Quadric

One surface Quadrica is a surface determined by a shape equation: P(x1,x2...xn)=0{displaystyle P(x_{1},x_{2}...x_{n})=0 }

where P is a second-degree polynomial at coordinates x1,x2...xn{displaystyle x_{1},x_{2}...x_{n} }.

When not specified, it is a surface of usual real three-dimensional space, in a unit orthogonal coordinate system, and the coordinates are called x, y, z.

Hyperboloid of a leaf.

History

It was the ancient Greek mathematicians who began the study of quadrics, with the cone (a quadric) and its sections, which are conics, curved in a two-dimensional plane, although they did not use equations.

Algebraic definition

One Quadrica or Quadrant surfaceIt's a hypersurface. D- dimensional represented by a second degree equation with spatial (coordinated) variables. If these coordinates are {x1,x2,...xD!{displaystyle {x_{1},x_{2},...x_{D}},}, then the typical quadric in that space is defined by the algebraic equation:

␡ ␡ i,j=1DQi,jxixj+␡ ␡ i=1DPixi+R=0{displaystyle sum _{i,j=1}^{D}Q_{i,j}x_{i}x_{j}_{j}+sum _{i=1}^{D}P_{i}x_{i}+R=0}

where Q is a square matrix of dimension (D), P is a vector of dimension (D) and R is a constant. Although Q, P and R are usually real or complex, a quadric can be defined in general on any ring.

Cartesian Equation

The Cartesian equation of a quadric surface is of the form:

Ax2+Band2+Cz2+Dxand+Eandz+Fxz+Gx+Hand+Iz+J=0{displaystyle Ax^{2}+By^{2}+Cz^{2}+Dxy+Eyz+Fxz+Gx+Hy+Iz+J=0,}

• The algebraic definition of the quadriques has the defect of including cases without geometric interest and without link to the topic.

For example, the equation:

x2+and2+z2+2xand+2andz+2xz=0{displaystyle x^{2}+y^{2}+z^{2}+2xy+2yz+2xz=0 }

is of second degree, but it can also be written as:

(x+and+z)2=0{displaystyle (x+y+z)^{2}=0 }

which is equivalent to:

x+and+z=0{displaystyle x+y+z=0 },

an equation of the first degree that corresponds to a plane, surface that does not have the properties related to the second degree. Generally, all quadratic polynomials that are square are discarded.

• Often, it is useful to remember that if the equation in its Cartesian form lacks cross terms, i.e., the coefficients D, E and F are equal to zero:

D=0,E=0,F=0{displaystyle D=0,E=0,F=0}

then the linear terms for each variable:

Gx,Hand,Iz{displaystyle Gx,Hy,Iz}

can be assimilated to quadratics:

Ax2,Band2,Cz2{displaystyle Ax^{2},By^{2},Cz^{2}}

by the method of completing squares, so that it is easy to interpret the equation as one of the "normalized" forms presented below, but "decent" or "transferred" (not centered on the origin, (0,0,0){displaystyle (0,0,0)}, but at a point of implicit coordinates in the new form).

Normalized Equation

The normalized equation of a three-dimensional quadric (D = 3), centered at the origin (0, 0, 0) of a three-dimensional space, is:

± ± x2a2± ± and2b2± ± z2c2± ± 1=0{displaystyle pm {x^{2} over a^{2}}pm {y^{2} over b^{2}{2}pm {z^{2} over c^{2}}}{2}}pm 1=0}

Types of quadrics

By means of translations and rotations any quadric can be transformed into one of the "normalized" forms. In Euclidean three-dimensional space, there are 16 standard shapes; the most interesting are the following:

aerobic → ellipsoid	x2a2+and2b2+z2c2− − 1=0{displaystyle {x^{2} over a^{2}}}+{y^{2} over b^{2}}}}} +{z^{2} over c^{2}}{2}}}1=0,}
aerob → spheroid (special case of ellipsoid)	x2a2+and2a2+z2b2− − 1=0{displaystyle {x^{2} over a^{2}}}+{y^{2} over a^{2}}}} +{z^{2} over b^{2}}}1=0,}
sphere (special case of spheroid)	x2a2+and2a2+z2a2− − 1=0{displaystyle {x^{2} over a^{2}}}+{y^{2} over a^{2}}}} +{z^{2} over a^{2}}1=0,}
paraboloid
→ hyperbolic paraboloid (special case of paraboloid)	x2a2− − and2b2− − z=0{displaystyle {x^{2} over a^{2}}-{y^{2} over b^{2}}}-z=0,}
→ elliptical paraboloid (special paraboloid case)	x2a2+and2b2− − z=0{displaystyle {x^{2} over a^{2}}}+{y^{2} over b^{2}}}-z=0,}
→ circular paraboloid (special case of elliptical paraboloid)	x2a2+and2a2− − z=0{displaystyle {x^{2} over a^{2}}}+{y^{2} over a^{2}}}-z=0,}
hyperboloid
→ elliptical hyperboloid of a leaf (special case of hyperboloid)	x2a2+and2b2− − z2c2− − 1=0{displaystyle {x^{2} over a^{2}}}+{y^{2} over b^{2}}}}-{z^{2} over c^{2}}{2}}}1=0,}
→ circular hyperboloid of a leaf (special case of hyperboloid)	x2a2+and2a2− − z2c2− − 1=0{displaystyle {x^{2} over a^{2}}}+{y^{2} over a^{2}}}}-{z^{2} over c^{2}}1=0,}
→ elliptical hyperboloid of two leaves (special case of hyperboloid)	x2a2+and2b2− − z2c2+1=0{displaystyle {x^{2} over a^{2}}}+{y^{2} over b^{2}}}}-{z^{2} over c^{2}} +1=0,}
→ circular hyperboloid of two leaves (special case of hyperboloid)	x2a2+and2a2− − z2c2+1=0{displaystyle {x^{2} over a^{2}}}+{y^{2} over a^{2}}}}-{z^{2} over c^{2}
Cylinder
→ elliptical cylinder (special cylinder case)	x2a2+and2b2− − 1=0{displaystyle {x^{2} over a^{2}}}+{y^{2} over b^{2}}}}1=0,}
→ Circular cylinder (special case of elliptical cylinder)	x2a2+and2a2− − 1=0{displaystyle {x^{2} over a^{2}}}+{y^{2} over a^{2}}}}1=0,}
→ hyperbolic cylinder (special cylinder case)	x2a2− − and2b2− − 1=0{displaystyle {x^{2} over a^{2}}-{y^{2} over b^{2}}}}1=0,}
→ Parabolic cylinder (special cylinder case)	x2+2aand=0{displaystyle x^{2}+2ay=0,}
elliptical cone	x2a2+and2b2− − z2c2=0{displaystyle {x^{2} over a^{2}}}+{y^{2} over b^{2}}}}-{z^{2} over c^{2}}=0,}
→ circular cone (special case of elliptical cone)	x2a2+and2a2− − z2c2=0{displaystyle {x^{2} over a^{2}}}+{y^{2} over a^{2}}}{z^{2} over c^{2}}=0,}

In real projective space, the ellipsoid, elliptic hyperboloid, and elliptic paraboloid are similar; the two hyperbolic paraboloids are also not different from each other (because they are ruled surfaces; the cone and the cylinder are also not different from each other (because they are "degenerate" quadrics). In the complex projective space all non-degenerate quadrics result indistinguishable from each other.