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- Canonical form of a elliptical paraboloid
- Canonical form of a double hyperboloid
- Canonical form of a imaginary ellipsoid
- Canonical form of a degenerate ellipse
- Canonical form of a parabola
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How to use it?
- Canonical form:
- 2*x-3/4*x-8=0
- 4x^2+25y^2-16x+50y-67=0
- x^2+y^2-xy+x+y=0
- x^2-y^2-6x-4y+1=0
- Plot:
- (y+4)^2+(x+2)^2=25
-
(x-1)^2+(y-1)^2=4
- ((|y|)+2)^2+((|x|)-4)^2=8
- y=-x2-4*x-4
-
Identical expressions
- (seven ^ two)*(x^ two)+((- one)^ six)*(six ^ two)*(y^ two)- two *(seven ^ two)* nine x+(seven ^ two)*((9^ two)-(six ^ two))
- (7 squared ) multiply by (x squared ) plus (( minus 1) to the power of 6) multiply by (6 squared ) multiply by (y squared ) minus 2 multiply by (7 squared ) multiply by 9x plus (7 squared ) multiply by ((9 squared ) minus (6 squared ))
- (seven to the power of two) multiply by (x to the power of two) plus (( minus one) to the power of six) multiply by (six to the power of two) multiply by (y to the power of two) minus two multiply by (seven to the power of two) multiply by nine x plus (seven to the power of two) multiply by ((9 to the power of two) minus (six to the power of two))
- (72)*(x2)+((-1)6)*(62)*(y2)-2*(72)*9x+(72)*((92)-(62))
- 72*x2+-16*62*y2-2*72*9x+72*92-62
- (7²)*(x²)+((-1)⁶)*(6²)*(y²)-2*(7²)*9x+(7²)*((9²)-(6²))
- (7 to the power of 2)*(x to the power of 2)+((-1) to the power of 6)*(6 to the power of 2)*(y to the power of 2)-2*(7 to the power of 2)*9x+(7 to the power of 2)*((9 to the power of 2)-(6 to the power of 2))
- (7^2)(x^2)+((-1)^6)(6^2)(y^2)-2(7^2)9x+(7^2)((9^2)-(6^2))
- (72)(x2)+((-1)6)(62)(y2)-2(72)9x+(72)((92)-(62))
- 72x2+-1662y2-2729x+7292-62
- 7^2x^2+-1^66^2y^2-27^29x+7^29^2-6^2
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Similar expressions
- (7^2)*(x^2)-((-1)^6)*(6^2)*(y^2)-2*(7^2)*9x+(7^2)*((9^2)-(6^2))
- (7^2)*(x^2)+((-1)^6)*(6^2)*(y^2)-2*(7^2)*9x-(7^2)*((9^2)-(6^2))
- (7^2)*(x^2)+((-1)^6)*(6^2)*(y^2)+2*(7^2)*9x+(7^2)*((9^2)-(6^2))
- (7^2)*(x^2)+((-1)^6)*(6^2)*(y^2)-2*(7^2)*9x+(7^2)*((9^2)+(6^2))
- (7^2)*(x^2)+((1)^6)*(6^2)*(y^2)-2*(7^2)*9x+(7^2)*((9^2)-(6^2))
(7^2)*(x^2)+((-1)^6)*(6^2)*(y^2)-2*(7^2)*9x+(7^2)*((9^2)-(6^2)) canonical form
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2 2 2205 - 882*x + 36*y + 49*x = 0
$$49 x^{2} - 882 x + 36 y^{2} + 2205 = 0$$
49*x^2 - 882*x + 36*y^2 + 2205 = 0
Detail solution
Given line equation of 2-order:
$$49 x^{2} - 882 x + 36 y^{2} + 2205 = 0$$
This equation looks like:
$$a_{11} x^{2} + 2 a_{12} x y + 2 a_{13} x + a_{22} y^{2} + 2 a_{23} y + a_{33} = 0$$
where
$$a_{11} = 49$$
$$a_{12} = 0$$
$$a_{13} = -441$$
$$a_{22} = 36$$
$$a_{23} = 0$$
$$a_{33} = 2205$$
To calculate the determinant
$$\Delta = \left|\begin{matrix}a_{11} & a_{12}\\a_{12} & a_{22}\end{matrix}\right|$$
