Semi-major and semi-minor axes

In <a href="/facts/Geometry/wTQf5ffQ">geometry</a>, the major axis of an <a href="/facts/Ellipse/1wUDnGxY">ellipse</a> is its longest <a href="/facts/Diameter/6uNn4VmU">diameter</a>: a <a href="/facts/Line_segment/V8E0M4qk">line segment</a> that runs through the center and both <a href="/facts/Focus_(geometry)/WFtFnqoa">foci</a>, with ends at the two most widely separated points of the <a href="/facts/Perimeter/yQv1AajA">perimeter</a>. The semi-major axis (major semiaxis) is the longest <a href="/facts/Semidiameter/yzsFRcgJ">semidiameter</a> or one half of the major axis, and thus runs from the centre, through a <a href="/facts/Focus_(geometry)/WFtFnqoa">focus</a>, and to the perimeter. The semi-minor axis (minor semiaxis) of an ellipse or <a href="/facts/Hyperbola/6R6ER0mY">hyperbola</a> is a line segment that is at <a href="/facts/Right_angle/psaslRq7">right angles</a> with the semi-major axis and has one end at the center of the <a href="/facts/Conic_section/teqgLptX">conic section</a>. For the special case of a circle, the lengths of the semi-axes are both equal to the <a href="/facts/Radius/9Gr9deYD">radius</a> of the circle.
The length of the semi-major axis a of an ellipse is related to the semi-minor axis's length b through the <a href="/facts/Eccentricity_(mathematics)/SzgpW5R7">eccentricity</a> e and the <a href="/facts/Semi-latus_rectum/teqgLptX">semi-latus rectum</a> 
 
 
 
 ℓ
 
 
 {\displaystyle \ell }
 
, as follows:

b
              
              
                
                =
                a
                
                  
                    1
                    −
                    
                      e
                      
                        2
                      
                    
                  
                
                ,
              
            
            
              
                ℓ
              
              
                
                =
                a
                (
                1
                −
                
                  e
                  
                    2
                  
                
                )
                ,
              
            
            
              
                a
                ℓ
              
              
                
                =
                
                  b
                  
                    2
                  
                
                .
              
            
          
        
      
    
    {\displaystyle {\begin{aligned}b&=a{\sqrt {1-e^{2}}},\\\ell &=a(1-e^{2}),\\a\ell &=b^{2}.\end{aligned}}}

The semi-major axis of a <a href="/facts/Hyperbola/6R6ER0mY">hyperbola</a> is, depending on the convention, plus or minus one half of the distance between the two branches. Thus it is the distance from the center to either <a href="/facts/Vertex_(curve)/FkoCoHhV">vertex</a> of the hyperbola.
A <a href="/facts/Parabola/Cweo3wgw">parabola</a> can be obtained as the limit of a sequence of ellipses where one focus is kept fixed as the other is allowed to move arbitrarily far away in one direction, keeping 
 
 
 
 ℓ
 
 
 {\displaystyle \ell }
 
 fixed. Thus a and b tend to infinity, a faster than b.
The major and minor axes are the <a href="/facts/Axis_of_symmetry/bpvJqcnS">axes of symmetry</a> for the curve: in an ellipse, the minor axis is the shorter one; in a hyperbola, it is the one that does not intersect the hyperbola.

Semi-major and semi-minor axes open-in-new

Semi-major and semi-minor axes