Mister Exam
Other calculators
- 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:
- x*y+x*z+x*k+y*z+y*k+z*k
- 4x^2+y^2/16=4
- 4x^2-5y^2+4z^2=-20
- x^2+y^2-2x+4y=0
- Plot:
- (x-1)^2*y=0
- (-y+3)/y+((x/3)-1)=0
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(-x^5+y^5)/(x^2*y^2+(x^2-y^2)^2)=0
- tan(sin(exp(-x)))=0
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Identical expressions
- 4x^ two +y^ two +z^ two +4xy+4xz+ two yz+2x+y+z-2= zero
- 4x squared plus y squared plus z squared plus 4xy plus 4xz plus 2yz plus 2x plus y plus z minus 2 equally 0
- 4x to the power of two plus y to the power of two plus z to the power of two plus 4xy plus 4xz plus two yz plus 2x plus y plus z minus 2 equally zero
- 4x2+y2+z2+4xy+4xz+2yz+2x+y+z-2=0
- 4x²+y²+z²+4xy+4xz+2yz+2x+y+z-2=0
- 4x to the power of 2+y to the power of 2+z to the power of 2+4xy+4xz+2yz+2x+y+z-2=0
- 4x^2+y^2+z^2+4xy+4xz+2yz+2x+y+z-2=O
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Similar expressions
- 4x^2+y^2+z^2+4xy+4xz+2yz+2x-y+z-2=0
- 4x^2+y^2+z^2+4xy+4xz+2yz+2x+y+z+2=0
- 4x^2+y^2-z^2+4xy+4xz+2yz+2x+y+z-2=0
- 4x^2-y^2+z^2+4xy+4xz+2yz+2x+y+z-2=0
- 4x^2+y^2+z^2+4xy+4xz+2yz-2x+y+z-2=0
- 4x^2+y^2+z^2+4xy+4xz+2yz+2x+y-z-2=0
- 4x^2+y^2+z^2+4xy-4xz+2yz+2x+y+z-2=0
- 4x^2+y^2+z^2+4xy+4xz-2yz+2x+y+z-2=0
- 4x^2+y^2+z^2-4xy+4xz+2yz+2x+y+z-2=0
4x^2+y^2+z^2+4xy+4xz+2yz+2x+y+z-2=0 canonical form
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The solution
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2 2 2 -2 + y + z + y + z + 2*x + 4*x + 2*y*z + 4*x*y + 4*x*z = 0
$$4 x^{2} + 4 x y + 4 x z + y^{2} + 2 y z + z^{2} + 2 x + y + z - 2 = 0$$
4*x^2 + 4*x*y + 4*x*z + 2*x + y^2 + 2*y*z + y + z^2 + z - 2 = 0
Invariants method
Given equation of the surface of 2-order:
$$4 x^{2} + 4 x y + 4 x z + y^{2} + 2 y z + z^{2} + 2 x + y + z - 2 = 0$$
This equation looks like:
$$a_{11} x^{2} + 2 a_{12} x y + 2 a_{13} x z + a_{22} y^{2} + 2 a_{23} y z + a_{33} z^{2} + 2 a_{14} x + 2 a_{24} y + 2 a_{34} z + a_{44} = 0$$
where
$$a_{11} = 4$$
$$a_{12} = 2$$
$$a_{13} = 2$$
$$a_{14} = 1$$
$$a_{22} = 1$$
$$a_{23} = 1$$
$$a_{24} = \frac{1}{2}$$
$$a_{33} = 1$$
$$a_{34} = \frac{1}{2}$$
$$a_{44} = -2$$
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
$$I_{1} = 6$$
$$I_{3} = \left|\begin{matrix}4 & 2 & 2\\2 & 1 & 1\\2 & 1 & 1\end{matrix}\right|$$
$$I_{4} = \left|\begin{matrix}4 & 2 & 2 & 1\\2 & 1 & 1 & \frac{1}{2}\\2 & 1 & 1 & \frac{1}{2}\\1 & \frac{1}{2} & \frac{1}{2} & -2\end{matrix}\right|$$
