Graphical Functionalization
Registration Number: 51105
School Name: Kiangsu-Chekiang College (Shatin)
Teacher Name: Cheng HoYin
Team Members’ Names: Or Ping Lam
Yu Chun Hin
Date of Report: 19th August, 2006
A report submitted to the Scientific Committee of the Hang Lung Mathematics Awards, 2006
Abstract
We used to learn the equation of circle in Additional Mathematics. However, we had a question: Is there an equation for square? We started the project base on this question.
The objective of our project is to find out the functions of all kinds of polygons and study their properties. We use absolute functions to achieve our aim as it can draw bended straight lines.
We have discovered the functions of different kinds of quadrilaterals, the general equations of 2n-sided polygon and the method on drawing regular (2n+1)-sided polygons. We also study the properties of Quadrilateral Equations.
Introduction
We discovered that a square could be represented by implicit absolute functions . We were motivated by this simple and beautiful function and started to study the method on constructing different polygons and shapes by functions. We generalized the study to the relationship between functions and polygons.
Firstly, we will introduce translation and rotation of curves to readers. Since we will study the polygons and its functions, in order to make the functions simpler, we will locate the ‘centre’ of polygon at origin. In case we want to express the functions of polygons which centres are not located at origin, we need the method of translation.
To introduce the regular polygons, we set one of their diagonal as x-axis. For expressing the regular polygon without x-axis as its diagonal, we need rotation.
Secondly, we will list out the functions of all kinds of quadrilaterals and also give the proofs. We found that the absolute functions could make bended lines. We studied and discovered that the functions of quadrilaterals can be formed by absolute functions.
Thirdly, we will deduce a set of construction rules so that we can find out the functions of specific quadrilaterals. Also we will discuss the relationship between parameters of functions and parameters of polygons.
Fourthly, we will generalize our study to n-sided polygon. We find that it is very difficult to deduce the functions of 2n+1-sided polygons, so we study 2n-sided polygons first. We discovered the general functions of 2n-sided regular polygons with proof. We also discuss how to find the function of 2n-sided polygons which passing through 2n given vertices.
Fifthly, we try to use another way to express the functions of regular (2n+1)-sided polygons. We will introduce the skills of combining curves, which can combine parts of the two curves together. After that, we will demonstrate how to create the functions of equilateral triangle and regular pentagon.
Finally, we will comment about our limitations and plan out further study.
Contents
Chapter 1 Basic knowledge
/5
Chapter 2 Functions of Quadrilaterals
/7
/ Chapter 2.1 Irregular Quadrilateral /7
/ Chapter 2.2 Parallelogram /10
/Chapter 2.3 Rectangle
/12
/Chapter 2.4 Rhombus
/14
/ Chapter 2.5 Square /15
/ Chapter 2.6 Kite /16
/Chapter 2.7 Trapezium
/18
/ Chapter 2.8 Relationship between Equations of Quadrilaterals /20
Chapter 3 Conditions Required to draw Specific Quadrilaterals
/21
Chapter 4 Equation for Quadrilateral with Four Given Vertices
/28
Chapter 5 Regular 2n-sided Polygon
/30
Chapter 6 Equation for a 2n-sided Polygon with 2n Given Vertices
/36
Chapter 7 Combination of Curves
/37
Chapter 8 Equations of Equilateral Triangle and Regular Pentagon
/43
/Chapter 8.1 Equation of Equilateral Triangle
/43
/Chapter 8.2 Equation of Regular Pentagon
/45
Chapter 1 Basic knowledge
In this chapter, we introduce transformation and usual notation in the report which is useful in other chapters.
Method of Translation:
By substituting x by (x – h) and y by (y – k), the graph will shift along the positive direction of x-axis by h units and along the positive direction of y-axis by k units.
Method of Rotation:
Refer to the figure,
and are the equations before and after rotation.
Let (x, y) and (, ) are the points on them with the same distance from origin, is the angle of rotation.
Express x, y in terms of , ,
Therefore,
In our study, we always divide the coordinate plane into several parts according the sign of Li. We define the function of straight line as
. .
