Downloadable Math Papers

This page was last updated on 11/28/2018.

The following is a list of 50 papers written by John Kennedy. The topics vary from pure math and logic to representations of integers, computer graphics, parse trees, and fundamental algorithms and techniques associated with numerical analysis and computing in general. Miscellaneous topics include solving cubic and quartic equations, Huffman coding related to data compression, number theory and modular arithmetic, the basic geometry of camera lenses and magnifying glasses, the geometry of a sextant, spherical trigonometry, an analysis of a piston, the mathematics of Vernier scales, rounding/truncating functions, regular polyhedra, trochoidal curves, dihedral groups and permutations, probability, and how to steer a pirate ship. For the most part these papers discuss applications of math or computer science.

The papers listed below are in no particular order and vary in terms of a required math background. Most of these papers are short and simple. The four major exceptions to this statement are the paper on error correcting codes, the paper on number theory, the paper on spherical trigonometry, and the paper containing notes on infinite series. These four papers are not simple, nor do any of these four make for a quick or a simple read.

Files of the type *.PDF are in the popular format known as the Portable Document Format. The *.PDF files may be read and/or printed using Adobe's free reader called Acrobat Reader version 9.0 or later. (See Adobe to download the free Acrobat Reader.)

The page counts do not include any title pages. All papers have been formatted for duplex or 2-sided printing. 

If your browser has been properly setup to load and display PDF files then to read a paper you can simply left-click on the PDF filename. An alternative is to right-click the PDF filename to cause a popup menu to appear and then you can choose "Save link as..." to actually save a copy anywhere you wish on your computer. Then you would use the Adobe Acrobat reader program to open the saved file to read and/or print it. 

RPN Perspective 
This paper provides a historical background for the Reverse Polish Notation and was originally published in the PPC Calculator Journal, Volume 9 Number 5, August 1982 pp. 26-29. This paper sets the historical record straight for when the Polish Notation was first invented. The time frame was in the year 1928, long before electronic computers or electronic calculators. This paper gives a logical background for understanding how a Reverse Polish Logic calculator works. It explains the relationship between logical and mathematical notations and explains the fundamentals of a parenthesis-free notation that can be used to represent mathematical expressions. It also discusses parallel meanings between logic and mathematics. 
11 pages. 
RPNBG.PDF  185,425 bytes   01/05/2012  7:59 pm. 

A Brief Introduction To Bezier Curves  
This paper describes what Bezier curves are and derives the basic cubic polynomial equations that are used to generate Bezier curves. It explains the amazing midpoint property of such curves and shows examples of different types of Bezier curves. It also shows how to make a smooth joining of two such curves and this discussion is a great aid in understanding why the mathematical definition of a smooth curve is one that requires a continuous first derivative. This paper also discusses the requirements for exact symmetry of Bezier curves and illustrates this with several more examples. The paper also shows how to construct points on a Bezier curve using a fixed ratio to make points that divide the lengths of several related vectors. The fixed ratio is a generalization of the midpoint property shown earlier. The algorithm described at the end of this paper is due to Casteljau and this algorithm is related to what are called the Bernstein polynomials. The paper concludes by showing relationships between text and Bezier curves. 
32 pages. 
Bezier.PDF  276,971 bytes   01/05/2012  8:01 pm. 

The Basic Geometry Behind A Camera Lens And A Magnifying Glass  
This paper describes the fundamental workings of a camera lens and a magnifying glass. The only math required is an understanding of the point-slope form for the equation of a line. This paper describes the physics behind a convex lens and derives what is called the Thin-Lens equation. It also describes the practical uses of the focal length. It explains the use of the magnification factor and its relationship to focal lengths. 
11 pages. 
CamLens.PDF  307,665 bytes   01/05/2012  8:02 pm. 

A General Divisibility Test and the Use of Synthetic Substitution 
This paper describes a little-known divisibility test for integer division that has wide applicability. In fact, the test can be applied to any integer with any integer divisor. The test makes use of synthetic substitution and takes advantage of viewing any integer as a special polynomial value. This paper also explains the relationship between this test and other popular divisibility tests such as the tests for even division by 2, 3, 5, 7, and 11. 
10 pages. 
DivTest.PDF  156,564 bytes   01/05/2012  8:04 pm. 

