Equations of Quadrilaterals

7/31/2019 Equations of Quadrilaterals

1/5

E q u a t i o n s o f q u a d r i l a t e r a l s | 1

Primarily, a quadrilateral may be represented by the combined equation of its four sides.For example, the equation of a quadrilateral having sides :

= 1 + 1 = 2 + 2 = 3 + 3 = 4 + 4can be represented as :1 + 1 2 + 2 3 + 3 4 + 4 = 0

However, a serious drawback of this equation is that along with the points on the sides ofthe quadrilateral, this equation is also satisfied for all other points lying on the four straight linesthat represent the sides of the quadrilateral and so, its graph looks like as follows :

Four straight lines form a rectangle

In this text, the or function (defined for real numbers) has been used indeducing equations of a quadrilateral. Subsequently, each of the graphs, obtained from theseequations has four non-differentiable points, which are the four vertices of a quadrilateral.

The greatest integer function could also have been used to

fix the problem. For example, including this function, the

equation of a square having vertices at (0,0), (0,1), (1,1) &

(1,0) would look like :1 = 1 1 ()--- where = 1 ( 1 ) , [] denotingthe greatest integer function (see figure aside). However, this

method is not handy for mathematical treatment, and this

equation cannot be easily generalized for any kind of

quadrilateral. So, I shall make no further discussion on this

equation. (Note that in this equation, the use of the modulus

function was also required.)

(0,0)

(0,0) (1,0)

(1,1)(0,1)

Fig : Square obtained from equation a

Uploaded By Mayukh Nath


2/5


[c is a constant]

Equation of square whose diagonals coincide with andaxesConsider the following equation : + =

It gives the following result.

Case 1: x>0, y>0

+

=

Case 2: x>0, y


3/5


Representation of equation() = When takes the value 4 , (Fig. 4) the equation (ii)

becomes : + + + =

2=

It is the equation of a square, whose sides are parallel to thecoordinate axes. In particular, when becomes the origin,then the equation (iii) becomes :

+ + + = 2 = Equation of any quadrilateral having axially aligned diagonals

The sides of the quadrilateral, shown in Fig. 5, havingarbitrary slopes cannot be represented by the modulus-

controlled equation (i). In this case, we need to introduce avariable quantity, outside the modulus brackets. Now,consider the following equation :

+ + + = 1This equation reduces as follows :

1st

quadrant: + + + = 12nd quadrant: + + = 13

rdquadrant: + = 1

4th

quadrant:

+

+

= 1

--- These also result a quadrilateral.

(0,0)

[ , ]

(0,0)

(0,a)

(d,0) (b,0)

(0,c)

[ , ]

(0,0)

= 4


4/5


Now, the set of equations for the sides of the quadrilateral in Fig. 5 is :

+ = 1 + = 1

+

= 1

+

= 1

Comparing this set of equations with the previous one, we get :

1 = + 1 = +1 = 1 = ( )

From the above relations, it is clear that = ( )2 , = ( + )2 , = ( )2 , = ( + )2

Now the quadrilateral in Fig. 5 can be represented as :

+

+

2 + +

+

2 = 1 () Any quadrilateral having mutually perpendicular diagonals

Equation (iv) can be generalized, just as the squareequation. Thus, the equation of the quadrilateral shown in Fig.6, and hence for any quadrilateral having mutually

perpendicular diagonals will be :

4

2

+

4 +

2

2

4 2+

3

1

+

3 +

1

2

3 1= 1

(

)

Here,i) = cos + sinii) = sin + cosiii) = cos + siniv) = sin + cos

Non - perpendicular - diagonal cases

Till now, only the cases, where the two diagonals of a quadrilateral were mutuallyperpendicular, have been considered. Quadrilaterals with non-perpendicular diagonals may be

handled as follow :

Step 1 : The origin is shifted to the point ,.

(1 ,1)(3, 3)

(2, 2)

(

4,

4)

P

[ , ]

= 1,2,3,4

, [old]New set of coordinate

axes

(0,0) [old]

(0,0)

(1, 1)

(3,3)(2, 2)

(4, 4)


5/5


Step 2 :

Let, the angles the diagonals make with the positiveaxis be

and

(

>

). Now the axis is rotated by an angle

, so that the angular bisector of the angle ( + ) coincidewith axis (Fig. 7-b). [ + is the angle, containing ,made by the diagonals.]

Step 3 :

Now, the scale of the

axis is modified by multiplying

the coordinates with a factor , such that in the new scale, thediagonals become mutually perpendicular.From Fig. 7-c, it is clear that, tan = = . Now, apoint ( , ) in the new coordinate system would take thefollowing form, when represented by (,) belonging to the oldsystem : = [ cos + sin ] = sin + cos This point ( , ) satisfies equation (iv) with = 4 .

So, the equation of a quadrilateral with non-perpendicular diagonals will be :4 2+ 4 +2242 + 3 1+ 3 + 12 31 = 1 ()

Where,

i) = ( + )ii) = ( + )iii) = ( + )iv)

= (

+

)

v) = [ cos + sin ]vi) = sin + cos At the end, It must be mentioned that all the equations discussed so far,

will work only in case of convex quadrilaterals, i.e. when the GC of the

quadrilateral will lie inside it. But in case of concave quadrilaterals, where

GC lies outside the quadrilateral, (e.g. Fig. 8) all the equations fail.

+

= +2

= 1,2,3,4

Fig. 8 : Concave quadrilateral

(0,0)

(

1

,

1

)

(2 , 2 )

(4 , 4 )(3 , 3 )

Documents

Equations of Quadrilaterals