This article describes translation, scaling and rotation of circles and ellipses.
First some definitions:
A function is a special case of an equation.
The "..." stand for operations and constants.
x is called the independent and y the dependent variable.
x is assigned a value and for each value of x y may be calculated.
A function has exactly one number as result so functions may be part of a formula.
Examples of formulas are:
3x2 + 5y2
7x - 8y - 10.
An example of an equation is:
3x2 + 5y2 = 2x - y + 25
Some examples of functions are:
y = 3sin(x)
y = 3x3 - 2x2 + x - 7
GraphicsGraphics are pictures of equations and functions.
Take equation x2 + y2 = 1
For a value of x the value of y may be calculated.
Writing these number pairs as (x,y) they become coordinates in a plane.
For each (x,y) pair we may paint a dot.
For the case above this results in:
A circle with center (0,0) and radius 1.
The Pythagoras theorem shows why.
TranslationThe center of the circle may be at another place than (0,0).
What would happen to the equation in this case?
We apprpoach the translation (shifting) ploblem in a general way.
Say we have an undefined equation ..x..y.. = ..x..y.. and point (x1,y1) is a point on the graph.
Next we replace (x) by (x+1) and (y) by (y+2).
The result is ..(x+1)..(y+2) = ..(x+1)..(y+2)
Say point (x2,y2) is on the new graph.
This will be the case if :
(y2 + 2) = y2
y2 = y1 - 2
The graph is shifted 1 left and 2 down.
In general, if in any equation :
y is replaced by (y+v) the graph will shift v positions down.
Note : For negative values of h and v the graph will shift right and up.
This is true for any function or equation.
Translation and equation
We start with circle x2+y2=1 and replace (x) by (x+h) and (y) by (y+v).
The factors of the squares of a and b do not change.
New are terms with x and y showing up as 2hx and 2vy.
This indicates a translation.
x2-10x+y2+6y=47 is a shifted circle.
2v = 6.......v = 3......shift of 3 down.
Scaling x,yIn equation x2+y2 = 1 we replace x by x/2.
What will happen?
Say (x1,y1) is on the circle and after the change this point has become (x2,y2).
This is true if:
x2 = 2x1
The result is doubling of the width.
Such a stretched (oval) circle is called an ellipse.
In general, if in any equation:
y is replaced by (y/b) the graph will stretch by a factor b in vertical direction.
The general equation of an ellipse is:
Scaling and the equation
Cicle x2 + y2 = 1 is stretched horizontally by a and vertically b resulting in (x/a)2 + (y/b)2 = 1.
Multiplication by a2b2 makes b2x2 + a2y2 = a2b2
Scaling changes the factors of the x, squares and the constant (without x or y)
No new terms with x, y are added.
Given is ellipse 9x2+49y2 = 441.
Calculate "a" and "b".
Divide both terms by 441 :
RotationWe rotate point P(x1,y1) counterclockwise by angle φ
For convenience we observe x and y separately.
Q is the rotated point P.
The center of the rotation is origin (0,0).
Rotation of x1 causes
Δy = x1sinφ.......infuence of x on y
Δx = y1sinφ........influence of y on x
y2 = y1cosφ + x1sinφ
If in any equation with variables x and y :
(y) is replaced by ycos(φ+xsin(φ)
(Remember: the (x,y) values which originally solved the equation were located at counterclockwise rotation over degree φ)
Rotation and equation
Ellipse b2x2+a2y2 = a2b2 is rotated right by φ degrees.
The rotation center is (0,0).
The equation changes into
Rotation also changes the terms with x, y, x2, y2 and the constant term.
Ellipse 4x2 + 9y2 = 36 is rotated.
The new equation shows the term 2,5xy.
Question: over which angle did rotation take place?
a = 3.
2(a2 - b2)sinφcosφ = 2,5
10sinφcosφ = 2,5
2sinφcosφ = 0,5
sin(2φ) = 0,5
φ = 15
Given is ellipse: 2x2+y2 = 2(x + y + xy -1 ).
Below the graph is pictured:
The x,y terms are the result of translation, the xy term indicates rotation.
First we try to shift the graph such that the x and y terms disappear.
replace x by (x+h) and y by (y+v).
This results in two equations which enables solving of h and v.
Extract the terms with x and y :
x terms: 4h = 2 + 2v
y terms: 2v = 2 + 2h
2h - v = 1
v - h = 1
h = 2
v = 3
The new equation becomes after shifting 2 left and 3 down:
2x2 + y2 - 2xy = 3
procedure to find the rotation
However, right of the "=" character another number then 1 may be found.
This means that in fact a and b need to be changed.
Before we noticed that rotation changes a and b, no constant change.
In the equation b2x2 + a2y2 = a2b2 the number right of the "=" is the product of x2 and y2 factors.
Why is this important?
We see 2x2 + y2 - 2xy = 3
Right of the "=" is no 1 but a 3. Later I come back at this pitfall.
q = sinφ
(y) is replaced by (py + qx)
This results in a new equation with different factors of x2, y2 and xy terms.
Next we equate these values to those in 2x2 + y2 - 2xy.
This results in a new set of equations with a,b,p,q.
The goal is to solve these equations to know the rotation and a,b, the ellipse dimensions.
The calculated a,b where derived from an equation with format ....= 1.
However our equation shows ....= 3.
Therefore we depart from equation b2x2 + a2y2 = 3a2b2.
(from the form without denominators)
Above we see the set of equations to be solved.
p2 + q2 = 1 we could add.
First we look after p and q.
Subtract row 2 from row 1 and combine terms with p2 en q2:
Note: φ is the angle of rotation. z = tanφ
Now we look after a and b.
Again the 3 previous equations with p,q,a,b are the start.
Add rows 1 and 2 and place p2 and q2 outside parenthesis:
Right of the "=" comes 3a2b2 = 2,946.
The equation of the ellise was before translation and rotattion:
what the next picture shows:
For clarity, two circles with radius a and b are included.
ExercisesWe started this example with equation 2x2+y2 = 2(x + y + xy - 1)
But what about nx2+y2 = 2(x + y + xy - 1) for cases where n = 3,4,...
In a math Facebook group I found this problem:
solve the equation
The plot of this equation however shows only one single point.
What happened? Which point?