A Gallery of Rounding and Truncating Functions 
This paper discusses many ways in which numbers can be rounded or truncated. It contains graphs of the five fundamental functions that may be called Int(X), Frac(X), Round(X), Ceiling(X), and Floor(X). This paper carefully discusses and shows the relationships between all of these functions. It also shows how to define a robust modulo operator for real numbers and shows why we might want to consider Int(X) and Frac(X) as the two fundamental rounding-related functions. It shows how to use the Int(X) and Frac(X) functions to round X to any decimal position and how to extract specific digits from X. It also shows how to define several utility type functions related to Int(X) and Frac(X). In total, there are 15 rounding/truncating related functions and this paper explains how all of these are related to the two fundamental Int(X) and Frac(X) functions. This paper is a must read for anyone whose does almost any kind of either discrete or floating point arithmetic using any kind of a computer or any kind of mathematical software. 
20 pages. 
GalleryRound.PDF  124,792 bytes   09/18/2015  12:13 pm.  

Negative Numbers and Binary Representations and the Significance of Complements 
This paper introduces the concept of representing integers using various bases. This paper shows how subtraction can be accomplished without borrowing using the concept of complements. It then introduces three popular formats for negative numbers that have the names Sign-Magnitude, One's Complement, and Two's Complement. The paper shows examples of how to detect both underflow and overflow with integer arithmetic. The paper also suggests the reader may continue by studying the IEEE 754 Floating-Point Standard. This paper finishes by very carefully defining the two standard types of radix complements and proving their range bound intervals are as described. 
10 pages. 
NegativesBinaryComplements.PDF  148,582 bytes   10/30/2016  9:22 am.  

Prime Factoring The Factorial of an Integer> 
This paper shows how to predict the multiplicity of any prime factor of n!. It shows practical applications in terms of how to compute large combinations and large binomial coefficients using multiple precision arithmetic. It also contains an intellectually interesting formula for the prime factorization of n! 
7 pages. 
NFact.PDF  149,112 bytes   01/05/2012  9:10 pm.  

An Efficient Algorithm For Computing Large Integer Powers of Any Base 
This paper discusses an efficient technique for computing b^n where n is a large integer, and b can be any type of base including a real number, a polynomial, a matrix, or any other base type for which multiplication is defined. The algorithm depends on the bit pattern found in the binary representation of the integer n. The paper also describes the relationship between this algorithm and one used to multiply any two integers that involves only shifting bits and adding and subtracting. This algorithm is also known as the Fast Exponential Method and has applications in modular arithmetic and cryptography. 
9 pages. 
NPwr.PDF  44,599 bytes   01/15/2015  12:18 am.  

Fractions And Decimals  
This paper was published in the PPC Calculator Journal, Volume 5 Number 6, July, 1978 pages 17-19. This paper explains the fundamental relationships between fractions and decimals and explains the relationship between finite decimals and periodic decimals. The results can be used to predict both the non-periodic and the periodic lengths of any repeating decimal and it explains when, why and how certain fractions result in finite decimals. It also contains the surprising result that any integer which does not have 2 or 5 as a prime factor has some multiple which consists of all 9's. This paper also mentions applications related to multiple precision arithmetic and cryptography. 
8 pages. 
FracDec.PDF  176,577 bytes   01/05/2012  9:58 pm.  

Algorithm To Convert A Decimal To A Fraction  
This paper describes an algorithm that can be used to convert an arbitrary decimal to a fraction. This technique depends on continued fractions and is of major importance in numerical analysis. The algorithm has been implemented on Hewlett-Packard scientific calculators and is also in Mathematica and MATLAB. 
4 pages. 
Dec2Frac.PDF  116,098 bytes   01/05/2012  10:04 pm.  

Parse Trees  
This paper discusses parse trees and shows how to construct on paper the parse tree for any mathematical or logical expression. Parse trees are recursive data structures and this paper also discusses recursive algorithms that can be used to traverse parse trees. Two examples are given of functions that can be used to compute any math expression or any symbolic logic expression. Another part of this paper discusses the relationship between a parse tree and a stack for the purpose of performing calculations like a Reverse Polish logic calculator. 
10 pages. 
PareseTrees.PDF  202,521 bytes   01/05/2012  9:12 pm.  

