Quadratic Surfaces and Level Curves

Quadratic Surfaces and Level Curves

• In $$$\mathbb{R}^3$ space, quadric surfaces are a set of points with general $(x,y,z)$ coordinates satisfying a polynomial equation.
• We can sketch these surfaces by using traces, which are curves generated by the intersection of the surface with the planes parallel to the coordinate planes.
• Traces are also known as level surfaces.
• To find level surfaces you should takes one variable constant.
• For example, if we take $z=k$, then we will have a curve in the plane $z=k$.

Ellipsoid

• Traces
• In the plane $z=k$, where $k\in\mathbb{R}$: an ellipse
• In the plane $y=m$, wh
• ere $m\in\mathbb{R}$: an ellipse
• In the plane $x=n$, where $n\in\mathbb{R}$: an ellipse
• If $a=b=c$, then this surface is a sphere

Hyperboloid of One Sheet

• Traces
• In the plane $z=k$, where $k\in\mathbb{R}$: an ellipse
• In the plane $y=m$, where $m\in\mathbb{R}$: a hyperbola
• In the plane $x=n$, where $n\in\mathbb{R}$: a hyperbola
• The axis of the surface/direction of opening corresponds to the variable with the negative coefficient

Hyperboloid of Two Sheets

• Traces
• In the plane $z=k$, where $k\in\mathbb{R}$: an ellipse or no trace
• In the plane $y=m$, where $m\in\mathbb{R}$: a hyperbola
• In the plane $x=n$, where $n\in\mathbb{R}$: a hyperbola
• The axis of the surface/direction of opening corresponds to the variable with the positive coefficient
• The surface does not intersect the coordinate plane perpendicular to the direction of opening

Elliptic Cone

• Traces
• In the plane $z=k$, where $k\in\mathbb{R}$: an ellipse
• In the plane $y=m$, where $m\in\mathbb{R}$: a hyperbola
• In the plane $x=n$, where $n\in\mathbb{R}$: a hyperbola
• In the $xz$ - plane ($y=0$): a pair of lines that intersect at the origin
• In the $yz$ - plane ($x=0$): a pair of lines that intersect at the origin
• The axis of the surface/direction of opening corresponds to the variable with the negative coefficient.
• The traces in the coordinate planes parallel to the direction of opening are intersecting lines

Elliptic Paraboloid

• Traces
• In the plane $z=k$, where $k\in\mathbb{R}$: an ellipse
• In the plane $y=m$, where $m\in\mathbb{R}$: a hyperbola
• In the plane $x=n$, where $n\in\mathbb{R}$: a parabola
• The axis of the surface/direction of opening corresponds to the linear variable

Hyperbolic Paraboloid

• Traces
• In the plane $z=k$, where $k\in\mathbb{R}$: an hyerbola
• In the plane $y=m$, where $m\in\mathbb{R}$: a parabola
• In the plane $x=n$, where $n\in\mathbb{R}$: a parabola
• The axis of the surface/direction of opening corresponds to the linear variable