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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)
  • Build a graph of the second order line, construct the center of the canonical system and the basis vectors of the canonical system


    Examples of simplified expressions

    Equation Canonical view 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 или 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