Mister exam

# Bring to Canonical Form

x: [, ]
y: [, ]
z: [, ]

#### Quality:

(Number of points on the axis)

### 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 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
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