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Ellipse edit


 
Figure 1: Ellipse (red curve) at origin with major axis horizontal.

Origin at point   .
Foci are points  
Line segment   is the  
 
Line segment   is the  
Each line   is a  
Each line   is a  
 

In cartesian geometry in two dimensions the ellipse is the locus of a point   that moves relative to two fixed points called    The distance   from one   to the other   is non-zero. The sum of the distances   from point to foci is constant.


  See figure 1.


The center of the ellipse is located at the origin   and the foci   are on the   at distance   from  


  has coordinates   has coordinates  . Line segments  


By definition  


  the length of the  


Each point   where the curve intersects the major axis is called a   are the vertices of the ellipse.


Line segment   perpendicular to the major axis at the midpoint of the major axis is the   with length  


 


Any line segment that intersects the curve in two places is a   A chord through the focus is a   Each focal chord   perpendicular to the major axis is a  


Equation of the ellipse



 

Let point   have coordinates  


 

 

 

 

 

 

 

 

 


Make appropriate substitutions, expand and the result is:

 

 

  or  


If the equation   is expressed as  

 


Length of latus rectum



 

 

 

 

 

 

 

Length of latus rectum  

The directrices edit


 
Figure 1: Ellipse (red curve) at origin with major axis horizontal.

Origin at point   .
Foci are points  
Directrices are lines    
 
  (yellow lines);   (purple lines)
To define ellipse specify  


See Figure 1. The vertical line through   with equation   is a   of the ellipse. Likewise for the vertical line  


A second definition of the ellipse: the locus of a point that moves relative to a point, the focus, and a fixed line called the directrix, so that the distance from point to focus and the distance from point to line form a constant ratio  


Consider the point   in the figure.  


Let the length  


Consider the point  :

 


 


 


Length  .


Distance from center to directrix  


Distance from center to directrix =  .

 .

 .

 .


Distance from focus to directrix  .


Ellipse using focus and directrix


 
Figure 2: Ellipse (red curve) at origin with major axis vertical.

Origin at point   .
Focus at point   
Directrix is line   
Point   has coordinates:  
  (yellow lines)


See Figure 2. Let the point   be  , the focus   be   and the directrix   have equation   where   The directrix is horizontal and the major axis vertical through the origin  .


 

 

 

 

 

 

 


Expand and the result is:


 

 

 

 

 

 

 

 


Compare this equation with the equation generated earlier:  .

When the equation is   the major axis is horizontal.

When the equation is   the major axis is vertical.

General ellipse at origin edit


 
Figure 1: Ellipse (red curve) at origin with major axis oblique.

Origin at point   .
Foci are points  
 
Line segment   is the  
 
Line   is the major axis extended.
Lines perpendicular to   at   are directrix, latus rectum. minor axis, latus rectum, directrix.
 
 
 


The general ellipse allows for the major axis to have slope other than horizontal or vertical. See Figure 1.


The line   is the major axis of the ellipse shown in red. Points   are the foci with coordinates   respectively.


Point   is any point on the curve. By definition  


Length  


Length  


 


 

 

 

 

 


Make appropriate substitutions, expand and the result is:


 .

This equation has the form   where:

 

 

 

  because center is at origin,

 


An example



Let   be (3,4) and  

The ellipse is:   or  


Reverse-engineering ellipse at origin



Given an ellipse in format   calculate   is non-zero. 


 

 

 

 


Coefficients provided could be, for example,   or  

or   where   is an arbitrary constant and all groups of coefficients define the same ellipse.


To produce consistent, correct values for   the equations become:


 

 

 

 


or:


 

 

 

 


Solutions are:


 . This formula for   is valid if both   are  .


  where  

You should see one positive value   and one negative value   for   Choose the positive value and  .


 


 


The solutions become simpler if K == 1. if ( K != 1 ) { A ← KA; B ← KB; C ← KC; } and the solutions for   become:


  where  


 


 


With values   available all the familiar values of the ellipse may be calculated:


 


Equation of major axis:   in normal form.


Equation of minor axis:   in normal form.


Equations of directrices:   in normal form.


An example using focus and directrix


 
Figure 2: Ellipse (red curve) at origin with major axis oblique.

Origin at point   .
Focus at point   
Directrix is line   
eccentricity  
Ellipse has equation:
 


Given focus   and directrix with equation  , calculate equation of the ellipse in form  


See Figure 2. Let point   be any point on the ellipse.


Distance from   to focus  


Distance from   to directrix  .


 .


 .


 .


 .


Expand and the result is:  


Because   the center of the ellipse is at the origin and the various lines have equations as follows.


Minor axis:  


Other directrix:  


Major axis:  


If you reverse-engineer the ellipse using the method above,  ,

the expression   becomes  , and

the expression   becomes  .



General Ellipse edit


 
Figure 1: The general ellipse (red curve).
Foci at   center at  
Point   is a vertex. Length  
 


The ellipse may assume any position and any orientation in Cartesian two-dimensional space. See the red curve in Figure 1.