For simplicity, we just write instead of in most cases.
Chapter 2 Functions of Quadrilaterals
In this chapter, we will introduce the functions of different quadrilaterals.
Chapter 2.1 Irregular Quadrilateral
All quadrilaterals can be represented by
where
L1 = ,
L2 = , ,
and ,
.
Proof:
A quadrilateral shows the following basic property:
For a pair of adjacent sides, the intersection of them should lie on the corresponding diagonal.As a quadrilateral must be composed of 4 different line segments, two implicit absolute functions are needed in order to form 4 cases. Therefore, neither a nor b is zero.
The coordinate plane is divided into 4 regions and 4 straight lines can be drawn:
Region 1:
…(1)
Region 2:
…(2)
Region 3:
…(3)
Region 4:
…(4)
Intersections on Diagonals:
(1) - (2), 2aL1 = 0
L1 = 0
Therefore, (1) and (2) intersect on L1 = 0, that means point A lies on L1 = 0.
(4) - (3), 2aL1 = 0
L1 = 0
Therefore, (3) and (4) intersect on L1 = 0, that means point C lies on L1 = 0.
(2) - (3), 2bL2 = 0
L2 = 0
Therefore, (2) and (3) intersect on L2 = 0, that means point B lies on L2 = 0.
(1) - (4), 2bL2 = 0
L2 = 0
Therefore, (1) and (4) intersect on L2 = 0, that means point D lies on L2 = 0.
Points A and C must lie on L1 = 0, B and D must lie on L2 = 0.
Remarks 1: Proof of
Firstly, .
Assume :
Case 1:
Consider Region 3:
Substitute B into BC
It contradicts with the proportion, .
Case 2:
Consider Region 1:
Substitute D into DA
It contradicts with the proportion, .
Therefore, it is impossible for .
Combining the result, .
Similar proof is used for b. .
Remarks 2: Proof of and .
Assume .
In region 1: ,.
DA : .
Substitute into DA,
, which contradicts in region 1.
So, .
Also, the same contradiction appears with assuming that ,
and . So that , , and .
Combining those results, and .
Remarks 3: Proof of .
If , then and represent the same diagonal. However, all the quadrilaterals have two diagonals, therefore, .
In conclusion, is the general equation for all quadrilaterals.
Chapter 2.2 Parallelogram
All parallelograms can be represented by
where
L1 = ,
L2 = , ,
.
Proof:
A parallelogram shows the following basic property:
The opposite sides are parallel to each other.The equation represents a quadrilateral since it is a form of the general equation of quadrilateral.
The coordinate plane is again divided into 4 regions.
Region 1:
Region 2:
Region 3:
Region 4:
Opposite Sides Parallel:
DA:
BC:
Obviously, .
AB:
CD:
Obviously, .
In conclusion, is the general equation for all parallelograms.
Chapter 2.3 Rectangle
All rectangles can be represented by
where
L1 = ,
L2 = , ,
.
Proof:
A rectangle shows the following basic properties:
1. The opposite sides are parallel to each other and2. The pair of adjacent sides is perpendicular to each other.
1. Opposite Sides Parallel:
The equation represents a quadrilateral, which must be a parallelogram since it is a form of the general equation of parallelogram. Thereforeand.
2. Adjacent Sides Perpendicular:
Region 1:
Region 2:
Slope of DA =
Slope of AB =
In conclusion, is the general equation for all rectangles.
Chapter 2.4 Rhombus
All rhombuses can be represented by
where
L1 = ,
L2 = ,
.
Proof:
A rhombus shows the following basic properties:
1. The opposite sides are parallel to each other.2. The diagonals are perpendicular to each other.
1. Opposite Sides Parallel
The equation represents a quadrilateral, which must be a parallelogram since it is
a form of the general equation of parallelogram. Therefore and
.
2. Diagonals Perpendicular
Slope of L1 =
Slope of L2 =
.
In conclusion, is the general equation for all rhombuses.
Chapter 2.5 Square
All squares can be represented by
where
L1 = ,
L2 = ,
.
