Quadric Surfaces
Quadric surfaces are a type of surface in three-dimensional space that are defined as the set of points whose coordinates satisfy a second-degree polynomial equation in three variables, typically in the form:
$$Ax^{2} + By^{2} + Cz^{2} + Dxy + Exz + Fyz + Gx + Hy + Iz + J = 0$$
Where \(A\), \(B\), \(C\), \(D\), \(E\), \(F\), \(G\), \(H\), \(I\), and \(J\) are real constants.
You can adjust the characteristics of the quadric surfaces charts by moving the sliders.
Ellipsoid
An ellipsoid is a three-dimensional closed surface that generalizes the shape of a sphere but with different radii along its three principal axes.
Equation
$$\frac{\left( x - h \right)^{2}}{a^{2}} + \frac{\left( y - k \right)^{2}}{b^{2}} + \frac{(z - l)^{2}}{c^{2}} = 1$$
Elliptic paraboloid
An elliptic paraboloid is a three-dimensional quadric surface whose cross-section is a parabola in two directions and an ellipse in the other.
Equation
$$\frac{\left( y - k \right)^{2}}{b^{2}} + \frac{\left( z - l \right)^{2}}{c^{2}} = \frac{(x - h)}{a}$$
$$\frac{\left( x - h \right)^{2}}{a^{2}} + \frac{\left( z - l \right)^{2}}{c^{2}} = \frac{(y - k)}{b}$$
$$\frac{\left( x - h \right)^{2}}{a^{2}} + \frac{\left( y - k \right)^{2}}{b^{2}} = \frac{(z - l)}{c}$$
Hyperbolic paraboloid
A hyperbolic paraboloid is a three-dimensional quadric surface characterized by parabolic cross-sections in two directions and a hyperbolic cross-section in the other.
Equation
$$\frac{\left( y - k \right)^{2}}{b^{2}} - \frac{\left( z - l \right)^{2}}{c^{2}} = \frac{(x - h)}{a}$$
$$\frac{\left( x - h \right)^{2}}{a^{2}} - \frac{\left( z - l \right)^{2}}{c^{2}} = \frac{(y - k)}{b}$$
$$\frac{\left( x - h \right)^{2}}{a^{2}} - \frac{\left( y - k \right)^{2}}{b^{2}} = \frac{(z - l)}{c}$$
Hyperboloid of one sheet
A hyperboloid of one sheet is a three-dimensional quadric surface with a characteristic hyperboloid shape, having elliptical cross-sections in one plane and hyperbolic cross-sections in the others.
Equation
$$-\frac{\left( x - h \right)^{2}}{a^{2}} + \frac{\left( y - k \right)^{2}}{b^{2}} + \frac{(z - l)^{2}}{c^{2}} = 1$$
$$\frac{\left( x - h \right)^{2}}{a^{2}} - \frac{\left( y - k \right)^{2}}{b^{2}} + \frac{(z - l)^{2}}{c^{2}} = 1$$
$$\frac{\left( x - h \right)^{2}}{a^{2}} + \frac{\left( y - k \right)^{2}}{b^{2}} - \frac{(z - l)^{2}}{c^{2}} = 1$$
Hyperboloid of two sheets
A hyperboloid of two sheets is a three-dimensional quadric surface that consists of two separate parts (or "sheets") and is characterized by hyperbolic cross-sections in two directions and elliptical cross-sections in the other.
Equation
$$\frac{\left( x - h \right)^{2}}{a^{2}} - \frac{\left( y - k \right)^{2}}{b^{2}} - \frac{(z - l)^{2}}{c^{2}} = 1$$
$$- \frac{\left( x - h \right)^{2}}{a^{2}} + \frac{\left( y - k \right)^{2}}{b^{2}} - \frac{(z - l)^{2}}{c^{2}} = 1$$
$$- \frac{\left( x - h \right)^{2}}{a^{2}} - \frac{\left( y - k \right)^{2}}{b^{2}} + \frac{(z - l)^{2}}{c^{2}} = 1$$
Elliptic cone
An elliptic cone is a three-dimensional surface generated by a family of straight lines passing through a fixed point, called the vertex, with elliptical cross-sections.
Equation
$$\frac{\left( y - k \right)^{2}}{b^{2}} + \frac{\left( z - l \right)^{2}}{c^{2}} = \frac{(x - h)^{2}}{a^{2}}$$
$$\frac{\left( x - h \right)^{2}}{a^{2}} + \frac{\left( z - l \right)^{2}}{c^{2}} = \frac{(y - k)^{2}}{b^{2}}$$
$$\frac{\left( x - h \right)^{2}}{a^{2}} + \frac{\left( y - k \right)^{2}}{b^{2}} = \frac{(z - l)^{2}}{c^{2}}$$
Elliptic cylinder
An elliptic cylinder is a three-dimensional surface formed by the set of all lines parallel to a fixed axis that pass through an ellipse in a plane perpendicular to that axis.
Equation
$$\frac{\left( x - h \right)^{2}}{a^{2}} + \frac{\left( y - k \right)^{2}}{b^{2}} = 1$$
Hyperbolic cylinder
An elliptic cylinder is a three-dimensional surface formed by the set of all lines parallel to a fixed axis that pass through an ellipse in a plane perpendicular to that axis.
Equation
$$\frac{\left( x - h \right)^{2}}{a^{2}} - \frac{\left( y - k \right)^{2}}{b^{2}} = 1$$
Parabolic cylinder
A parabolic cylinder is a three-dimensional surface generated by the set of all lines parallel to a fixed axis that pass through a parabola in a plane perpendicular to that axis.
Equation
$$(x-h)^{2} - 2a(y-k) = 0$$