or, substitute
$$\Delta = \left|\begin{matrix}49 & 0\\0 & 36\end{matrix}\right|$$
$$\Delta = 1764$$
Because
$$\Delta$$
is not equal to 0, then
find the center of the canonical coordinate system. To do it, solve the system of equations
$$a_{11} x_{0} + a_{12} y_{0} + a_{13} = 0$$
$$a_{12} x_{0} + a_{22} y_{0} + a_{23} = 0$$
substitute coefficients
$$49 x_{0} - 441 = 0$$
$$36 y_{0} = 0$$
then
$$x_{0} = 9$$
$$y_{0} = 0$$
Thus, we have the equation in the coordinate system O'x'y'
$$a'_{33} + a_{11} x'^{2} + 2 a_{12} x' y' + a_{22} y'^{2} = 0$$
where
$$a'_{33} = a_{13} x_{0} + a_{23} y_{0} + a_{33}$$
or
$$a'_{33} = 2205 - 441 x_{0}$$
$$a'_{33} = -1764$$
then equation turns into
$$49 x'^{2} + 36 y'^{2} - 1764 = 0$$
Given equation is ellipse
- reduced to canonical form
The center of canonical coordinate system at point O
Basis of the canonical coordinate system
$$\vec e_1 = \left( 1, \ 0\right)$$
$$\vec e_2 = \left( 0, \ 1\right)$$
$$49 x^{2} - 882 x + 36 y^{2} + 2205 = 0$$
This equation looks like:
$$a_{11} x^{2} + 2 a_{12} x y + 2 a_{13} x + a_{22} y^{2} + 2 a_{23} y + a_{33} = 0$$
where
$$a_{11} = 49$$
$$a_{12} = 0$$
$$a_{13} = -441$$
$$a_{22} = 36$$
$$a_{23} = 0$$
$$a_{33} = 2205$$
To calculate the determinant
$$\Delta = \left|\begin{matrix}a_{11} & a_{12}\\a_{12} & a_{22}\end{matrix}\right|$$
or, substitute
$$\Delta = \left|\begin{matrix}49 & 0\\0 & 36\end{matrix}\right|$$
$$\Delta = 1764$$
Because
$$\Delta$$
is not equal to 0, then
find the center of the canonical coordinate system. To do it, solve the system of equations
$$a_{11} x_{0} + a_{12} y_{0} + a_{13} = 0$$
$$a_{12} x_{0} + a_{22} y_{0} + a_{23} = 0$$
substitute coefficients
$$49 x_{0} - 441 = 0$$
$$36 y_{0} = 0$$
then
$$x_{0} = 9$$
$$y_{0} = 0$$
Thus, we have the equation in the coordinate system O'x'y'
$$a'_{33} + a_{11} x'^{2} + 2 a_{12} x' y' + a_{22} y'^{2} = 0$$
where
$$a'_{33} = a_{13} x_{0} + a_{23} y_{0} + a_{33}$$
or
$$a'_{33} = 2205 - 441 x_{0}$$
$$a'_{33} = -1764$$
then equation turns into
$$49 x'^{2} + 36 y'^{2} - 1764 = 0$$
Given equation is ellipse
2 2
\tilde x \tilde y
--------- + --------- = 1
2 2
/ 1 \ / 1 \
|------| |------|
\7*1/42/ \6*1/42/ - reduced to canonical form
The center of canonical coordinate system at point O
(9, 0)
Basis of the canonical coordinate system
$$\vec e_1 = \left( 1, \ 0\right)$$
$$\vec e_2 = \left( 0, \ 1\right)$$
Invariants method
Given line equation of 2-order:
$$49 x^{2} - 882 x + 36 y^{2} + 2205 = 0$$
This equation looks like:
$$a_{11} x^{2} + 2 a_{12} x y + 2 a_{13} x + a_{22} y^{2} + 2 a_{23} y + a_{33} = 0$$
where
$$a_{11} = 49$$
$$a_{12} = 0$$
$$a_{13} = -441$$
$$a_{22} = 36$$
$$a_{23} = 0$$
$$a_{33} = 2205$$
The invariants of the equation when converting coordinates are determinants:
$$I_{1} = a_{11} + a_{22}$$