$$I{\left(\lambda \right)} = \left|\begin{matrix}- \lambda + 4 & 2 & 2\\2 & - \lambda + 1 & 1\\2 & 1 & - \lambda + 1\end{matrix}\right|$$
$$I_{1} = 6$$
$$I_{2} = 0$$
$$I_{3} = 0$$
$$I_{4} = 0$$
$$I{\left(\lambda \right)} = - \lambda^{3} + 6 \lambda^{2}$$
$$K_{2} = - \frac{27}{2}$$
$$K_{3} = 0$$
Because
$$I_{2} = 0 \wedge I_{3} = 0 \wedge I_{4} = 0 \wedge K_{3} = 0 \wedge I_{1} \neq 0$$
then by type of surface:
you need to
then the canonical form of the equation will be
$$I_{1} \tilde x^{2} + \frac{K_{2}}{I_{1}} = 0$$
$$6 \tilde x^{2} - \frac{9}{4} = 0$$
$$\left(\tilde x + 0\right)^{2} = \frac{3}{8}$$
this equation is fora type two parallel planes
- reduced to canonical form
$$4 x^{2} + 4 x y + 4 x z + y^{2} + 2 y z + z^{2} + 2 x + y + z - 2 = 0$$
This equation looks like:
$$a_{11} x^{2} + 2 a_{12} x y + 2 a_{13} x z + a_{22} y^{2} + 2 a_{23} y z + a_{33} z^{2} + 2 a_{14} x + 2 a_{24} y + 2 a_{34} z + a_{44} = 0$$
where
$$a_{11} = 4$$
$$a_{12} = 2$$
$$a_{13} = 2$$
$$a_{14} = 1$$
$$a_{22} = 1$$
$$a_{23} = 1$$
$$a_{24} = \frac{1}{2}$$
$$a_{33} = 1$$
$$a_{34} = \frac{1}{2}$$
$$a_{44} = -2$$
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
$$I_{1} = 6$$
|4 2| |1 1| |4 2|
I2 = | | + | | + | |
|2 1| |1 1| |2 1|$$I_{3} = \left|\begin{matrix}4 & 2 & 2\\2 & 1 & 1\\2 & 1 & 1\end{matrix}\right|$$
$$I_{4} = \left|\begin{matrix}4 & 2 & 2 & 1\\2 & 1 & 1 & \frac{1}{2}\\2 & 1 & 1 & \frac{1}{2}\\1 & \frac{1}{2} & \frac{1}{2} & -2\end{matrix}\right|$$
$$I{\left(\lambda \right)} = \left|\begin{matrix}- \lambda + 4 & 2 & 2\\2 & - \lambda + 1 & 1\\2 & 1 & - \lambda + 1\end{matrix}\right|$$
|4 1 | | 1 1/2| | 1 1/2|
K2 = | | + | | + | |
|1 -2| |1/2 -2 | |1/2 -2 | |4 2 1 | | 1 1 1/2| |4 2 1 |
| | | | | |
K3 = |2 1 1/2| + | 1 1 1/2| + |2 1 1/2|
| | | | | |
|1 1/2 -2 | |1/2 1/2 -2 | |1 1/2 -2 |$$I_{1} = 6$$
$$I_{2} = 0$$
$$I_{3} = 0$$
$$I_{4} = 0$$
$$I{\left(\lambda \right)} = - \lambda^{3} + 6 \lambda^{2}$$
$$K_{2} = - \frac{27}{2}$$
$$K_{3} = 0$$
Because
$$I_{2} = 0 \wedge I_{3} = 0 \wedge I_{4} = 0 \wedge K_{3} = 0 \wedge I_{1} \neq 0$$
then by type of surface:
you need to
then the canonical form of the equation will be
$$I_{1} \tilde x^{2} + \frac{K_{2}}{I_{1}} = 0$$
$$6 \tilde x^{2} - \frac{9}{4} = 0$$
$$\left(\tilde x + 0\right)^{2} = \frac{3}{8}$$
this equation is fora type two parallel planes
- reduced to canonical form