future a part of the General trigonometry topic. Thus, the definition of this original part is presented. The rectangular trigonometric functions are also defined. The importance of these functions is by producing multi signal forms by varying some parameters of a single function. Different signals and forms are analyzed and discussed. The concept of the rectangular Trigonometry is completely different from the traditional trigonometry in which the study of angles is not the relation between sides of a right triangle that describes a circle as the previous one, but the idea here is to use the relation between angles and sides of a rectangular form with the internal and external circles formed by the intersection of the rectangular form and the positive parts of x'ox and y'oy axis in the Euclidian 2D space and their projections. This new concept of relations will open a huge gate in the mathematical domain and it can resolve many complicated problems that are difficult or almost impossible to solve with the traditional trigonometry, and it can describe a huge number of multi form periodic signals.
In industrial electronic systems, power converters with power components are used. Each controlled component has its own control circuit. In this paper, the authors propose an original control circuit in order to replace the different existing circuits. The proposed circuit is the representation of an elliptical trigonometry function. Thus, by varying the value of one of its parameter, the output waveform will change. Finally, Labview and Matlab simulation results of the studied circuit are presented and discussed.
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