$$I_{3} = \left|\begin{matrix}a_{11} & a_{12} & a_{13}\\a_{12} & a_{22} & a_{23}\\a_{13} & a_{23} & a_{33}\end{matrix}\right|$$
$$I{\left(\lambda \right)} = \left|\begin{matrix}a_{11} - \lambda & a_{12}\\a_{12} & a_{22} - \lambda\end{matrix}\right|$$
substitute coefficients
$$I_{1} = 85$$
$$I_{3} = \left|\begin{matrix}49 & 0 & -441\\0 & 36 & 0\\-441 & 0 & 2205\end{matrix}\right|$$
$$I{\left(\lambda \right)} = \left|\begin{matrix}49 - \lambda & 0\\0 & 36 - \lambda\end{matrix}\right|$$
$$I_{1} = 85$$
$$I_{2} = 1764$$
$$I_{3} = -3111696$$
$$I{\left(\lambda \right)} = \lambda^{2} - 85 \lambda + 1764$$
$$K_{2} = -7056$$
Because
$$I_{2} > 0 \wedge I_{1} I_{3} < 0$$
then by line type:
this equation is of type : ellipse
Make the characteristic equation for the line:
$$- I_{1} \lambda + I_{2} + \lambda^{2} = 0$$
or
$$\lambda^{2} - 85 \lambda + 1764 = 0$$
$$\lambda_{1} = 49$$
$$\lambda_{2} = 36$$
then the canonical form of the equation will be
$$\tilde x^{2} \lambda_{1} + \tilde y^{2} \lambda_{2} + \frac{I_{3}}{I_{2}} = 0$$
or
$$49 \tilde x^{2} + 36 \tilde y^{2} - 1764 = 0$$
- reduced to canonical form
$$49 x^{2} - 882 x + 36 y^{2} + 2205 = 0$$
This equation looks like:
$$a_{11} x^{2} + 2 a_{12} x y + 2 a_{13} x + a_{22} y^{2} + 2 a_{23} y + a_{33} = 0$$
where
$$a_{11} = 49$$
$$a_{12} = 0$$
$$a_{13} = -441$$
$$a_{22} = 36$$
$$a_{23} = 0$$
$$a_{33} = 2205$$
The invariants of the equation when converting coordinates are determinants:
$$I_{1} = a_{11} + a_{22}$$
|a11 a12|
I2 = | |
|a12 a22|$$I_{3} = \left|\begin{matrix}a_{11} & a_{12} & a_{13}\\a_{12} & a_{22} & a_{23}\\a_{13} & a_{23} & a_{33}\end{matrix}\right|$$
$$I{\left(\lambda \right)} = \left|\begin{matrix}a_{11} - \lambda & a_{12}\\a_{12} & a_{22} - \lambda\end{matrix}\right|$$
|a11 a13| |a22 a23|
K2 = | | + | |
|a13 a33| |a23 a33|substitute coefficients
$$I_{1} = 85$$
|49 0 |
I2 = | |
|0 36|$$I_{3} = \left|\begin{matrix}49 & 0 & -441\\0 & 36 & 0\\-441 & 0 & 2205\end{matrix}\right|$$
$$I{\left(\lambda \right)} = \left|\begin{matrix}49 - \lambda & 0\\0 & 36 - \lambda\end{matrix}\right|$$
| 49 -441| |36 0 |
K2 = | | + | |
|-441 2205| |0 2205|$$I_{1} = 85$$
$$I_{2} = 1764$$
$$I_{3} = -3111696$$
$$I{\left(\lambda \right)} = \lambda^{2} - 85 \lambda + 1764$$
$$K_{2} = -7056$$
Because
$$I_{2} > 0 \wedge I_{1} I_{3} < 0$$
then by line type:
this equation is of type : ellipse
Make the characteristic equation for the line:
$$- I_{1} \lambda + I_{2} + \lambda^{2} = 0$$
or
$$\lambda^{2} - 85 \lambda + 1764 = 0$$
$$\lambda_{1} = 49$$
$$\lambda_{2} = 36$$
then the canonical form of the equation will be
$$\tilde x^{2} \lambda_{1} + \tilde y^{2} \lambda_{2} + \frac{I_{3}}{I_{2}} = 0$$
or
$$49 \tilde x^{2} + 36 \tilde y^{2} - 1764 = 0$$
2 2
\tilde x \tilde y
--------- + --------- = 1
2 2
/ 1 \ / 1 \
|------| |------|
\7*1/42/ \6*1/42/ - reduced to canonical form