A Piston Model  
This paper shows a model of a piston head and derives the basic equation that governs the motion of a piston head as it moves in a periodic way, back and forth, inside a cylinder. This paper explains why the periodic motion of a piston head is NOT a pure sine wave, even though at first glance the graph may look very similar to a sine wave. We also calculate and analyze the velocity of the piston head and determine where it moves the fastest and where it moves the slowest. We also show how the angular velocity of a turning crankshaft relates to the linear speed of the piston head as it traverses back and forth. A piston moving back and forth inside a cylinder provides a means of converting linear motion to circular motion and vice versa. Such a model is fundamental to understanding just one aspect of both fluid pumps and internal combustion gasoline engines. 
5 pages. 
PistonModel.PDF  182,212 bytes   01/05/2012  9:41 pm.  

Subtraction Using Complements  
This short paper describes a simple algorithm for performing subtraction of integers. The algorithm is unusual because it describes subtraction as a special process involving only complements and addition. Examples are given in base 10 and the general algorithm is explained using any base b where b is 2 or larger. When b=2 the algorithm is used to explain how a computer actually performs subtraction by performing a special kind of addition. This paper also briefly discusses problems that must be avoided when computers perform computations with numbers. 
4 pages. 
SubtractionUC.PDF  39,377 bytes   01/25/2012  2:12 pm.  

How Does A Sextant Work?  
This paper describes the basic geometry behind a sextant which is a simple navigational device with mirrors that is used to determine the angle between an object in the sky and the horizon. 
5 pages. 
Sextant.PDF  844,908 bytes   01/05/2012  9:40 pm.  

Pythagorean Triples  
This paper contains 12 items related to Pythagorean Triples. Of significance is a formula and technique for generating primitive Pythagorean triples. 
4 pages. 
PythagT.PDF  105,300 bytes   01/05/2012  10:29 pm.  

A Remarkable Half-Angle Relationship  
This paper shows a simple figure related to the unit circle and a special line that intersects that circle. The slope of that line is related to points on the unit circle and three application examples are given. One of those applications involves a special integration technique by Karl Weierstrass (1815-1897), commonly taught in 2nd semester calculus. Another application is related to finding rational points on the unit circle and suggests a way of thinking about a class of Diophantine equations. A third application is a ruler and compass construction using the unit circle and a couple of special lines to construct the reciprocal of a number. 
8 pages. 
HalfAngleSpecial.PDF  181,774 bytes   01/05/2012  9:52 pm.  

The Simplex Algorithm Divorced From Inequalities  
This paper was written to explain the basis for the Simplex Algorithm and to show how the algorithm arises naturally in the context of using matrices to solve linear systems. A comparison is made with the steps used to find the row reduced echelon form of a matrix. The algorithm is carefully explained and an example is given of cycling. This paper also discusses nonstandard problems and the use of artificial variables. 
18 pages. 
SimplexP.PDF  244,605 bytes   01/05/2012  9:53 pm.  

Permutations, Regular Polygons, Symmetries, Rotations, Reflections, Dihedral groups, Orthogonal Matrices, Isometries, Cycles, Transpositions, Parity, And All That Jazz  
This paper develops some of the fundamental properties of permutations by considering the properties of linear transformations that map regular polygons onto themselves. This paper shows relationships among permutations, regular polygons, symmetries, rotations, reflections, orthogonal matrices, isometries, and dihedral groups. It also discusses permutations as products of cycles and transpositions and introduces the idea of a Cauchy number for a permutation. This in turn leads to the parity properties of permutations. This paper assumes a minimal background in linear algebra. Some experience with groups or group theory would also be helpful. 
24 pages. 
PermsAndRots.PDF  553,419 bytes   01/15/2012  9:09 am.  

How Does A Vernier Scale Work?  
This paper describes the basic math behind a Vernier scale. Vernier scales are often found on scientific instruments and make it possible to read values accurately to within 1/100 of a unit. 
7 pages. 
VernierScale.PDF  861,718 bytes   01/09/2012  12:10 pm.  

Huffman Coding  
This paper introduces the concept of Huffman coding that is concerned with data compression. This paper also discusses Morse Code, an early example of data compression that pre-dates the computer by 100 years. The discussion includes a very detailed analysis of how to encode the algorithm that can be used to create an efficient coding scheme for any kind of data. 
21 pages. 
Huffman.PDF  535,679 bytes   11/17/2018  1:01 pm.

