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:
- 4𝑥1𝑥3−6𝑥2𝑥4
- 8*x^2-4*x*y+5*y^2-52*x+22*y+53=0
- 8x^2+5y^2+15x+6y-7=0
- 2x^2+y^2-z^2-6y=2z+80
Bring to Canonical Form
What can a canonical calculator do?
- For a given equation it finds:
- Canonical form of the equation (for lines and surfaces of second order)
- Basis-vector of canonical coordinate system (for 2nd order lines)
- Center of canonical coordinate system (for 2nd order lines)
- Detailed Solution in Two Ways:
- Direct method with transition to a new center of coordinates and rotation around a new center of coordinates (for lines)
- Method of invariants with calculation of a set of determinants (for lines and surfaces)
- Plot a graph of the second order line, plot the center of the canonical system and the basis vectors of the canonical system
Examples
- Two parallel straight lines
9x^2+12xy+4y^2-24x-16y+3=0
- Parabola
x^2-2xy+y^2-10x-6y+25=0
- Degenerate Ellipse
5x^2+4xy+y^2-6x-2y+2=0
- Ellipse
5*x^2+4*x*y+8*y^2+8*x+14*y+5=0
- Imaginary Ellipsoid
2*x^2+4*y^2+z^2-4*x*y-4*y-2*z+5=0
- Double Hyperboloid
x^2+y^2-z^2-2*x-2*y+2*z+2=0
- Elliptical Paraboloid
x^2+y^2-6*x+6*y-4*z+18=0
- Two Parallel Planes
x^2+4*y^2+9*z^2+4*x*y+12*y*z+6*x*z-4*x-8*y-12*z+3=0
Examples of equations of lines and surfaces of the second order
| Equation | Canonical form | Type | Measurement |
|---|---|---|---|
| 9x^2+12xy+4y^2-24x-16y+3=0 | x^2=1 | Two parallel straight lines | Line |
| x^2-2xy+y^2-10x-6y+25=0 | y^2=4*sqrt(2)*x | Parabola | Line |
| 5x^2+4xy+y^2-6x-2y+2=0 | x^2/(1/sqrt(2*sqrt(2)+3))^2 + y^2/(1/sqrt(-2*sqrt(2)+3))^2=0 | Degenerate Ellipse | Line |
| 5*x^2+4*x*y+8*y^2+8*x+14*y+5=0 | x^2/(3/4)^2+y^2/(1/2)^2=1 | Ellipse | Line |
| 2*x^2+4*y^2+z^2-4*x*y-4*y-2*z+5=0 | z^2/(2/sqrt(2)/sqrt(3-sqrt(5)))^2+x^2/(2/sqrt(2)/sqrt(3+sqrt(5)))^2+y^2/(2/sqrt(2))^2=-1 | Imaginary Ellipsoid | Surface |
| x^2+y^2-z^2-2*x-2*y+2*z+2=0 | x^2/1^2+y^2-z^2=-1 | Double Hyperboloid | Surface |
| x^2+y^2-6*x+6*y-4*z+18=0 | x^2/2+y^2-2*z=0 or x^2/2+y^2+2*z=0 | Elliptical Paraboloid | Surface |
| x^2+4*y^2+9*z^2+4*x*y+12*y*z+6*x*z-4*x-8*y-12*z+3=0 | x^2/=1/14 | Two Parallel Planes | Surface |