Proof:
A square shows the following basic properties:
1. The opposite sides are parallel to each other.2. A pair of adjacent sides is perpendicular to each other.
3. The diagonals are perpendicular to each other.
1. Opposite Sides Parallel and Adjacent Sides Perpendicular
The equation represents a quadrilateral, which must be a rectangle since it is a form of the general equation of rectangle. Thereforeand, .
2. Diagonals Perpendicular
The equation represents a rhombus since it is a form of the general equation of rhombus. Therefore.
In conclusion, is the general equation for all squares.
Chapter 2.6 Kite
All kites can be represented by
or
where
L1 = ,
L2 = ,
,
for ,
for .
Proof:
A kite shows the following basic properties:
1. The diagonals perpendicular to each other.2. One of the diagonal bisects another.
Consider .
The equation represents a quadrilateral since it is a form of the general equation of quadrilateral.
1. Diagonals Perpendicular
Slope of L1 =
Slope of L2 =
2. One of the Diagonal Bisects another
Consider ,
Substitute point into the equation:
Point D gives the same result,
Therefore, diagonal L1 bisects diagonal L2.
Similar proof is used for equation .
Remarks: Axis of Symmetry
Consider and its regions.
Region 1:
Region 4:
Substitute points and into the equation AD, CD respectively:
On the other hand, Point C gives a result of
when except .
Therefore, is a kite with L1 as the axis of symmetry.
Similar proving is used for equation .
In conclusion, is the general equation for all kites with L2 as the axis of symmetry; is the general equation for all kites with L1 as the axis of symmetry. When , the equation becomes an equation of rhombus, and both L1and L2 are the axes of symmetry.
Chapter 2.7 Trapezium
All trapeziums can be represented by
or
where
L1 = ,
L2 = , ,
, .
Proof:
A trapezium shows the following basic property:
A pair of opposite sides is parallel to each otherConsider .
The equation represents a quadrilateral since it is a form of the general equation of quadrilateral.
The coordinate plane is divided into 4 regions.
Region 1:
…(1)
Region 2:
…(2)
Region 3:
…(3)
Region 4:
…(4)
A pair of Opposite Sides Parallel:
Obviously, .
Therefore, represents a trapezium.
Similar proof is used for equation .
Remarks: The pair of parallel lines
Consider ,
.
It is obvious that AB does not parallel to CD expect p = 0
Similar proof is used for equation .
Therefore
If is used, .
If is used, .
When p = 0, the equation becomes an equation of parallelogram, and both ,.
In conclusion, are the general equations for all trapeziums.
Chapter 2.8 Relationship between Equations of Quadrilaterals
All quadrilaterals can be expressed by the equation:
where
L1 = ,
L2 = , ,
and ,
.
If () then the quadrilateral is a Parallelogram.
If ( and ) then the quadrilateral is a Rectangle.
If ( and ) then the quadrilateral is a Rhombus.
If ( and and ) then the quadrilateral is a
Square.
If ( or but not both and )
then the quadrilateral is a Kite.
If ( or ) then the quadrilateral is a
Trapezium.
Chapter 3 Conditions Required to Draw Specific Quadrilaterals
Now, there are a number of equations of quadrilateral, however, if we want to draw a specific quadrilateral with some given conditions, how can we make use of them?
In this chapter, we would like to discuss this problem. We would introduce the properties of variables in those equations and simplify the equations.
Properties of :
For all equations, and are the diagonals of the quadrilateral drawn. As
,
and are the slopes of the diagonals, are the angles of inclination.
Properties of a, b, r:
The properties of the coefficients a, b of and , and the constant r can be discovered by substituting the vertices of the quadrilateral into the corresponding equations. The process is demonstrated by using the equations of rectangle () and that of parallelogram ().
Substitute into the equation , it becomes
as . Obviously, r is the distance between point A and L2.
Similar process is repeated for equation . All the results are illustrated on the graph on next few pages.
From those results, we can conclude that a, b, r affect the perpendicular distances
between vertices and diagonals.