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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:
- kx^2-3y^2-8z^2-14yz+6y+14z-3=0
- 10x^2+40(x-y)+4y^2+120=0
- x1^2+5x2^2+x3^2+2x1x2+6x1x3+2x2x3
- (3x−3–√y−3+3–√)+(−3–√x+5y+3–√−5)=0
- Plot:
-
x^2+4*x−3*y−2=0
- x+3*y-z-5=0
- tg(2*(x^2)+3)
- sqrt(2*x+1)*x-y-1=0
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Identical expressions
- 4x^ two +9y^ two +36z^ two -8x-18y-72z+ thirteen = zero
- 4x squared plus 9y squared plus 36z squared minus 8x minus 18y minus 72z plus 13 equally 0
- 4x to the power of two plus 9y to the power of two plus 36z to the power of two minus 8x minus 18y minus 72z plus thirteen equally zero
- 4x2+9y2+36z2-8x-18y-72z+13=0
- 4x²+9y²+36z²-8x-18y-72z+13=0
- 4x to the power of 2+9y to the power of 2+36z to the power of 2-8x-18y-72z+13=0
- 4x^2+9y^2+36z^2-8x-18y-72z+13=O
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Similar expressions
- 4x^2+9y^2+36z^2-8x-18y-72z-13=0
- 4x^2+9y^2+36z^2+8x-18y-72z+13=0
- 4x^2+9y^2+36z^2-8x+18y-72z+13=0
- 4x^2+9y^2-36z^2-8x-18y-72z+13=0
- 4x^2+9y^2+36z^2-8x-18y+72z+13=0
- 4x^2-9y^2+36z^2-8x-18y-72z+13=0
4x^2+9y^2+36z^2-8x-18y-72z+13=0 canonical form
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The solution
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2 2 2 13 - 72*z - 18*y - 8*x + 4*x + 9*y + 36*z = 0
$$4 x^{2} - 8 x + 9 y^{2} - 18 y + 36 z^{2} - 72 z + 13 = 0$$
4*x^2 - 8*x + 9*y^2 - 18*y + 36*z^2 - 72*z + 13 = 0
Invariants method
Given equation of the surface of 2-order:
$$4 x^{2} - 8 x + 9 y^{2} - 18 y + 36 z^{2} - 72 z + 13 = 0$$
This equation looks like:
$$a_{11} x^{2} + 2 a_{12} x y + 2 a_{13} x z + 2 a_{14} x + a_{22} y^{2} + 2 a_{23} y z + 2 a_{24} y + a_{33} z^{2} + 2 a_{34} z + a_{44} = 0$$
where
$$a_{11} = 4$$
$$a_{12} = 0$$
$$a_{13} = 0$$
$$a_{14} = -4$$
$$a_{22} = 9$$
$$a_{23} = 0$$
$$a_{24} = -9$$
$$a_{33} = 36$$
$$a_{34} = -36$$
$$a_{44} = 13$$
The invariants of the equation when converting coordinates are determinants:
$$I_{1} = a_{11} + a_{22} + a_{33}$$
$$I_{3} = \left|\begin{matrix}a_{11} & a_{12} & a_{13}\\a_{12} & a_{22} & a_{23}\\a_{13} & a_{23} & a_{33}\end{matrix}\right|$$
$$I_{4} = \left|\begin{matrix}a_{11} & a_{12} & a_{13} & a_{14}\\a_{12} & a_{22} & a_{23} & a_{24}\\a_{13} & a_{23} & a_{33} & a_{34}\\a_{14} & a_{24} & a_{34} & a_{44}\end{matrix}\right|$$
$$I{\left(\lambda \right)} = \left|\begin{matrix}a_{11} - \lambda & a_{12} & a_{13}\\a_{12} & a_{22} - \lambda & a_{23}\\a_{13} & a_{23} & a_{33} - \lambda\end{matrix}\right|$$
substitute coefficients
$$I_{1} = 49$$
$$I_{3} = \left|\begin{matrix}4 & 0 & 0\\0 & 9 & 0\\0 & 0 & 36\end{matrix}\right|$$
$$I_{4} = \left|\begin{matrix}4 & 0 & 0 & -4\\0 & 9 & 0 & -9\\0 & 0 & 36 & -36\\-4 & -9 & -36 & 13\end{matrix}\right|$$
$$I{\left(\lambda \right)} = \left|\begin{matrix}4 - \lambda & 0 & 0\\0 & 9 - \lambda & 0\\0 & 0 & 36 - \lambda\end{matrix}\right|$$
$$I_{1} = 49$$
$$I_{2} = 504$$
$$I_{3} = 1296$$
$$I_{4} = -46656$$
$$I{\left(\lambda \right)} = - \lambda^{3} + 49 \lambda^{2} - 504 \lambda + 1296$$
$$K_{2} = -756$$
$$K_{3} = -14256$$
Because
then by type of surface:
you need to
Make the characteristic equation for the surface:
$$- I_{1} \lambda^{2} + I_{2} \lambda - I_{3} + \lambda^{3} = 0$$
or