Some Polynomial Theorems  
This paper contains a collection of 31 theorems, lemmas, and corollaries that help explain some fundamental properties of polynomials. The statements of all these theorems can be understood by students at the precalculus level, even though a few of these theorems do not appear in any precalculus text. However, to understand the proofs requires a much more substantial and more mature mathematical background, including proof by mathematical induction and some simple calculus. Of significance are the Division Algorithm and theorems about the sum and product of the roots, two theorems about the bounds of roots, a theorem about conjugates of irrational roots, a theorem about integer roots, a theorem about the equality of two polynomials, theorems related to the Euclidean Algorithm for finding the G.C.D. of two polynomials, and theorems about the Partial Fraction Decomposition of a rational function and Descartes' Rule of Signs. It is rare to find proofs of either of these last two major theorems in any precalculus text. 
30 pages. 
PolyTheorems.PDF  123,152 bytes   10/14/2012  7:28 pm.  

Signed Angles and the Perpendicular Vector Concept Applied to Find the Area of a Polygon  
This paper introduces the concept of the signed angle between two vectors and combines this with the idea of a particular vector perpendicular to a given vector to obtain a most simple and efficient formula that computes the area of any polygon in two-dimensional space. A special sum of dot products using the polygon's vertex point coordinates is all that is required to compute the area enclosed by the polygon. 
14 pages. 
VectorPerp.PDF  179,050 bytes   01/05/2012  9:56 pm.  

Equivalence Relations And Partitions  
This paper develops the fundamental properties of equivalence relations. It also establishes the connections between equivalence relations and partitions of a set. This paper shows how every equivalence relation determines a unique partition, and vice versa, it shows how every partition determines a unique equivalence relation. In fact, this paper begins by defining an ordered pair in terms of sets. This paper is for mature students who would like to see applications within the theory of sets. 
4 pages. 
EquivalenceRelations.PDF  130,660 bytes   01/15/2012  9:12 am.  

Some Very Special Trigonometric Function Values  
This paper develops the exact values of both sine and cosine functions for angles that are multiples of 9 degrees. The values are given in an exact radical form. Then this paper shows how these values relate to the Golden Ratio and to a pentagon and also how they relate to 3D polyhedra such as the dodecahedron and the icosahedron. 
11 pages. 
SpecialTrigValues.PDF  158,799 bytes   02/05/2018  5:39 pm.  

Establishing Regular Polyhedra Information  
This paper develops formulas for the surface area and volume of regular polyhedra as well as formulas for the radii of both inscribed and circumscribed spheres. Even more important it develops the exact values of the dihedral angles for all five of the regular Platonic solids. The five regular polyhedra are the tetrahedron, the hexahedron or cube, the octahedron, the dodecahedron, and the icosahedron. 
28 pages. 
EstablishingPolyhedraInfo.PDF  190,092 bytes   01/15/2015  12:13 am.  

Continued Fractions  
This paper is an introduction to continued fractions. It introduces the notation and terminology related to continued fractions and presents two Fundamental Theorems about continued fractions. It shows how to solve a quadratic equation using continued fractions. It also illustrates how to convert an arbitrary decimal to a fraction. This algorithm is of fundamental importance. This paper also introduces a matrix notation for both simple and general continued fractions and shows examples of constants and function expressions in continued fraction forms. These forms are interesting alternatives when compared to infinite series for the same constants and functions. 
20 pages. 
ConFrac.PDF  190,685 bytes   01/05/2012  10:14 pm.  

Earth Navigation and the Estimation of Dip  
This paper describes some basic earth navigational facts and uses some mathematics to derive an estimation for what is called Dip. Dip is a phenomena that is associated with the refraction of light. It is essential to know about Dip in order to make effective use of a sextant. 
6 pages. 
Navigation01.PDF  183,225 bytes   01/05/2012  10:07 pm.  

How To Steer A Pirate Ship  
This paper is all about setting up and solving a differential equation related to a classical problem of pursuit. Although the problem is stated in terms of a pirate ship, the same result could be applied to the case of a dog chasing a cat or a lion chasing a zebra. This paper assumes you have completed two semesters of calculus. 
13 pages. 
PirateShip.PDF  170,356 bytes   01/05/2012  10:08 pm.  

Statistical Summations  
This paper develops a set of alternative formulas which can be used to accumulate standard statistical sums. The formulas are recursive in nature. The paper first summarizes all the standard sums that are used in performing linear regression. Then it introduces the alternative formulas and thereby a means of accumulating sums that is more accurate than using the standard formulas. Next it derives the equivalences of the standard regression formulas and demonstrates how to compute the slope and y-intercept of the line of best fit. 
14 pages. 
StatSum.PDF  139,080 bytes   06/21/2018  1:00 pm.  

