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
-
How to use it?
- Canonical form:
- -6x1x2+3x1x3
- 25x^2+9y^2-9z^2=225
- z^2+2y^2-x=0
- 5x^2+y^2+10x-6y-10z+14=0
- Plot:
-
-0.47y-1.02y^3+0.5ln((1+y)/(1-y))=x(1-exp(2.2*x/2))
- y=1/3*x^3-2*x^2
- x^2+y^2=16
- x^2*(y-x^2/3)^2=1
- Sum of series:
-
225
-
Identical expressions
- two 5x^ two +9y^ two -9z^2= two hundred and twenty-five
- 25x squared plus 9y squared minus 9z squared equally 225
- two 5x to the power of two plus 9y to the power of two minus 9z squared equally two hundred and twenty minus five
- 25x2+9y2-9z2=225
- 25x²+9y²-9z²=225
- 25x to the power of 2+9y to the power of 2-9z to the power of 2=225
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Similar expressions
- 25x^2+9y^2+9z^2=225
- 25x^2-9y^2-9z^2=225
25x^2+9y^2-9z^2=225 canonical form
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The solution
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2 2 2 -225 - 9*z + 9*y + 25*x = 0
$$25 x^{2} + 9 y^{2} - 9 z^{2} - 225 = 0$$
25*x^2 + 9*y^2 - 9*z^2 - 225 = 0
Invariants method
Given equation of the surface of 2-order:
$$25 x^{2} + 9 y^{2} - 9 z^{2} - 225 = 0$$
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
$$a_{11} = 25$$
$$a_{12} = 0$$
$$a_{13} = 0$$
$$a_{14} = 0$$
$$a_{22} = 9$$
$$a_{23} = 0$$
$$a_{24} = 0$$
$$a_{33} = -9$$
$$a_{34} = 0$$
$$a_{44} = -225$$
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
$$I_{1} = 25$$
$$I_{3} = \left|\begin{matrix}25 & 0 & 0\\0 & 9 & 0\\0 & 0 & -9\end{matrix}\right|$$
$$I_{4} = \left|\begin{matrix}25 & 0 & 0 & 0\\0 & 9 & 0 & 0\\0 & 0 & -9 & 0\\0 & 0 & 0 & -225\end{matrix}\right|$$
$$I{\left(\lambda \right)} = \left|\begin{matrix}25 - \lambda & 0 & 0\\0 & 9 - \lambda & 0\\0 & 0 & - \lambda - 9\end{matrix}\right|$$
$$I_{1} = 25$$
$$I_{2} = -81$$
$$I_{3} = -2025$$
$$I_{4} = 455625$$
$$I{\left(\lambda \right)} = - \lambda^{3} + 25 \lambda^{2} + 81 \lambda - 2025$$
$$K_{2} = -5625$$
$$K_{3} = 18225$$
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
$$\lambda^{3} - 25 \lambda^{2} - 81 \lambda + 2025 = 0$$
$$\lambda_{1} = 25$$
$$\lambda_{2} = 9$$
$$\lambda_{3} = -9$$
then the canonical form of the equation will be
$$\left(\tilde z^{2} \lambda_{3} + \left(\tilde x^{2} \lambda_{1} + \tilde y^{2} \lambda_{2}\right)\right) + \frac{I_{4}}{I_{3}} = 0$$
$$25 \tilde x^{2} + 9 \tilde y^{2} - 9 \tilde z^{2} - 225 = 0$$
this equation is fora type one-sided hyperboloid
- reduced to canonical form
$$25 x^{2} + 9 y^{2} - 9 z^{2} - 225 = 0$$
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
$$a_{11} = 25$$
$$a_{12} = 0$$
$$a_{13} = 0$$
$$a_{14} = 0$$
$$a_{22} = 9$$
$$a_{23} = 0$$
$$a_{24} = 0$$
$$a_{33} = -9$$
$$a_{34} = 0$$
$$a_{44} = -225$$
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
$$I_{1} = 25$$
|25 0| |9 0 | |25 0 |
I2 = | | + | | + | |
|0 9| |0 -9| |0 -9|$$I_{3} = \left|\begin{matrix}25 & 0 & 0\\0 & 9 & 0\\0 & 0 & -9\end{matrix}\right|$$
$$I_{4} = \left|\begin{matrix}25 & 0 & 0 & 0\\0 & 9 & 0 & 0\\0 & 0 & -9 & 0\\0 & 0 & 0 & -225\end{matrix}\right|$$
$$I{\left(\lambda \right)} = \left|\begin{matrix}25 - \lambda & 0 & 0\\0 & 9 - \lambda & 0\\0 & 0 & - \lambda - 9\end{matrix}\right|$$
|25 0 | |9 0 | |-9 0 |
K2 = | | + | | + | |
|0 -225| |0 -225| |0 -225| |25 0 0 | |9 0 0 | |25 0 0 |
| | | | | |
K3 = |0 9 0 | + |0 -9 0 | + |0 -9 0 |
| | | | | |
|0 0 -225| |0 0 -225| |0 0 -225|$$I_{1} = 25$$
$$I_{2} = -81$$
$$I_{3} = -2025$$
$$I_{4} = 455625$$
$$I{\left(\lambda \right)} = - \lambda^{3} + 25 \lambda^{2} + 81 \lambda - 2025$$
$$K_{2} = -5625$$
$$K_{3} = 18225$$
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
$$\lambda^{3} - 25 \lambda^{2} - 81 \lambda + 2025 = 0$$
$$\lambda_{1} = 25$$
$$\lambda_{2} = 9$$
$$\lambda_{3} = -9$$
then the canonical form of the equation will be
$$\left(\tilde z^{2} \lambda_{3} + \left(\tilde x^{2} \lambda_{1} + \tilde y^{2} \lambda_{2}\right)\right) + \frac{I_{4}}{I_{3}} = 0$$
$$25 \tilde x^{2} + 9 \tilde y^{2} - 9 \tilde z^{2} - 225 = 0$$
2 2 2
\tilde x \tilde y \tilde z
--------- + --------- - --------- = 1
2 2 2
/ 1 \ / 1 \ / 1 \
|------| |------| |------|
\5*1/15/ \3*1/15/ \3*1/15/ this equation is fora type one-sided hyperboloid
- reduced to canonical form