Mister Exam

Bring to Canonical Form

The teacher will be very surprised to see your correct solution 😉


The graph:

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


 (Number of points on the axis)

Plot type:

    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


    Learn more about Canonical form.

    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