You Can't Do That  
This paper presents a clever technique for exchanging the contents of any two variables in computer memory, but without using a temporary third variable. The problem solution surprises most programmers who think the solution requires using of a temporary third memory variable. The algorithm employs the XOR operator three times and works with any two same size memory variables. We raise the equivalent question of how to exchange the contents of a glass of milk and a glass of wine. Most people can only imagine doing this by using a third glass to temporarily hold and exchange one of the fluids. This only seems to be the same problem to exchange the contents of two computer variables. The power and simplicity of XOR is made evident by the technique that is described in just 3 lines. An explanatory example is also included for this brain-teaser that is completely described on just one page! 
1 page. 
Cant.PDF  74,981 bytes   09/22/2017  5:52 pm.  

3-Dimensional Graphing Transformations  
This paper discusses the fundamental mathematics and algorithms behind making 3-dimensional perspective drawings. It describes how to transform world coordinates to eye coordinates and how to perform 3D clipping with respect to a viewing pyramid and how to map eye coordinates onto the viewing plane and from there to the final display device plane. 
18 pages. 
3DSetup.PDF  338,168 bytes   01/05/2012  10:09 pm.  

Bresenham's Integer Only Line Drawing Algorithm  
This is a fundamental paper on how to efficiently draw a line on any display device that is composed of discrete pixels. The fundamental ideas in this paper can also be applied to drawing circles and ellipses. This contains what is perhaps the first and most fundamental techniques of all computer-based graphics. See also the related papers for drawing circles and ellipses. 
8 pages. 
BresenL.PDF  166,142 bytes   01/05/2012  10:11 pm.  

A Fast Bresenham Type Algorithm For Drawing Circles  
This paper describes how to quickly and efficiently draw a circle on any display device that is composed of discrete pixels. It contains an algorithm that takes advantage of the circle's 8-point symmetry and minimizes the errors associated with finding the nearest pixel for each next point to be plotted. The algorithm uses exact integer arithmetic only. See also the related papers for drawing ellipses and lines. 
7 pages. 
BCircle.PDF  170,125 bytes   01/05/2012  10:12 pm.  

A Fast Bresenham Type Algorithm For Drawing Ellipses  
This paper describes how to quickly and efficiently draw an ellipse on any display device that is composed of discrete pixels. It contains an algorithm that takes advantage of the ellipse's 4-point symmetry and minimizes the errors associated with finding the nearest pixel for each next point to be plotted. The algorithm uses exact integer arithmetic only. See also the related papers for drawing circles and lines. 
7 pages. 
BEllipse.PDF  176,462 bytes   01/05/2012  10:13 pm.  

Some Simple Number Theory  
This is a collection of notes on number theory that begins with a discussion of the Well Ordering Principle and the Principle of Mathematical Induction. Then we discuss binomial coefficients and prove the Binomial Theorem and introduce Pascal's Triangle. Then we discuss divisors and state the Division Algorithm for real numbers before introducing a non-traditional form of the modulo operator. We discuss Greatest Common Divisors and show the importance of writing the GCD as a linear combination of its two arguments. We introduce prime numbers and prove the Fundamental Theorem of Arithmetic. We discuss Euclid's Algorithm and show the computer code for that algorithm. We discuss the simple linear Diophantine equation. Then we start the major topic of modular arithmetic and explain the use of equivalence classes. This paper is particularly strong on its coverage of modular arithmetic. We give Fermat's Little Theorem and discuss the Euler Phi function. Then we prove Euler's Theorem and Wilson's Theorem. We define a group and derive the basic properties of finite cyclic groups. We describe how the RSA Cryptographic system works. Then we discuss nth roots of unity and the cyclotomic polynomials. We actually show how to manually construct the first ten of these. We define the sigma function that is the sum of all the divisors of an integer and we prove this function has the multiplicative property. We introduce the Mobius function and eventually we prove the Mobius Inversion Formula for both sums and products. 
85 pages. 
SomeSimpleNumberTheory.PDF  507,531 bytes   11/28/2018  9:40 pm.  