$$\lambda^{3} - 49 \lambda^{2} + 504 \lambda - 1296 = 0$$
$$\lambda_{1} = 36$$
$$\lambda_{2} = 9$$
$$\lambda_{3} = 4$$
then the canonical form of the equation will be
$$\left(\tilde z^{2} \lambda_{3} + \left(\tilde x^{2} \lambda_{1} + \tilde y^{2} \lambda_{2}\right)\right) + \frac{I_{4}}{I_{3}} = 0$$
$$36 \tilde x^{2} + 9 \tilde y^{2} + 4 \tilde z^{2} - 36 = 0$$
this equation is fora type ellipsoid
- reduced to canonical form
$$4 x^{2} - 8 x + 9 y^{2} - 18 y + 36 z^{2} - 72 z + 13 = 0$$
This equation looks like:
$$a_{11} x^{2} + 2 a_{12} x y + 2 a_{13} x z + 2 a_{14} x + a_{22} y^{2} + 2 a_{23} y z + 2 a_{24} y + a_{33} z^{2} + 2 a_{34} z + a_{44} = 0$$
where
$$a_{11} = 4$$
$$a_{12} = 0$$
$$a_{13} = 0$$
$$a_{14} = -4$$
$$a_{22} = 9$$
$$a_{23} = 0$$
$$a_{24} = -9$$
$$a_{33} = 36$$
$$a_{34} = -36$$
$$a_{44} = 13$$
The invariants of the equation when converting coordinates are determinants:
$$I_{1} = a_{11} + a_{22} + a_{33}$$
|a11 a12| |a22 a23| |a11 a13|
I2 = | | + | | + | |
|a12 a22| |a23 a33| |a13 a33|$$I_{3} = \left|\begin{matrix}a_{11} & a_{12} & a_{13}\\a_{12} & a_{22} & a_{23}\\a_{13} & a_{23} & a_{33}\end{matrix}\right|$$
$$I_{4} = \left|\begin{matrix}a_{11} & a_{12} & a_{13} & a_{14}\\a_{12} & a_{22} & a_{23} & a_{24}\\a_{13} & a_{23} & a_{33} & a_{34}\\a_{14} & a_{24} & a_{34} & a_{44}\end{matrix}\right|$$
$$I{\left(\lambda \right)} = \left|\begin{matrix}a_{11} - \lambda & a_{12} & a_{13}\\a_{12} & a_{22} - \lambda & a_{23}\\a_{13} & a_{23} & a_{33} - \lambda\end{matrix}\right|$$
|a11 a14| |a22 a24| |a33 a34|
K2 = | | + | | + | |
|a14 a44| |a24 a44| |a34 a44| |a11 a12 a14| |a22 a23 a24| |a11 a13 a14|
| | | | | |
K3 = |a12 a22 a24| + |a23 a33 a34| + |a13 a33 a34|
| | | | | |
|a14 a24 a44| |a24 a34 a44| |a14 a34 a44|substitute coefficients
$$I_{1} = 49$$
|4 0| |9 0 | |4 0 |
I2 = | | + | | + | |
|0 9| |0 36| |0 36|$$I_{3} = \left|\begin{matrix}4 & 0 & 0\\0 & 9 & 0\\0 & 0 & 36\end{matrix}\right|$$
$$I_{4} = \left|\begin{matrix}4 & 0 & 0 & -4\\0 & 9 & 0 & -9\\0 & 0 & 36 & -36\\-4 & -9 & -36 & 13\end{matrix}\right|$$
$$I{\left(\lambda \right)} = \left|\begin{matrix}4 - \lambda & 0 & 0\\0 & 9 - \lambda & 0\\0 & 0 & 36 - \lambda\end{matrix}\right|$$
|4 -4| |9 -9| |36 -36|
K2 = | | + | | + | |
|-4 13| |-9 13| |-36 13 | |4 0 -4| |9 0 -9 | |4 0 -4 |
| | | | | |
K3 = |0 9 -9| + |0 36 -36| + |0 36 -36|
| | | | | |
|-4 -9 13| |-9 -36 13 | |-4 -36 13 |$$I_{1} = 49$$
$$I_{2} = 504$$
$$I_{3} = 1296$$
$$I_{4} = -46656$$
$$I{\left(\lambda \right)} = - \lambda^{3} + 49 \lambda^{2} - 504 \lambda + 1296$$
$$K_{2} = -756$$
$$K_{3} = -14256$$
Because
I3 != 0
then by type of surface:
you need to
Make the characteristic equation for the surface:
$$- I_{1} \lambda^{2} + I_{2} \lambda - I_{3} + \lambda^{3} = 0$$
or
$$\lambda^{3} - 49 \lambda^{2} + 504 \lambda - 1296 = 0$$
$$\lambda_{1} = 36$$
$$\lambda_{2} = 9$$
$$\lambda_{3} = 4$$
then the canonical form of the equation will be
$$\left(\tilde z^{2} \lambda_{3} + \left(\tilde x^{2} \lambda_{1} + \tilde y^{2} \lambda_{2}\right)\right) + \frac{I_{4}}{I_{3}} = 0$$
$$36 \tilde x^{2} + 9 \tilde y^{2} + 4 \tilde z^{2} - 36 = 0$$
2 2 2
\tilde x \tilde y \tilde z
--------- + --------- + --------- = 1
2 2 2
/ 1 \ / 1 \ / 1 \
|-----| |-----| |-----|
\6*1/6/ \3*1/6/ \2*1/6/ this equation is fora type ellipsoid
- reduced to canonical form