index_5.html

Some examples.

  Assume the Conic is:

     + xy + + x + y + = 0

  Then the program is starting by the command:

       Conic[ + xy + + x + y + ]

   The variables of the Conic must be x, y. Besides, x, y must not be assigned to a value, otherwise the execution stops.

First example of ellipse: reducing = 0 to the canonical form.

In[48]:=

$\n 1. The Original Conic \n$

$\t Symmetric matrix A out of degree 2 coefficients: \n$

$\t The Conic is an Ellipse. \n$

$\t Eigenvalues of A: \n$

$\t Eigenvectors of A: \n$

$\n 2. Rotating the Conic \n$

$RowBox[{\t We apply the rotation of , , 45., , degrees: \n}]$

${x (x - y)/2^(1/2), y (x + y)/2^(1/2)}$

$\t Equation of the rotated conic: \n$

$\n 3. Translating the Conic: the canonical form \n$

$RowBox[{\t We apply the translation of , , 1.41421, , units \n}]$

${x2^(1/2) + x, yy}$

$\t Canonical form (after rotating and translating): \n$

$\n 4. Grafico delle Coniche \n\t 1. Original Conic\t 2. Rotated Conic\t 3. The Canonical Form$

[Graphics:../HTMLFiles/index_101.gif]

Out[48]=

Second example of ellipse: reducing the vertical ellipse -4.

In[49]:=

$\n 1. The Original Conic \n$

$\t Symmetric matrix A out of degree 2 coefficients: \n$

$\t The Conic is an Ellipse. \n$

$\t Eigenvalues of A: \n$

$\t Eigenvectors of A: \n$

$\n 2. Rotating the Conic \n$

$RowBox[{\t We apply the rotation of , , 90., , degrees: \n}]$

$\t Equation of the rotated conic: \n$

$\n 3. Translating the Conic: the canonical form \n$

$RowBox[{\t We apply the translation of , , 0., , units \n}]$

$\t Canonical form (after rotating and translating): \n$

$\n 4. Grafico delle Coniche \n\t 1. Original Conic\t 2. Rotated Conic\t 3. The Canonical Form$

[Graphics:../HTMLFiles/index_125.gif]

Out[49]=

First example of parabola: reducing to the canonical form.

In[50]:=

$\n 1. The Original Conic \n$

$\t Symmetric matrix A out of degree 2 coefficients: \n$

$\t The Conic is a Parabola. \n$

$\t Eigenvalues of A: \n$

$\t Eigenvectors of A: \n$

$\n 2. Rotating the Conic \n$

$RowBox[{\t We apply the rotation of , , 135., , degrees: \n}]$

${x -(x + y)/2^(1/2), y (x - y)/2^(1/2)}$

$\t Equation of the rotated conic: \n$

$\n 3. Translating the Conic: the canonical form \n$

$RowBox[{\t We apply the translation of , , 7.07107, , units \n}]$

${x -5 2^(1/2) + x, yy}$

$\t Canonical form (after rotating and translating): \n$

$\n 4. Grafico delle Coniche \n\t 1. Original Conic\t 2. Rotated Conic\t 3. The Canonical Form$

[Graphics:../HTMLFiles/index_149.gif]

Out[50]=

Second example of parabola: reducing the horizontal parabola .

In[51]:=

$\n 1. The Original Conic \n$

$\t Symmetric matrix A out of degree 2 coefficients: \n$

$\t The Conic is a Parabola. \n$

$\t Eigenvalues of A: \n$

$\t Eigenvectors of A: \n$

$\n 2. Rotating the Conic \n$

$RowBox[{\t We apply the rotation of , , 0., , degrees: \n}]$

$\t Equation of the rotated conic: \n$

$\n 3. Translating the Conic: the canonical form \n$

$RowBox[{\t We apply the translation of , , 0., , units \n}]$

$\t Canonical form (after rotating and translating): \n$

$\n 4. Grafico delle Coniche \n\t 1. Original Conic\t 2. Rotated Conic\t 3. The Canonical Form$

[Graphics:../HTMLFiles/index_173.gif]

Out[51]=

First example of hyperbola: reducing to the canonical form.