The Theory of Identity  
This is a technical paper written primarily in the first order predicate calculus. It explains in a very subtle way the difference in meaning of “equals” in mathematics and logic. Using only one axiom this paper proves the Reflexive Law for equality (i.e., x=x) and it proves the Law of Symmetry (i.e., if x=y then y=x), and the Transitive Law for Equality (i.e. if x=y and y=z then x=z. Yes these need to be proved! It also discusses the difference between equality in logic and equality in mathematics. 
3 pages. 
Identity.PDF  116,721 bytes   01/05/2012  10:14 pm.  

Implementing Trigonometric and Hyperbolic Functions and Their Inverses For Both Real and Complex Arguments  
This paper discusses how to implement all the trigonometric and hyperbolic functions with either real or complex arguments. The paper basically uses only Tan to generate all the inverse functions. This paper carefully discusses the complex exponential and logarithmic functions and shows how to compute complex nth roots and how to define a general complex power function. 
11 pages. 
InvTH.PDF  165,853 bytes   01/05/2012  10:15 pm.  

The Number e and Compound Interest  
This paper discusses how the number e arises naturally in the computation of compound interest. Using a limit formula from calculus, this paper shows how e arises by considering instantaneous compounding. It also shows how to compute the Future Value of a series of payments that results in an integral formula. It also discusses the case where the compounding and payment periods are different. 
6 pages. 
NumberE.PDF  150,991 bytes   01/05/2012  10:18 pm.  

Probability Notes  
This is a series of notes that lay the foundation for studying probability. This paper consists of 44 items that are mostly definitions and theorems. Of significance are the definitions for effectively impossible events and effectively certain events. The paper gives two examples of events, one of which has probability 0 and the other has probability 1. The first event is not impossible and the second event is not certain to happen. 
11 pages. 
Prob.PDF  181,788 bytes   01/05/2012  10:28 pm.  

Series Notes  
This is a collection of 97 theorems and definitions related to infinite series.  This set of notes includes most of the standard results but also contains proofs of several results that are left incomplete in many calculus textbooks.  These notes are fundamental for studying real analysis in more detail.  Included are proofs that the positive and the negative terms of a conditionally convergent series must both be divergent, that all re-arrangements of an absolutely convergent series converge to the same sum, that some re-arrangement of a conditionally convergent series can be made to converge to any real number, that a function defined by a power series is continuous and that it is differentiable and integrable term by term and that such a function has an infinite number of derivatives.  An example is also given of an infinitely differentiable function that cannot be represented by a power series.  Cauchy products and quotients of series are developed and both the Nested Interval Property and the Cauchy Convergence Criterion are given.  This paper also has a short and elegant proof that the sum of the reciprocals of the squares of consecutive positive integers converges to pi-squared divided by 6. 
42 pages. 
SeriesP.PDF  362,447 bytes   01/05/2012  10:31 pm.  

Sequences and Infinite Series of Functions  
This paper introduces the limit concept for a sequence of functions and pre-supposes a knowledge of infinite series. Examples are given which show that a sequence of continuous functions can have a discontinuous limit and that the limit of the derivatives or integrals do not necessarily converge to the derivative or integral of the limit. The concept of uniform convergence is presented and then six theorems are given that establish the correct limit properties that one would expect. The paper concludes by giving a proof of the Weierstrass M-test for an infinite series of functions. 
8 pages. 
SequencesFOfXP.PDF  149,882 bytes   01/05/2012   10:30 pm.  

Synthetic Substitution Versus Long Division  
This is a short paper whose only purpose is to explain the connection between long division and synthetic substitution. In fact, this may be considered to form the basis for the technique popularly known and misunderstood as Synthetic Division. 
3 pages. 
SynthD.PDF  117,203 bytes   01/05/2012  10:42 pm.  

Trochoidal Curves  
This paper develops the parametric formulas that can be used to generate 2D curves that are in the family of what are called trochoids. Such curves include hypotrochoids, epitrochoids, peritrochoids and trochoidal roses. Special cases of these curves include what are called deltoids, astroids, cardioids, nephroids, and limacons. These curves can be used to make some stunning artistic figures by varying some of the graph parameters for the curves. A free computer program called Koch is available that can be used to make computer art related to these curves. 
13 pages. 
Trochoids.PDF  446,160 bytes   01/03/2018  4:28 pm.  