The equation of this curve may be derived as follows:

Given two     of length   and point  


 


 


The expansion of this expression is somewhat complicated because it contains 5 variables  


The expansion may be simplified by reducing the number of variables from   to  , the familiar  


Let  , the center of the ellipse, have coordinates   where   and  


Then     and


 


The expansion is:   where:

 

 

 

 

 

 


A different approach



Begin with ellipse at origin:   where:

 

 

 

 


By translation of coordinates, move the ellipse so that the new center of the ellipse is:    


The equation above becomes:   where:


 


 

the same as the value of   in the method above.


The expansion of   will show that this method produces the same results as the method above.


Given foci and major axis, perhaps the simplest way to produce   is to calculate   and move the center from   to  .


Center of ellipse



Given   we know from above that  


Therefore   where the point   is the center of the ellipse.


An Example of the General Ellipse edit

 
Figure 1: Translation of coordinate axes.
Red curve and green curve have same size, shape and orientation. The only difference is that center of green curve   has been moved to   where it is the center of the red curve.
In both curves  


See Figure 1. Given foci   and   calculate the equation of the ellipse in form  


Calculate the center:  


 


Equation of ellipse at origin:  


Move ellipse from origin   to  

 


Reverse-engineering the general ellipse



Given ellipse   calculate the foci and the major axis.


Calculate the center of the ellipse (point  ):  


 

 

 

 


Solutions are:


 


where  

In this example,  


If   !=     


The values   may be calculated as in "Reverse-engineering ellipse at origin" above.


 


Length of major axis  


Focus  


Focus  


Significant lines of the Ellipse


 
Figure 2. Graph of ellipse illustrating axes, each latus rectum, each directrix.
Directrix through  
Directrix through  
Latus rectum through  
Latus rectum through  
Major axis  
Minor axis  


The significant lines of the ellipse are:   each   each  


Consider the ellipse in Figure 2. Given foci   and   calculate the equations of all the significant lines.


Slope of major axis  

Major axis has equation   and it passes through  

Therefore,  

Major axis   has equation:  


Center of ellipse  


Minor axis is perpendicular to major axis. Therefore, minor axis has equation:   and it passes through the center  

 

Equation of minor axis (orange line through  ):   or   or  


 


Using the equation of the minor axis, the fact that each latus rectum is parallel to the minor axis, and that the distance from minor axis to latus rectum   each latus rectum has equation:  

Equation of latus rectum (blue line) through  

Equation of latus rectum (blue line) through  


Using the equation of the minor axis, the fact that each directrix is parallel to the minor axis, and that the distance from minor axis to directrix   each directrix has equation:  

Equation of directrix (red line) through  

Equation of directrix (red line) through  

K and "Standard Form" edit


 
Figure 1: Three ellipses illustrating "standard form."

For green curve  
For red curve  
For blue curve  


Everything about the ellipse can be derived from   the last three of which   are contained within:

 

 

 

  for ellipse at origin.


Consider the ellipse:   the green curve in Figure 1. It is tempting to say that  


These values satisfy  


However,   These values for   are not correct.


Put the equation of the ellipse into "standard form." In this context "standard form" means that  


For ellipse at origin  


In fact  


     


 


The equation of the ellipse becomes:   and


 


The equation of the ellipse is in "standard form" and:


 


 

 

 

 

 


The values   are correct.


Example 2. Consider the ellipse   the red curve in Figure 1.


In this example,   and the equation in "standard form" is:


 .


Example 3. Consider the ellipse   the blue curve in Figure 1.


The center  


In this example   and the equation in "standard form" is:


 .


Tangent at latus rectum edit


 
Figure 1: Ellipse and tangent at Latus Rectum.

Origin at point   .
Red curve is ellipse at origin with major axis vertical.
Line   is directrix:  
Line   passes through point  
Line   is tangent to curve at Latus Rectum:  

See Figure 1. The red curve is that of an ellipse at the origin with major axis vertical:  

The line   is the directrix with equation:  

The green line   has equation:  

The aim of this section is to show that the line   is tangent to the ellipse at the  


Let the line intersect the curve. The   coordinates of the point of intersection are given by:

 


If the line   is a tangent,   has one value and the discriminant is  

  or:

 

 

 


The tangent   has slope  and equation:  


Let this line intersect the curve. The   coordinates of the points of intersection are given by:  

Discriminant =  


  half length of latus rectum.


The tangent   touches the curve where  , the point   where   is the latus rectum.

Reflectivity of ellipse edit


 
Figure 1: Ellipse (red curve) with major axis horizontal.

Origin at point   .
Foci are points  
Line   tangent to curve at  .
Angle of incidence = angle of reflection:  

See Figure 1.


The curve (red line) is an ellipse with equation:   where  


 


Foci   have coordinates  

Line   is tangent to the curve at point  

A ray of light emanating from focus   is reflected from the inside surface of the ellipse at point   and passes through the other focus  


The aim is to prove that  


Point   has coordinates  

At point  

Slope of line  

Slope of line  

Slope of curve at  


Using  ,


 

 


if   then:

 ,

 ,

  and

  where  


If you make the substitutions and expand, the result is  .


Therefore, angle of reflection   angle of incidence   and the reflected ray   passes through the other focus