Editing Ellipsoid (section)

=== Determining the ellipse of a plane section ===
[[File:Ellipso-eb-beisp.svg|thumb|Plane section of an ellipsoid (see example)]]
'''Given:''' Ellipsoid {{math|{{sfrac|''x''<sup>2</sup>|''a''<sup>2</sup>}} + {{sfrac|''y''<sup>2</sup>|''b''<sup>2</sup>}} + {{sfrac|''z''<sup>2</sup>|''c''<sup>2</sup>}} {{=}} 1}} and the plane with equation {{math|''n<sub>x</sub>x'' + ''n<sub>y</sub>y'' + ''n<sub>z</sub>z'' {{=}} ''d''}}, which have an ellipse in common.

'''Wanted:''' Three vectors {{math|'''f'''<sub>0</sub>}} (center) and {{math|'''f'''<sub>1</sub>}}, {{math|'''f'''<sub>2</sub>}} (conjugate vectors), such that the ellipse can be represented by the parametric equation 
:<math>\mathbf x = \mathbf f_0 + \mathbf f_1\cos t + \mathbf f_2\sin t</math>
(see [[Ellipse#Ellipse as an affine image of the unit circle x²+y²=1|ellipse]]).

[[File:Ellipso-eb-ku.svg|300px|thumb|Plane section of the unit sphere (see example)]]
'''Solution:''' The scaling {{math|1=''u'' = {{sfrac|''x''|''a''}}, ''v'' = {{sfrac|''y''|''b''}}, ''w'' = {{sfrac|''z''|''c''}}}} transforms the ellipsoid onto the unit sphere {{math|''u''<sup>2</sup> + ''v''<sup>2</sup> + ''w''<sup>2</sup> {{=}} 1}} and the given plane onto the plane with equation
:<math>\ n_x au + n_y bv + n_z cw = d. </math>
Let {{math|''m<sub>u</sub>u'' + ''m<sub>v</sub>v'' + ''m<sub>w</sub>w'' {{=}} ''&delta;''}} be the [[Hesse normal form]] of the new plane and
:<math>\;\mathbf m = \begin{bmatrix} m_u \\ m_v \\ m_w \end{bmatrix}\;</math>
its unit normal vector. Hence
:<math>\mathbf e_0 = \delta \mathbf m \;</math>
is the  ''center'' of the intersection circle and
:<math>\;\rho = \sqrt{1 - \delta^2}\;</math>
its radius (see diagram).

Where {{math|''m<sub>w</sub>'' {{=}} ±1}} (i.e. the plane is horizontal), let
:<math>\ \mathbf e_1 = \begin{bmatrix} \rho \\ 0 \\ 0 \end{bmatrix},\qquad \mathbf e_2 = \begin{bmatrix} 0 \\ \rho \\ 0 \end{bmatrix}.</math>

Where {{math|''m<sub>w</sub>'' ≠ ±1}}, let
:<math >\mathbf e_1 = \frac{\rho}{\sqrt{m_u^2 + m_v^2}}\, \begin{bmatrix} m_v \\ -m_u \\ 0 \end{bmatrix}\, ,\qquad \mathbf e_2 = \mathbf m \times \mathbf e_1\ .</math>

In any case, the vectors {{math|'''e'''<sub>1</sub>, '''e'''<sub>2</sub>}} are orthogonal, parallel to the intersection plane and have length {{mvar|&rho;}} (radius of the circle). Hence the intersection circle can be described by the parametric equation
:<math>\;\mathbf u = \mathbf e_0 + \mathbf e_1\cos t + \mathbf e_2\sin t\;.</math>

The reverse scaling (see above) transforms the unit sphere back to the ellipsoid and the vectors {{math|'''e'''<sub>0</sub>, '''e'''<sub>1</sub>, '''e'''<sub>2</sub>}} are mapped onto vectors {{math|'''f'''<sub>0</sub>, '''f'''<sub>1</sub>, '''f'''<sub>2</sub>}}, which were wanted for the parametric representation of the intersection ellipse. 

How to find the vertices and semi-axes of the ellipse is described in [[Ellipse#Ellipse as an affine image of the unit circle x²+y²=1|ellipse]].

'''Example:''' The diagrams show an ellipsoid with the semi-axes {{math|1=''a'' = 4, ''b'' = 5, ''c'' = 3}} which is cut by the plane {{math|1=''x'' + ''y'' + ''z'' = 5}}.
{{clear}}