An Angle Problem and an Analytic Geometry Solution  
This paper presents a simple geometry problem in which a point P lies in the first quadrant but on the unit circle in the xy-plane. The point's x-coordinate is (1+SQRT(5))/4. The problem is to prove that the angle between the positive x-axis and the line connecting the origin point to P must measure 36 degrees exactly. The solution to this problem involves constructing a regular pentagon inscribed in the unit circle. The points of that regular pentagon are constructed one at a time, using analytic geometry. This paper demonstrates the power of analytic geometry because it is very difficult to solve this problem using only plane geometry. 
7 pages. 
AnAngleProblem.PDF  148,604 bytes   01/05/2012  11:13 pm.  

The Pure Cubic Polynomial and Intersections With Linear Functions  
This paper contains five fundamental theorems that describe the three distinct ways in which a linear function may intersect the graph of Y=X^3. The results are simple, but somewhat surprising because we have never seen them summarized as they are here, in any PreCalculus text. This paper includes a ruler and compass construction method for drawing a tangent line to the curve Y=X^3 at any point on that curve. After reading this paper you will be better prepared to read the paper on Solving Cubic Equations. 
8 pages. 
CubicAndLinearRelationShip.PDF  51,117 bytes   01/15/2015  12:26 am.  

Solving Cubic Equations  
This paper demonstrates the techniques used by del Ferro, Cardano, Bombelli and Viete to solve cubic equations. This paper contains examples showing exact radical solutions and shows other solutions that are difficult to put in an exact form. The paper finishes by showing a technique that uses both trigonometric and hyperbolic functions to solve cubics. See also the paper that discusses the relationship between linear functions and the pure cubic curve Y=X^3.
10 pages. 
SolvingCubics.PDF  66,288 bytes   01/15/2015  12:24 am.  

Solving Quartic Equations  
This paper demonstrates using completing the square techniques to solve 4th degree polynomial equations. Such solutions also involve using the techniques to solve cubic equations. See also the paper that discusses how the pure quartic Y=X^4 can intersect with a general quadratic polynomial. 
6 pages. 
SolvingQuarticEquations.PDF  40,614 bytes   01/15/2015  12:20 am.  

The Pure Quartic and Intersections With Quadratic Polynomials  
This paper contains six fundamental theorems that describe and summarize the various ways in which a quadratic polynomial can intersect with the pure quartic Y=X^4. These results should be in any PreCalculus text, but we have never seen them. This paper is related to another paper on solving quartic equations. 
15 pages. 
QuarticQuadraticIntersections.PDF  190,898 bytes   01/05/2012  9:09 pm.  

JK Notes On Spherical Trigonometry  
This set of notes provides a basis for studying spherical trigonometry. We carefully define the terminology associated with spherical triangles and we develop the concept of spherical Excess for a spherical triangle. We demonstrate the relationships between lunes and spherical triangles. Then we derive the fundamental spherical trigonometric identity that is also known as the Law of Cosines. We introduce the norm concept and we develop the Law of Sines. We derive four Analogies by Delambre and follow that with four Napier Analogies. We prove both Cagnoli's Theorem and L'Huilier's Theorem. We develop a formula for the tangent of half the spherical excess and we discuss how to use triangulation to find the area of convex spherical polygon. We introduce haversines and the polar triangle. We show 19 cases of triangle inputs and discuss how to find all triangle parts. Then we introduce the GPS (Global Positioning Satellite) coordinate system using latitudes and longitudes. We explain the special technique known as slerping. Slerping is the equivalent of doing linear interpolation, but slerping applies to a great circle arc on a sphere. We give computer code that shows how to carefully compute everything related to GPS coordinates. 
61 pages. 
JKNotesOnSphericalTrigonometry.PDF  469,537 bytes   10/06/2018  7:46 am.  

Abstract Algebraic Theories Applied To Error Correcting Codes 
This paper is an extensive set of notes that develops basic properties of the two abstract algebra constructs known as groups and fields. In particular, it develops some of the basic properties of finite fields known as Galois fields. After establishing a mathematical foundation, these notes discuss examples of error detecting and error correcting codes. Included are linear codes, Hamming codes, cyclic codes, BCH (Bose-Chadhuri-Hocquenghem) codes, and Reed-Solomon codes. This is a very long paper and requires a strong math background and some mathematical maturity. It is recommended for students who have completed at least one course in Linear Algebra and who are interested in advanced applications of mathematical theories. 
189 pages. 
ErrorCodesTheory.PDF  850,561 bytes   01/14/2015  5:10 pm.