In[52]:=

$\n 1. The Original Conic \n$

$\t Symmetric matrix A out of degree 2 coefficients: \n$

$\t The Conic is an Hyperbola. \n$

$\t Eigenvalues of A: \n$

${-5^(1/2)/2, 5^(1/2)/2}$

$\t Eigenvectors of A: \n$

${{2 + 5^(1/2), 1}, {2 - 5^(1/2), 1}}$

$\n 2. Rotating the Conic \n$

$RowBox[{\t We apply the rotation of , , 103.283, , degrees: \n}]$

${x -(1/2 - 1/5^(1/2))^(1/2) x - (1/2 + 1/5^(1/2))^(1/2) y, y (1/2 + 1/5^(1/2))^(1/2) x - (1/2 - 1/5^(1/2))^(1/2) y}$

$\t Equation of the rotated conic: \n$

$\n 3. Translating the Conic: the canonical form \n$

$RowBox[{\t We apply the translation of , , 0.632456, , units \n}]$

${x (5 - 5^(1/2))^(1/2)/5 + x, y (5 + 5^(1/2))^(1/2)/5 + y}$

$\t Canonical form (after rotating and translating): \n$

$\n 4. Grafico delle Coniche \n\t 1. Original Conic\t 2. Rotated Conic\t 3. The Canonical Form$

[Graphics:../HTMLFiles/index_197.gif]

Out[52]=

Second example of hyperbola: reducing the vertical hyperbola +4.

In[53]:=

$\n 1. The Original Conic \n$

$\t Symmetric matrix A out of degree 2 coefficients: \n$

$\t The Conic is an Hyperbola. \n$

$\t Eigenvalues of A: \n$

$\t Eigenvectors of A: \n$

$\n 2. Rotating the Conic \n$

$RowBox[{\t We apply the rotation of , , 90., , degrees: \n}]$

$\t Equation of the rotated conic: \n$

$\n 3. Translating the Conic: the canonical form \n$

$RowBox[{\t We apply the translation of , , 0., , units \n}]$

$\t Canonical form (after rotating and translating): \n$

$\n 4. Grafico delle Coniche \n\t 1. Original Conic\t 2. Rotated Conic\t 3. The Canonical Form$

[Graphics:../HTMLFiles/index_221.gif]

Out[53]=

A degenerated Conic.

In[54]:=

$\n 1. The Original Conic \n$

$\t Symmetric matrix A out of degree 2 coefficients: \n$

$\t The Conic is an Hyperbola. \n$

$\t Eigenvalues of A: \n$

${5/2 (1 + 2^(1/2)), 5/2 (1 - 2^(1/2))}$

$\t Eigenvectors of A: \n$

${{7 - 5 2^(1/2), 1}, {7 + 5 2^(1/2), 1}}$

$\n 2. Rotating the Conic \n$

$RowBox[{\t We apply the rotation of , , 4.06505, , degrees: \n}]$

${x1/10 ((50 + 35 2^(1/2))^(1/2) x - (50 - 35 2^(1/2))^(1/2) y), y1/10 ((50 - 35 2^(1/2))^(1/2) x + (50 + 35 2^(1/2))^(1/2) y)}$

$\t Equation of the rotated conic: \n$

$\n 3. Translating the Conic: the canonical form \n$

$RowBox[{\t We apply the translation of , , 3.82099, , units \n}]$

${x (73/10 + 103/(10 2^(1/2)))^(1/2) + x, y7/(1460 + 1030 2^(1/2))^(1/2) + y}$

$\t Canonical form (after rotating and translating): \n$

$\n 4. Grafico delle Coniche \n\t 1. Original Conic\t 2. Rotated Conic\t 3. The Canonical Form$

[Graphics:../HTMLFiles/index_244.gif]

Out[54]=

Another degenerated Conic.

In[55]:=

$\n 1. The Original Conic \n$

$\t Symmetric matrix A out of degree 2 coefficients: \n$

$\t The Conic is a Parabola. \n$

$\t Eigenvalues of A: \n$

$\t Eigenvectors of A: \n$

$\n 2. Rotating the Conic \n$

$RowBox[{\t We apply the rotation of , , 45., , degrees: \n}]$

${x (x - y)/2^(1/2), y (x + y)/2^(1/2)}$

$\t Equation of the rotated conic: \n$

$\n 3. Translating the Conic: the canonical form \n$

$RowBox[{\t We apply the translation of , , 0.353553, , units \n}]$

${xx, y1/(2 2^(1/2)) + y}$

$\t Canonical form (after rotating and translating): \n$

$\n 4. Grafico delle Coniche \n\t 1. Original Conic\t 2. Rotated Conic\t 3. The Canonical Form$

[Graphics:../HTMLFiles/index_267.gif]

Out[55]=

Created by Mathematica (August 4, 2004)